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Author  Title  Year  Journal/Proceedings  Reftype  DOI/URL 

Aach, T. & Kunz, D.  A lapped directional transform for spectral image analysis and its application to restoration and enhancement [BibTeX] 
2000  Signal Process. Vol. 80(11), pp. 23472364 
article  
BibTeX:
@article{Aach_T_2000_sp_lap_dtsiaare, author = {Aach, T. and Kunz, D.}, title = {A lapped directional transform for spectral image analysis and its application to restoration and enhancement}, journal = {Signal Process.}, year = {2000}, volume = {80}, number = {11}, pages = {23472364} } 

Abrial, P., Moudden, Y., Starck, J.L., Bobin, J., Afeyan, B. & Nguyen, M.K.  Morphological Component Analysis and Inpainting on the sphere: Application in Physics and Astrophysics [BibTeX] 
2007  J. Fourier Anal. Appl. Vol. 13(6), pp. 729748 
article  
BibTeX:
@article{Abrial_P_2007_jfouranalappl_mor_caisapa, author = {Abrial, P. and Moudden, Y. and Starck, J.L. and Bobin, J. and Afeyan, B. and Nguyen, M. K.}, title = {Morphological Component Analysis and Inpainting on the sphere: Application in Physics and Astrophysics}, journal = {J. Fourier Anal. Appl.}, year = {2007}, volume = {13}, number = {6}, pages = {729748}, note = {Special issue: "Analysis on the Sphere'"} } 

Abry, P. & Flandrin, P.  Multiresolution transient detection  1994  Proc. Int. Symp. on TimeFreq. and TimeScale Analysis, pp. 225228  inproceedings  DOI 
Abstract: Designs and studies the performance of a multiresolutionbased transient detector. The transients the authors are interested in consist of wideband, pulselike, coherent structures in a turbulent flow. To take advantage of the fast pyramidal wavelet algorithm, an important point when processing large amounts of experimental data, the detector makes use of the discrete wavelet transform. The authors show how the lack of time invariance drawback of the discrete transform can be efficiently overcome by using relevant analytic wavelets. They thus compare this detection technique with one based on a continuous wavelet transform, as well as with other standard methods and show that wavelets perform best when the transients are superimposed on a colored 1/f background noise. This description is very close to that of turbulence and relevant also in many other situations  
BibTeX:
@inproceedings{Abry_P_1994_stfts_mul_td, author = {Abry, P. and Flandrin, P.}, title = {Multiresolution transient detection}, booktitle = {Proc. Int. Symp. on TimeFreq. and TimeScale Analysis}, year = {1994}, pages = {225228}, doi = {http://dx.doi.org/10.1109/TFSA.1994.467252} } 

Adelson, E.H., Anderson, C.H., Bergen, J.R., Burt, P.J. & Ogden, J.M.  Pyramid Method in Image Processing  1984  RCA Eng. Vol. 29(6), pp. 3341 
article  
Abstract: The data structure used to represent image information can be critical to the successful completion of an image processing task. One structure that has attracted considerable attention is the image pyramid This consists of a set of lowpass or bandpass copies of an image, each representing pattern information of a different scale. Here we describe a variety of pyramid methods that we have developed for image data compression, enhancement, analysis and graphics.  
BibTeX:
@article{Adelson_E_1984_jrcaeng_pyr_mip, author = {E. H. Adelson and C. H. Anderson and J. R. Bergen and P. J. Burt and J. M. Ogden}, title = {Pyramid Method in Image Processing}, journal = {RCA Eng.}, year = {1984}, volume = {29}, number = {6}, pages = {3341} } 

Allen, J.  Shortterm spectral analysis, synthesis, and modification by discrete Fourier transform  1977  IEEE Trans. Acous., Speech Signal Process. Vol. 25(3), pp. 235238 
article  
Abstract: A theory of short term spectral analysis, synthesis, and modification is presented with an attempt at pointing out certain practical and theoretical questions. The methods discussed here are useful in designing filter banks when the filter bank outputs are to be used for synthesis after multiplicative modifications are made to the spectrum.  
BibTeX:
@article{Allen_J_1977_tassp_sho_tsasmdft, author = {Allen, J.}, title = {Shortterm spectral analysis, synthesis, and modification by discrete Fourier transform}, journal = {IEEE Trans. Acous., Speech Signal Process.}, year = {1977}, volume = {25}, number = {3}, pages = {235238} } 

Andres, E. & Carré, P.  Ridgelet transform based on Reveillès Discrete Lines  2002  Vol. 2301Proc. IAPR Int. Conf. Discrete Geom. Comput. Imagery (DGCI), pp. 417427 
inproceedings  
Abstract: In this paper we present a new discrete implementation of ridgelet transforms based on Reveillès discrete 2D lines. Ridgelet transforms are particular invertible wavelet transforms. Our approach uses the arithmetical thickness parameter of Reveillès lines to adapt the Ridgelet transform to specific applications. We illustrate this with a denoising and a compression algorithm. The broader aim of this paper is to show how results of discrete analytical geometry can be sucessfully used in image analysis.  
BibTeX:
@inproceedings{Andres_E_2002_pdgci_rid_trdl, author = {Andres, E. and Carré, P.}, title = {Ridgelet transform based on Reveillès Discrete Lines}, booktitle = {Proc. IAPR Int. Conf. Discrete Geom. Comput. Imagery (DGCI)}, year = {2002}, volume = {2301}, pages = {417427} } 

Antoine, J.P., Bogdanova, I. & Vandergheynst, P.  The continuous wavelet transform on conic sections  2008  Int. J. Wavelets Multidim. Inform. Proc. Vol. 6(2), pp. 137156 
article  DOI 
Abstract: We review the coherent state (or grouptheoretical) construction of the continuous wavelet transform (CWT) on the twosphere. Next, we describe the construction of a CWT on the upper sheet of a twosheeted hyperboloid, emphasizing the similarities between the two cases. Finally, we give some indications on the CWT on a paraboloid and we introduce a unified approach to the CWT on conic sections.  
BibTeX:
@article{Antoine_J_2008_jijwmip_con_wtcs, author = {Antoine, J.P. and Bogdanova, I. and Vandergheynst, P.}, title = {The continuous wavelet transform on conic sections}, journal = {Int. J. Wavelets Multidim. Inform. Proc.}, year = {2008}, volume = {6}, number = {2}, pages = {137156}, doi = {http://dx.doi.org/10.1142/S0219691308002288} } 

Antoine, J.P., Carrette, P., Murenzi, R. & Piette, B.  Image analysis with twodimensional continuous wavelet transform [BibTeX] 
1993  Signal Process. Vol. 31(3), pp. 241272 
article  
BibTeX:
@article{Antoine_J_1993_jsp_ima_atdcwt, author = {Antoine, J.P. and Carrette, P. and Murenzi, R. and Piette, B.}, title = {Image analysis with twodimensional continuous wavelet transform}, journal = {Signal Process.}, publisher = {Elsevier Science}, year = {1993}, volume = {31}, number = {3}, pages = {241272} } 

Antoine, J.P., Demanet, L., Jacques, L. & Vandergheynst, P.  Wavelets on the sphere: implementation and approximations  2002  Appl. Comp. Harm. Analysis Vol. 13(3), pp. 177200 
article  DOI URL 
Abstract: We continue the analysis of the continuous wavelet transform on the 2sphere, introduced in a previous paper. After a brief review of the transform, we define and discuss the notion of directional spherical wavelet, i.e., wavelets on the sphere that are sensitive to directions. Then we present a calculation method for data given on a regular spherical grid . This technique, which uses the FFT, is based on the invariance of under discrete rotations around the z axis preserving the [phi] sampling. Next, a numerical criterion is given for controlling the scale interval where the spherical wavelet transform makes sense, and examples are given, both academic and realistic. In a second part, we establish conditions under which the reconstruction formula holds in strong Lp sense, for 1[lessthanorequals, slant]p<[infinity]. This opens the door to techniques for approximating functions on the sphere, by use of an approximate identity, obtained by a suitable dilation of the mother wavelet.  
BibTeX:
@article{Antoine_J_2002_jacha_wav_sia, author = {J.P. Antoine and L. Demanet and L. Jacques and P. Vandergheynst}, title = {Wavelets on the sphere: implementation and approximations}, journal = {Appl. Comp. Harm. Analysis}, year = {2002}, volume = {13}, number = {3}, pages = {177200}, url = {http://www.sciencedirect.com/science/article/B6WB3474DMCF3/2/0fe29bdaef0d485c199c1b1218aef397}, doi = {DOI: 10.1016/S10635203(02)005079} } 

Antoine, J.P., Murenzi, R. & Vandergheynst, P.  Directional Wavelets Revisited: Cauchy Wavelets and Symmetry Detection in Patterns  1999  Appl. Comp. Harm. Analysis Vol. 6(3), pp. 314345 
article  DOI URL 
Abstract: The analysis of oriented features in images requires twodimensional directional wavelets. Among these, we study in detail the class of Cauchy wavelets, which are strictly supported in a (narrow) convex cone in spatial frequency space. They have excellent angular selectivity, as shown by a standard calibration test, and they have minimal uncertainty. In addition, we present a new application of directional wavelets, namely a technique for determining the symmetries of a given pattern with respect to rotations and dilation.  
BibTeX:
@article{Antoine_J_1999_jacha_dir_wrcwsdp, author = {J.P. Antoine and R. Murenzi and P. Vandergheynst}, title = {Directional Wavelets Revisited: Cauchy Wavelets and Symmetry Detection in Patterns}, journal = {Appl. Comp. Harm. Analysis}, year = {1999}, volume = {6}, number = {3}, pages = {314345}, url = {http://www.sciencedirect.com/science/article/B6WB345HR77TF/2/48f6b27610c784c0a3fcbcf1905d75bb}, doi = {DOI: 10.1006/acha.1998.0255} } 

Antoine, J.P., Murenzi, R., Vandergheynst, P. & Twareque Ali, S.  Twodimensional wavelets and their relatives  2004  book  
Abstract: Twodimensional wavelets offer a number of advantages over discrete wavelet transforms when processing rapidly varying functions and signals. In particular, they offer benefits for realtime applications such as medical imaging, fluid dynamics, shape recognition, image enhancement and target tracking. This book introduces the reader to 2D wavelets via 1D continuous wavelet transforms, and includes a long list of useful applications. The authors then describe in detail the underlying mathematics before moving on to more advanced topics such as matrix geometry of wavelet analysis, threedimensional wavelets and wavelets on a sphere. Throughout the book, practical applications and illustrative examples are used extensively, ensuring the book?s value to engineers, physicists and mathematicians alike. The first of its kind in print dealing with the two  and higher  dimensional continuous wavelet transforms, with extensive examples of applications. Gradual introduction of the underlying mathematical tools, with very few prerequisites, yet leading the reader to the research frontier. Covers both the continuous and the discrete wavelet transforms  
BibTeX:
@book{Antoine_J_2004_book_two_dwr, author = {Antoine, J.P. and Murenzi, R. and Vandergheynst, P. and Twareque Ali, S. }, title = {Twodimensional wavelets and their relatives}, publisher = {Cambridge University Press}, year = {2004} } 

Antoine, J.P., Roşca, D. & Vandergheynst, P.  Wavelet transform on manifolds: Old and new approaches  2010  Appl. Comp. Harm. Analysis Vol. 28(2), pp. 189202 
article  DOI URL 
Abstract: Given a twodimensional smooth manifold and a bijective projection from on a fixed plane (or a subset of that plane), we explore systematically how a wavelet transform (WT) on may be generated from a plane WT by the inverse projection . Examples where the projection maps the whole manifold onto a plane include the twosphere, the upper sheet of the twosheeted hyperboloid and the paraboloid. When no such global projection is available, the construction may be performed locally, i.e., around a given point on . We apply this procedure both to the continuous WT, already treated in the literature, and to the discrete WT. Finally, we discuss the case of a WT on a graph, for instance, the graph defined by linking the elements of a discrete set of points on the manifold.  
BibTeX:
@article{Antoine_J_2010_jacha_wav_tmona, author = {J.P. Antoine and Roşca, D. and Vandergheynst, P.}, title = {Wavelet transform on manifolds: Old and new approaches}, journal = {Appl. Comp. Harm. Analysis}, year = {2010}, volume = {28}, number = {2}, pages = {189202}, note = {Special Issue on Continuous Wavelet Transform in Memory of Jean Morlet, Part I}, url = {http://www.sciencedirect.com/science/article/B6WB34XF83YM1/2/f8f4a90ab76eead584134a5741d579ab}, doi = {DOI: 10.1016/j.acha.2009.10.002} } 

Antoine, J.P. & Vandergheynst, P.  Wavelets on the 2sphere: A grouptheoretical approach [BibTeX] 
1999  Appl. Comp. Harm. Analysis Vol. 7(3), pp. 262291 
article  DOI URL 
BibTeX:
@article{Antoine_J_1999_jacha_wav_2sgta, author = {Antoine, J.P. and Vandergheynst, P.}, title = {Wavelets on the 2sphere: A grouptheoretical approach}, journal = {Appl. Comp. Harm. Analysis}, publisher = {Elsevier}, year = {1999}, volume = {7}, number = {3}, pages = {262291}, url = {http://www.sciencedirect.com/science/article/B6WB345HR76Y2/2/04d716fb4bd326464c0e65e73911f461}, doi = {DOI: 10.1006/acha.1999.0272} } 

Aujol, J.F., Aubert, G., BlancFeraud, L. & Chambolle, A.  Image Decomposition into a Bounded Variation Component and an Oscillating Component  2005  J. Math. Imaging Vis. Vol. 22(1), pp. 7188 
article  DOI 
Abstract: We construct an algorithm to split an image into a sum u + v of a bounded variation component and a component containing the textures and the noise. This decomposition is inspired from a recent work of Y. Meyer. We find this decomposition by minimizing a convex functional which depends on the two variables u and v, alternately in each variable. Each minimization is based on a projection algorithm to minimize the total variation. We carry out the mathematical study of our method. We present some numerical results. In particular, we show how the u component can be used in nontextured SAR image restoration.  
BibTeX:
@article{Aujol_J_2005_jmathimagingvis_ima_dbvcoc, author = {Aujol, J.F. and Aubert, G. and BlancFeraud, L. and Chambolle, A.}, title = {Image Decomposition into a Bounded Variation Component and an Oscillating Component}, journal = {J. Math. Imaging Vis.}, year = {2005}, volume = {22}, number = {1}, pages = {7188}, doi = {http://dx.doi.org/10.1007/s1085100547838} } 

Auscher, P.  Wavelet bases for $L^2(R)$ with rational dilation factor [BibTeX] 
1992  Wavelets and their applications, pp. 439452  incollection  
BibTeX:
@incollection{Auscher_P_1992_incollwav_bl2rrdf, author = {Auscher, P.}, title = {Wavelet bases for $L^2(R)$ with rational dilation factor}, booktitle = {Wavelets and their applications}, publisher = {Jones and Bartlett}, year = {1992}, pages = {439452} } 

Averbuch, A., Coifman, R.R., Donoho, D.L., Elad, M. & Israeli, M.  Fast and accurate Polar Fourier transform [BibTeX] 
2006  Appl. Comp. Harm. Analysis Vol. 21, pp. 145167 
article  
BibTeX:
@article{Averbuch_A_2006_jacha_fas_apft, author = {A. Averbuch and R. R. Coifman and D. L. Donoho and M. Elad and M. Israeli}, title = {Fast and accurate Polar Fourier transform}, journal = {Appl. Comp. Harm. Analysis}, year = {2006}, volume = {21}, pages = {145167} } 

Ayache, A.  Some Methods for Constructing Nonseparable, Orthonormal, Compactly Supported Wavelet Bases  2001  Appl. Comp. Harm. Analysis Vol. 10(1), pp. 99111 
article  DOI URL 
Abstract: We first show that by combining monodimensional filter banks one can obtain nonseparable filter banks. We then give necessary conditions for these filter banks to generate orthonormal and regular wavelets. Finally, we establish that some of these filter banks lead to arbitrarily smooth, nonseparable, orthonormal, compactly supported wavelet bases.  
BibTeX:
@article{Ayache_A_2001_jacha_som_mcnocswb, author = {A. Ayache}, title = {Some Methods for Constructing Nonseparable, Orthonormal, Compactly Supported Wavelet Bases}, journal = {Appl. Comp. Harm. Analysis}, year = {2001}, volume = {10}, number = {1}, pages = {99111}, url = {http://www.sciencedirect.com/science/article/B6WB345BT4NK1B/2/20fcc16958f02d3ae2bc58c0acdd9a6a}, doi = {DOI: 10.1006/acha.2000.0325} } 

Babaud, J., Witkin, A.P., Baudin, M. & Duda, R.O.  Uniqueness of the Gaussian Kernel for ScaleSpace Filtering  1986  IEEE Trans. Patt. Anal. Mach. Int. Vol. 8(1), pp. 2633 
article  DOI 
Abstract: Scalespace filtering constructs hierarchic symbolic signal descriptions by transforming the signal into a continuum of versions of the original signal convolved with a kernal containing a scale or bandwidth parameter. It is shown that the Gaussian probability density function is the only kernel in a broad class for which firstorder maxima and minima, respectively, increase and decrease when the bandwidth of the filter is increased. The consequences of this result are explored when the signal or its image by a linear differential operator is analyzed in terms of zerocrossing contours of the transform in scalespace.  
BibTeX:
@article{Babaud_J_1986_tpami_uni_gkssf, author = {Babaud, J. and Witkin, A. P. and Baudin, M. and Duda, R. O.}, title = {Uniqueness of the Gaussian Kernel for ScaleSpace Filtering}, journal = {IEEE Trans. Patt. Anal. Mach. Int.}, year = {1986}, volume = {8}, number = {1}, pages = {2633}, doi = {http://dx.doi.org/10.1109/TPAMI.1986.4767749} } 

Bamberger, R.H. & Smith, M.J.T.  A filter bank for the directional decomposition of images: theory and design [BibTeX] 
1992  IEEE Trans. Signal Process. Vol. 40(4), pp. 882893 
article  
BibTeX:
@article{Bamberger_R_1992_jieeetsp_fil_bdditd, author = {Bamberger, R. H. and Smith, M. J. T.}, title = {A filter bank for the directional decomposition of images: theory and design}, journal = {IEEE Trans. Signal Process.}, year = {1992}, volume = {40}, number = {4}, pages = {882893} } 

Baussard, A., Nicolier, F. & Truchetet, F.  Rational multiresolution analysis and fast wavelet transform: application to wavelet shrinkage denoising  2004  Signal Process. Vol. 84(10), pp. 17351747 
article  DOI URL 
Abstract: This paper presents a contribution to rational multiresolution analysis (MRA). The rational analysis allows a better adaptation of scale factors to signal components than the dyadic one. The theory of rational MRA is reviewed and a pyramidal algorithm for fast rational orthogonal wavelet transform is proposed. Both, the analysis and synthesis parts of the process are detailed. Examples of scaling and wavelet functions and associated filters are given. Moreover, dealing with filters defined in Fourier domain, the implementation of the algorithm in this domain is described. Then, the study is extended to the 2D separable case in order to give a more conclusive presentation of the rational MRA. In order to illustrate the potential of rational analysis for signal and image processing, some results given by wavelet shrinkage denoising based on the [`]SURE' thresholding method are presented.  
BibTeX:
@article{Baussard_A_2004_jsp_rat_mafwtawsd, author = {A. Baussard and F. Nicolier and F. Truchetet}, title = {Rational multiresolution analysis and fast wavelet transform: application to wavelet shrinkage denoising}, journal = {Signal Process.}, year = {2004}, volume = {84}, number = {10}, pages = {17351747}, url = {http://www.sciencedirect.com/science/article/B6V184CS4G0S1/2/90023d217aeef798c7fc727c9b8cd0a6}, doi = {DOI: 10.1016/j.sigpro.2004.06.001} } 

Bayram, I. & Selesnick, I.W.  FrequencyDomain Design of Overcomplete RationalDilation Wavelet Transforms  2009  IEEE Trans. Signal Process. Vol. 57(8), pp. 29572972 
article  DOI 
Abstract: The dyadic wavelet transform is an effective tool for processing piecewise smooth signals; however, its poor frequency resolution (its low Qfactor) limits its effectiveness for processing oscillatory signals like speech, EEG, and vibration measurements, etc. This paper develops a more flexible family of wavelet transforms for which the frequency resolution can be varied. The new wavelet transform can attain higher Qfactors (desirable for processing oscillatory signals) or the same low Qfactor of the dyadic wavelet transform. The new wavelet transform is modestly overcomplete and based on rational dilations. Like the dyadic wavelet transform, it is an easily invertible 'constantQ' discrete transform implemented using iterated filter banks and can likewise be associated with a wavelet frame for L2(R). The wavelet can be made to resemble a Gabor function and can hence have good concentration in the timefrequency plane. The construction of the new wavelet transform depends on the judicious use of both the transform's redundancy and the flexibility allowed by frequencydomain filter design.  
BibTeX:
@article{Bayram_I_2009_jieeetsp_fre_ddordwt, author = {Bayram, I. and Selesnick, I. W.}, title = {FrequencyDomain Design of Overcomplete RationalDilation Wavelet Transforms}, journal = {IEEE Trans. Signal Process.}, year = {2009}, volume = {57}, number = {8}, pages = {29572972}, doi = {http://dx.doi.org/10.1109/TSP.2009.2020756} } 

Bayram, I. & Selesnick, I.W.  On the DualTree Complex Wavelet Packet and $M$Band Transforms  2008  IEEE Trans. Signal Process. Vol. 56(6), pp. 22982310 
article  DOI 
Abstract: The twoband discrete wavelet transform (DWT) provides an octaveband analysis in the frequency domain, but this might not be ldquooptimalrdquo for a given signal. The discrete wavelet packet transform (DWPT) provides a dictionary of bases over which one can search for an optimal representation (without constraining the analysis to an octaveband one) for the signal at hand. However, it is well known that both the DWT and the DWPT are shiftvarying. Also, when these transforms are extended to 2D and higher dimensions using tensor products, they do not provide a geometrically oriented analysis. The dualtree complex wavelet transform , introduced by Kingsbury, is approximately shiftinvariant and provides directional analysis in 2D and higher dimensions. In this paper, we propose a method to implement a dualtree complex wavelet packet transform , extending the as the DWPT extends the DWT. To find the best complex wavelet packet frame for a given signal, we adapt the basis selection algorithm by Coifman and Wickerhauser, providing a solution to the basis selection problem for the . Lastly, we show how to extend the twoband to an band (provided that ) using the same method.  
BibTeX:
@article{Bayram_I_2008_tsp_dua_tcwpmbt, author = {Bayram, I. and Selesnick, I. W.}, title = {On the DualTree Complex Wavelet Packet and $M$Band Transforms}, journal = {IEEE Trans. Signal Process.}, year = {2008}, volume = {56}, number = {6}, pages = {22982310}, doi = {http://dx.doi.org/10.1109/TSP.2007.916129} } 

Belzer, B., Lina, J.M. & Villasenor, J.  Complex, linearphase filters for efficient image coding [BibTeX] 
1995  IEEE Trans. Signal Process. Vol. 43(10), pp. 24252427 
article  
BibTeX:
@article{Belzer_B_1995_tsp_com_lpfeic, author = {Belzer, B. and Lina, J.M. and Villasenor, J.}, title = {Complex, linearphase filters for efficient image coding}, journal = {IEEE Trans. Signal Process.}, year = {1995}, volume = {43}, number = {10}, pages = {24252427} } 

Bergeaud, F. & Mallat, S.  Matching pursuit: Adaptive representations of images and sounds [BibTeX] 
1996  Comput. Appl. Math. Vol. 15(2), pp. 97109 
article  
BibTeX:
@article{Bergeaud_F_1996_jcomputapplmathbirkhauser_mat_paris, author = {F. Bergeaud and S. Mallat}, title = {Matching pursuit: Adaptive representations of images and sounds}, journal = {Comput. Appl. Math.}, year = {1996}, volume = {15}, number = {2}, pages = {97109} } 

Beylkin, G. & Torrésani, B.  Implementation of Operators via Filter Banks: Autocorrelation Shell and Hardy Wavelets [BibTeX] 
1996  Appl. Comp. Harm. Analysis Vol. 3, pp. 164185 
article  
BibTeX:
@article{Beylkin_G_1996_acha_imp_ofbashw, author = {Beylkin, G. and Torrésani, B.}, title = {Implementation of Operators via Filter Banks: Autocorrelation Shell and Hardy Wavelets}, journal = {Appl. Comp. Harm. Analysis}, year = {1996}, volume = {3}, pages = {164185} } 

Beylkin, G. & Torrésani, B.  Transformation de Hilbert et Bancs de Filtres [BibTeX] 
1994  Vol. 25Colloque tempsfréquence, ondelettes et multirésolution : théorie, modèles et applications, pp. 14 
inproceedings  
BibTeX:
@inproceedings{Beylkin_G_1994_tfomtma_tra_hbf, author = {Beylkin, G. and Torrésani, B.}, title = {Transformation de Hilbert et Bancs de Filtres}, booktitle = {Colloque tempsfréquence, ondelettes et multirésolution : théorie, modèles et applications}, year = {1994}, volume = {25}, pages = {14} } 

Bharath, A.A. & Ng, J.  A steerable complex wavelet construction and its application to image denoising  2005  IEEE Trans. Image Process. Vol. 14(7), pp. 948959 
article  DOI 
Abstract: This work addresses the design of a novel complex steerable wavelet construction, the generation of transformspace feature measurements associated with corner and edge presence and orientation properties, and the application of these measurements directly to image denoising. The decomposition uses pairs of bandpass filters that display symmetry and antisymmetry about a steerable axis of orientation. While the angular characterization of the bandpass filters is similar to those previously described, the radial characteristic is new, as is the manner of constructing the interpolation functions for steering. The complex filters have been engineered into a multirate system, providing a synthesis and analysis subband filtering system with good reconstruction properties. Although the performance of our proposed denoising strategy is currently below that of recently reported stateoftheart techniques in denoising, it does compare favorably with wavelet coring approaches employing global thresholds and with an "Oracle" shrinkage technique, and presents a very promising avenue for exploring structurebased denoising in the wavelet domain.  
BibTeX:
@article{Bharath_A_2005_tip_ste_cwcaid, author = {Bharath, A. A. and Ng, J.}, title = {A steerable complex wavelet construction and its application to image denoising}, journal = {IEEE Trans. Image Process.}, year = {2005}, volume = {14}, number = {7}, pages = {948959}, doi = {http://dx.doi.org/10.1109/TIP.2005.849295} } 

Blu, T.  A new design algorithm for twoband orthonormal rational filterbanks and orthonormal rational wavelets  1998  IEEE Trans. Acous., Speech Signal Process. Vol. 46(6), pp. 14941504 
article  DOI 
Abstract: Abstract We present a new algorithm for the design of orthonormal twoband rational filter banks. Owing to the connection between iterated rational filter banks and rational wavelets, this is also a design algorithm for orthonormal rational wavelets. It is basically a simple iterative procedure, which explains its exponential convergence and adaptability under various linear constraints (e,g., regularity). Although the filters obtained from this algorithm are suboptimally designed, they show excellent frequency selectivity. After an indepth account of the algorithm, we discuss the properties of the rational wavelets generated by some designed filters. In particular, we stress the possibility to design ?almost? shift errorfree wavelets, which allows the implementation of a rational wavelet transform  
BibTeX:
@article{Blu_T_1998_tsp_new_datborforw, author = {Blu, T.}, title = {A new design algorithm for twoband orthonormal rational filterbanks and orthonormal rational wavelets}, journal = {IEEE Trans. Acous., Speech Signal Process.}, year = {1998}, volume = {46}, number = {6}, pages = {14941504}, doi = {http://dx.doi.org/10.1109/78.678463} } 

Blu, T.  Iterated filter banks with rational rate changes connection with discrete wavelet transforms  1993  IEEE Trans. Acous., Speech Signal Process. Vol. 41(12), pp. 32323244 
article  DOI 
Abstract: Some properties of twoband filter banks with rational rate changes (?rational filter banks?) are first reviewed. Focusing then on iterated rational filter banks, compactly supported limit functions are obtained, in the same manner as previously done for dyadic schemes, allowing a characterization of such filter banks. These functions are carefully studied and the properties they share with the dyadic case are highlighted. They are experimentally observed to verify a ?shift property? (strictly verified in the dyadic ease) up to an error which can be made arbitrarily small when their regularity increases. In this case, the highpass outputs of an iterated filter bank can be very close to samples of a discrete wavelet transform with the same rational dilation factor. Straightforward extension of the formalism of multiresolution analysis is also made. Finally, it is shown that if one is ready to put up with the loss of the shift property, rational iterated filter banks can be used in the same manner as if they were dyadic filter banks, with the advantage that rational dilation factors can be chosen closer to 1  
BibTeX:
@article{Blu_T_1993_tsp_ite_fbrccdwt, author = {Blu, T.}, title = {Iterated filter banks with rational rate changes connection with discrete wavelet transforms}, journal = {IEEE Trans. Acous., Speech Signal Process.}, year = {1993}, volume = {41}, number = {12}, pages = {32323244}, doi = {http://dx.doi.org/10.1109/78.258070} } 

Blu, T. & Unser, M.  The Fractional Spline Wavelet Transform: Definition and Implementation  2000  Vol. IProc. Int. Conf. Acoust. Speech Signal Process., pp. 512515 
inproceedings  
Abstract: We define a new wavelet transform that is based on a recently defined family of scaling functions: the fractional Bsplines. The interest of this family is that they interpolate between the integer degrees of polynomial Bsplines and that they allow a fractional order of approximation. The orthogonal fractional spline wavelets essentially behave as a fractional differentiators. This property seems promising for the analysis of 1/f^aplha noise that can be whitened by an appropriate choice of the degree of the spline transform. We present a practical FFTbased algorithm for the implementation of these fractional wavelet transforms, and give some examples of processing.  
BibTeX:
@inproceedings{Blu_T_2000_icassp_fra_swtdi, author = {Blu, T. and Unser, M.}, title = {The Fractional Spline Wavelet Transform: Definition and Implementation}, booktitle = {Proc. Int. Conf. Acoust. Speech Signal Process.}, year = {2000}, volume = {I}, pages = {512515} } 

Bogdanova, I., Vandergheynst, P., Antoine, J.P., Jacques, L. & Morvidone, M.  Stereographic wavelet frames on the sphere [BibTeX] 
2005  Appl. Comp. Harm. Analysis Vol. 19(2), pp. 223252 
article  
BibTeX:
@article{Bogdanova_I_2005_jacha_ste_wfs, author = {Bogdanova, I. and Vandergheynst, P. and Antoine, J.P. and Jacques, L. and Morvidone, M.}, title = {Stereographic wavelet frames on the sphere}, journal = {Appl. Comp. Harm. Analysis}, year = {2005}, volume = {19}, number = {2}, pages = {223252} } 

Bracewell, R.N.  The Fourier transform and its applications [BibTeX] 
1986  book  
BibTeX:
@book{Bracewell_R_1986_book_fou_ta, author = {R. N. Bracewell}, title = {The Fourier transform and its applications}, publisher = {McGrawHill}, year = {1986}, edition = {2nd} } 

Bredies, K., Lorenz, D.A. & Maass, P.  Mathematical concepts of multiscale smoothing  2005  Appl. Comp. Harm. Analysis Vol. 19(2), pp. 141161 
article  DOI URL 
Abstract: The starting point for this paper is the wellknown equivalence between convolution filtering with a rescaled Gaussian and the solution of the heat equation. In the first sections we analyze the equivalence between multiscale convolution filtering, linear smoothing methods based on continuous wavelet transforms and the solutions of linear diffusion equations. This means we determine a wavelet [psi], respectively a convolution filter [phi], which is associated with a given linear diffusion equation and vice versa. This approach has an extension to nonlinear smoothing techniques. The main result of this paper is the derivation of a differential equation, whose solution is equivalent to nonlinear multiscale smoothing based on soft shrinkage methods applied to Fourier or continuous wavelet transforms.  
BibTeX:
@article{Bredies_K_2005_jacha_mat_cms, author = {Bredies, K. and Lorenz, D. A. and Maass, P.}, title = {Mathematical concepts of multiscale smoothing}, journal = {Appl. Comp. Harm. Analysis}, year = {2005}, volume = {19}, number = {2}, pages = {141161}, url = {http://www.sciencedirect.com/science/article/B6WB34G1PKMR1/2/89406ed97397578cd8b8dada02c8f7af}, doi = {DOI: 10.1016/j.acha.2005.02.007} } 

Breiman, L., Friedman, J.H., Olshen, R.A. & Stone, C.J.  Classification and Regression Trees  1984  book  
Abstract: The methodology used to construct tree structured rules is the focus of this monograph. Unlike many other statistical procedures, which moved from pencil and paper to calculators, this text's use of trees was unthinkable before computers. Both the practical and theoretical sides have been developed in the authors' study of tree methods. Classification and Regression Trees reflects these two sides, covering the use of trees as a data analysis method, and in a more mathematical framework, proving some of their fundamental properties.  
BibTeX:
@book{Breiman_L_1984_book_cla_rt, author = {Breiman, L. and Friedman, J. H. and Olshen, R. A. and Stone, C. J.}, title = {Classification and Regression Trees}, publisher = {Wadsworth}, year = {1984} } 

Bresenham, J.E.  Algorithm for computer control of a digital plotter [BibTeX] 
1965  IBM Syst. J. Vol. 4(1), pp. 2530 
article  
BibTeX:
@article{Bresenham_J_1998_jibmsystj_alg_ccdp, author = {Bresenham, J. E.}, title = {Algorithm for computer control of a digital plotter}, journal = {IBM Syst. J.}, year = {1965}, volume = {4}, number = {1}, pages = {2530} } 

Bruekers, F.A.M.L. & van den Enden, A.W.M.  New networks for perfect inversion and perfect reconstruction  1992  IEEE J. Selected Areas Comm. Vol. 10(1), pp. 129137 
article  DOI 
Abstract: The authors present a new network structure that realizes perfect inversion networks (PINs) and perfect reconstruction networks (PRNs). In some applications, such as transform source coders, it is important that the cascade of the forward and the inverse transform give the identity exactly (perfect inversion), although the coefficients and the intermediate results are quantized. In subband coders, for example, the split and merge filter banks should preferably have perfect reconstruction. It is advantageous if perfect reconstruction can be accomplished even when the coefficients and the intermediate results are quantized. The proposed network has a ladderlike shape and a predescribed symmetry between the forward and inverse network or between the split and merge bank. In some parts of this ladder network almost any function is allowed. Due to the prescribed symmetry, the property of perfect inversion or perfect reconstruction is structurally assured  
BibTeX:
@article{Bruekers_F_1992_jieeeselareascom, author = {Bruekers, F. A. M. L. and van den Enden, A. W. M.}, title = {New networks for perfect inversion and perfect reconstruction}, journal = {IEEE J. Selected Areas Comm.}, year = {1992}, volume = {10}, number = {1}, pages = {129137}, doi = {http://dx.doi.org/10.1109/49.124464} } 

Brémaud, P.  Mathematical principles of signal processing: Fourier and wavelet analysis [BibTeX] 
2002  book  
BibTeX:
@book{Bremaud_P_2002_book_mat_pspfwa, author = {Brémaud, P.}, title = {Mathematical principles of signal processing: Fourier and wavelet analysis}, publisher = {SpringerVerlag}, year = {2002} } 

Burt, P.J. & Adelson, E.H.  The Laplacian Pyramid as a compact image code  1983  IEEE Trans. Commun. Vol. 31(4), pp. 532540 
article  
Abstract: We describe a technique for image encoding in which local operators of many scales but identical shape serve as the basis functions. The representation differs from established techniques in that the code elements are localized in spatial frequency as well as in space. Pixeltopixel correlations are first removed by subtracting a lowpass filtered copy of the image from the image itself. The result is a net data compression since the difference, or error, image has low variance and entropy, and the lowpass filtered image may represented at reduced sample density. Further data compression is achieved by quantizing the difference image. These steps are then repeated to compress the lowpass image iteration of the process at appropriately expanded scales generates a pyramid data structure. The encoding process is equivalent to sampling the image with Laplacian operators of many scales. Thus, the code tends to enhance salient image features. A further advantage of the present code is that it is well suited for many image analysis tasks as well as for image compression. Fast algorithms are described for coding and decoding.  
BibTeX:
@article{Burt_P_1983_tcom_lap_pcic, author = {Burt, P. J. and Adelson, E. H.}, title = {The Laplacian Pyramid as a compact image code}, journal = {IEEE Trans. Commun.}, year = {1983}, volume = {31}, number = {4}, pages = {532540} } 

Bülow, T.  Multiscale image processing on the sphere [BibTeX] 
2002  Proc. DAGM Symp. Patt. Recogn., pp. 609617  inproceedings  
BibTeX:
@inproceedings{Bulow_T_2002_pdagm_mul_ips, author = {Bülow, T.}, title = {Multiscale image processing on the sphere}, booktitle = {Proc. DAGM Symp. Patt. Recogn.}, publisher = {Springer}, year = {2002}, pages = {609617} } 

Bülow, T. & Sommer, G.  Hypercomplex signals  a novel extension of the analytic signal to the multidimensional case  2001  IEEE Trans. Signal Process. Vol. 49(11), pp. 28442852 
article  DOI 
Abstract: The construction of Gabor's (1946) complex signalwhich is also known as the analytic signalprovides direct access to a real onedimensional (1D) signal's local amplitude and phase. The complex signal is built from a real signal by adding its Hilbert transformwhich is a phaseshifted version of the signalas an imaginary part to the signal. Since its introduction, the complex signal has become an important tool in signal processing, with applications, for example, in narrowband communication. Different approaches to an nD analytic or complex signal have been proposed in the past. We review these approaches and propose the hypercomplex signal as a novel extension of the complex signal to nD. This extension leads to a new definition of local phase, which reveals information on the intrinsic dimensionality of the signal. The different approaches are unified by expressing all of them as combinations of the signal and its partial and total Hilbert transforms. Examples that clarify how the approaches differ in their definitions of local phase and amplitude are shown. An example is provided for the twodimensional (2D) hypercomplex signal, which shows how the novel phase concept can be used in texture segmentation  
BibTeX:
@article{Bulow_T_2001_jieeetsp_hyp_sneasmc, author = {Bülow, T. and Sommer, G.}, title = {Hypercomplex signals  a novel extension of the analytic signal to the multidimensional case}, journal = {IEEE Trans. Signal Process.}, year = {2001}, volume = {49}, number = {11}, pages = {28442852}, doi = {http://dx.doi.org/10.1109/78.960432} } 

Cai, T.  Adaptive Wavelet Estimation: A Block Thresholding And Oracle Inequality Approach  1999  Ann. Stat. Vol. 27, pp. 898924 
article  
Abstract: We study wavelet function estimation via the approach of block thresholding and ideal adaptation with oracle. Oracle inequalities are derived and serve as guides for the selection of smoothing parameters. Based on an oracle inequality and motivated by the data compression and localization properties of wavelets, an adaptive wavelet estimator for nonparametric regression is proposed and the optimality of the procedure is investigated. We show that the estimator achieves simultaneously three objectives: adaptivity, spatial adaptivity, and computational efficiency. Specifically, it is proved that the estimator attains the exact optimal rates of convergence over a range of Besov classes and the estimator achieves adaptive local minimax rate for estimating functions at a point. The estimator is easy to implement, at the computational cost of $O(n)$. Simulation shows that the estimator has excellent numerical performance relative to more traditional wavelet estimators.  
BibTeX:
@article{Cai_T_1999_annstat_ada_webtoia, author = {Cai, T.}, title = {Adaptive Wavelet Estimation: A Block Thresholding And Oracle Inequality Approach}, journal = {Ann. Stat.}, year = {1999}, volume = {27}, pages = {898924} } 

Candès, E.J. & Donoho, D.L.  New tight frames of curvelets and optimal representations of objects with piecewise $C^2$ singularities [BibTeX] 
2004  Comm. Pure Applied Math. Vol. 57(2), pp. 219266 
article  
BibTeX:
@article{Candes_E_2004_jcommpureapplmath_new_tfcoropc2s, author = {E. J. Candès and D. L. Donoho}, title = {New tight frames of curvelets and optimal representations of objects with piecewise $C^2$ singularities}, journal = {Comm. Pure Applied Math.}, year = {2004}, volume = {57}, number = {2}, pages = {219266} } 

Candès, E.J.  Compressive sampling  2006  Vol. 3Proc. Int. Congr. Mathematicians, pp. 14331452 
inproceedings  
Abstract: Conventional wisdom and common practice in acquisition and reconstruction of images from frequency data follow the basic principle of the Nyquist density sampling theory. This principle states that to reconstruct an image, the number of Fourier samples we need to acquire must match the desired resolution of the image, i.e. the number of pixels in the image. This paper surveys an emerging theory which goes by the name of ?compressive sampling? or ?compressed sensing,? and which says that this conventional wisdom is inaccurate. Perhaps surprisingly, it is possible to reconstruct images or signals of scientific interest accurately and sometimes even exactly from a number of samples which is far smaller than the desired resolution of the image/signal, e.g. the number of pixels in the image. It is believed that compressive sampling has far reaching implications. For example, it suggests the possibility of new data acquisition protocols that translate analog information into digital form with fewer sensors than what was considered necessary. This new sampling theory may come to underlie procedures for sampling and compressing data simultaneously. In this short survey, we provide some of the key mathematical insights underlying this new theory, and explain some of the interactions between compressive sampling and other fields such as statistics, information theory, coding theory, and theoretical computer science. 

BibTeX:
@inproceedings{Candes_E_2006_pintcongressmath_com_s, author = {E. J. Candès}, title = {Compressive sampling}, booktitle = {Proc. Int. Congr. Mathematicians}, year = {2006}, volume = {3}, pages = {14331452} } 

Candès, E.J., Demanet, L., Donoho, D.L. & Ying, L.  Fast discrete curvelet transforms  2006  Multiscale Model. Simul. Vol. 5(3), pp. 861899 
article  DOI 
Abstract: This paper describes two digital implementations of a new mathematical transform, namely, the second generation curvelet transform in two and three dimensions. The first digital transformation is based on unequally spaced fast Fourier transforms, while the second is based on the wrapping of specially selected Fourier samples. The two implementations essentially differ by the choice of spatial grid used to translate curvelets at each scale and angle. Both digital transformations return a table of digital curvelet coefficients indexed by a scale parameter, an orientation parameter, and a spatial location parameter. And both implementations are fast in the sense that they run in O(n^2 log n) flops for n by n Cartesian arrays; in addition, they are also invertible, with rapid inversion algorithms of about the same complexity. Our digital transformations improve upon earlier implementations?based upon the first generation of curvelets?in the sense that they are conceptually simpler, faster, and far less redundant. The software CurveLab, which implements both transforms presented in this paper, is available at http://www.curvelet.org.  
BibTeX:
@article{Candes_E_2006_siammms_fas_dct, author = {Candès, E. J. and Demanet, L. and Donoho, D. L. and Ying, L.}, title = {Fast discrete curvelet transforms}, journal = {Multiscale Model. Simul.}, year = {2006}, volume = {5}, number = {3}, pages = {861899}, doi = {http://dx.doi.org/10.1137/05064182X} } 

Candès, E.J. & Donoho, D.L.  Curvelets  a surprisingly effective nonadaptive representation for objects with edges  1999  Curves and Surfaces, pp. 105120  incollection  
Abstract: It is widely believed that to efficiently represent an otherwise smooth object with discontinuities along edges, one must use an adaptive representation that in some sense 'tracks' the shape of the discontinuity set. This folkbelief  some would say folktheorem  is incorrect. At the very least, the possible quantitative advantage of such adaptation is vastly smaller than commonly believed. We have recently constructed a tight frame of curvelets which provides stable, efficient, and nearoptimal representation of otherwise smooth objects having discontinuities along smooth curves. By applying naive thresholding to the curvelet transform of such an object, one can form mterm approximations with rate of L2 approximation rivaling the rate obtainable by complex adaptive schemes which attempt to `track' the discontinuity set. In this article we explain the basic issues of efficient mterm approximation, the construction of efficient adaptive representation, the construction of the curvelet frame, and a crude analysis of the performance of curvelet schemes.  
BibTeX:
@incollection{Candes_E_1999_curvessurfaces_cur_senroe, author = {Candès, E. J. and Donoho, D. L.}, title = {Curvelets  a surprisingly effective nonadaptive representation for objects with edges}, booktitle = {Curves and Surfaces}, publisher = {Vanderbilt University Press}, year = {1999}, pages = {105120} } 

Candès, E.J. & Donoho, D.L.  Continuous curvelet transform: I. Resolution of the wavefront set [BibTeX] 
2003  Appl. Comp. Harm. Analysis Vol. 19, pp. 162197 
article  
BibTeX:
@article{Candes_E_2003_jacha_con_ct1rws, author = {E. J. Candès and D. L. Donoho}, title = {Continuous curvelet transform: I. Resolution of the wavefront set}, journal = {Appl. Comp. Harm. Analysis}, year = {2003}, volume = {19}, pages = {162197} } 

Candès, E.J. & Donoho, D.L.  Continuous curvelet transform: II. Discretization and frames [BibTeX] 
2003  Appl. Comp. Harm. Analysis Vol. 19, pp. 198222 
article  
BibTeX:
@article{Candes_E_2003_jacha_con_ct2df, author = {E. J. Candès and D. L. Donoho}, title = {Continuous curvelet transform: II. Discretization and frames}, journal = {Appl. Comp. Harm. Analysis}, year = {2003}, volume = {19}, pages = {198222} } 

Candès, E.J. & Donoho, D.L.  New tight frames of curvelets and optimal representations of objects with piecewise $C^2$ singularities  2003  Comm. Pure Applied Math. Vol. 57(2), pp. 219266 
article  
Abstract: This paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along piecewise C2 edges. Conceptually, the curvelet transform is a multiscale pyramid with many directions and positions at each length scale, and needleshaped elements at fine scales. These elements have many useful geometric multiscale features that set them apart from classical multiscale representations such as wavelets. For instance, curvelets obey a parabolic scaling relation which says that at scale 2j, each element has an envelope that is aligned along a ridge of length 2j/2 and width 2j. We prove that curvelets provide an essentially optimal representation of typical objects f that are C2 except for discontinuities along piecewise C2 curves. Such representations are nearly as sparse as if f were not singular and turn out to be far more sparse than the wavelet decomposition of the object. For instance, the nterm partial reconstruction fCn obtained by selecting the n largest terms in the curvelet series obeys ?f  fCn?2L2 ? C . n2 . (log n)3, n ? ?. This rate of convergence holds uniformly over a class of functions that are C2 except for discontinuities along piecewise C2 curves and is essentially optimal. In comparison, the squared error of nterm wavelet approximations only converges as n1 as n ? ?, which is considerably worse than the optimal behavior.  
BibTeX:
@article{Candes_E_2003_jcommpureapplmath_new_tfcoropc2s, author = {Candès, E. J. and Donoho, D. L.}, title = {New tight frames of curvelets and optimal representations of objects with piecewise $C^2$ singularities}, journal = {Comm. Pure Applied Math.}, year = {2003}, volume = {57}, number = {2}, pages = {219266} } 

Candès, E.J. & Donoho, D.L.  Ridgelets: a key to higherdimensional intermittency? [BibTeX] 
1999  Phil. Trans. R. Soc. Lond. A Vol. 357, pp. 24952509 
article  
BibTeX:
@article{Candes_E_1999_philtransrsoclond_rid_khdi, author = {Candès, E. J. and Donoho, D. L.}, title = {Ridgelets: a key to higherdimensional intermittency?}, journal = {Phil. Trans. R. Soc. Lond. A}, year = {1999}, volume = {357}, pages = {24952509} } 

Casazza, P.G.  The art of frame theory [BibTeX] 
2000  Taiwanese J. of Math. Vol. 15(4), pp. 129201 
article  
BibTeX:
@article{Casazza_P_2000_tjm_art_ft, author = {Casazza, P. G.}, title = {The art of frame theory}, journal = {Taiwanese J. of Math.}, year = {2000}, volume = {15}, number = {4}, pages = {129201} } 

Cayón, L., Sanz, J.L., Barreiro, R.B., MartínezGonzález, E., Vielva, P., Toffolatti, L., Silk, J., Diego, J.M. & Argüeso, F.  Isotropic wavelets: a powerful tool to extract point sources from cosmic microwave background maps  2000  Mon. Not. Roy. Astron. Soc. Vol. 315(4), pp. 757761 
article  DOI URL 
Abstract: It is the aim of this paper to introduce the use of isotropic wavelets to detect and determine the flux of point sources appearing in cosmic microwave background (CMB) maps. The most suitable wavelet to detect point sources filtered with a Gaussian beam is the 'Mexican Hat'. An analytical expression of the wavelet coefficient obtained in the presence of a point source is provided and used in the detection and flux estimation methods presented. For illustration the method is applied to two simulations (assuming Planck mission characteristics) dominated by CMB (100 GHz) and dust (857 GHz), as these will be the two signals dominating at low and high frequencies respectively in the Planck channels. We are able to detect bright sources above 1.58 Jy at 857 GHz (82 per cent of all sources) and above 0.36 Jy at 100 GHz (100 per cent of all), with errors in the flux estimation below 25 per cent. The main advantage of this method is that nothing has to be assumed about the underlying field, i.e. about the nature and properties of the signal plus noise present in the maps. This is not the case in the detection method presented by Tegmark & OliveiraCosta. Both methods are compared, producing similar results.  
BibTeX:
@article{Cayon_L_2000_jmonnotroyastronsoc_iso_wptepscmbm, author = {Cayón, L. and Sanz, J. L. and Barreiro, R. B. and MartínezGonzález, E. and Vielva, P. and Toffolatti, L. and Silk, J. and Diego, J. M. and Argüeso, F.}, title = {Isotropic wavelets: a powerful tool to extract point sources from cosmic microwave background maps}, journal = {Mon. Not. Roy. Astron. Soc.}, year = {2000}, volume = {315}, number = {4}, pages = {757761}, url = {http://dx.doi.org/10.1046/j.13658711.2000.03462.x}, doi = {http://dx.doi.org/10.1046/j.13658711.2000.03462.x} } 

Chambolle, A. & Lucier, B.J.  Interpreting translationinvariant wavelet shrinkage as a new image smoothing scale space  2001  IEEE Trans. Image Process. Vol. 10(7), pp. 9931000 
article  DOI 
Abstract: Coifman and Donoho (1995) suggested translationinvariant wavelet shrinkage as a way to remove noise from images. Basically, their technique applies wavelet shrinkage to a twodimensional (2D) version of the semidiscrete wavelet representation of Mallat and Zhong (1992), Coifman and Donoho also showed how the method could be implemented in O(Nlog N) operations, where there are N pixels. In this paper, we provide a mathematical framework for iterated translationinvariant wavelet shrinkage, and show, using a theorem of Kato and Masuda (1978), that with orthogonal wavelets it is equivalent to gradient descent in L 2(I) along the seminorm for the Besov space B1 1(L1(I)), which, in turn, can be interpreted as a new nonlinear waveletbased image smoothing scale space. Unlike many other scale spaces, the characterization is not in terms of a nonlinear partial differential equation  
BibTeX:
@article{Chambolle_A_2001_jieeetip_int_tiwsnisss, author = {Chambolle, A. and Lucier, B. J.}, title = {Interpreting translationinvariant wavelet shrinkage as a new image smoothing scale space}, journal = {IEEE Trans. Image Process.}, year = {2001}, volume = {10}, number = {7}, pages = {9931000}, doi = {http://dx.doi.org/10.1109/83.931093} } 

Chan, W., Choi, H. & Baraniuk, R.G.  Directional hypercomplex wavelets for multidimensional signal analysis and processing  2004  Vol. 3Proc. Int. Conf. Acoust. Speech Signal Process., pp. 996999 
inproceedings  DOI 
Abstract: We extend the wavelet transform to handle multidimensional signals that are smooth save for singularities along lowerdimensional manifolds. We first generalize the complex wavelet transform to higher dimensions using a multidimensional Hilbert transform. Then, using the resulting hypercomplex wavelet transform (HWT) as a building block, we construct new classes of nearly shiftinvariant wavelet frames that are oriented along lowerdimensional subspaces. The HWT can be computed efficiently using a 1D dualtree complex wavelet transform along each signal axis. We demonstrate how the HWT can be used for fast line detection in 3D.  
BibTeX:
@inproceedings{Chan_W_2004_icassp_dir_hwmsap, author = {Chan, W. and Choi, H. and Baraniuk, R. G.}, title = {Directional hypercomplex wavelets for multidimensional signal analysis and processing}, booktitle = {Proc. Int. Conf. Acoust. Speech Signal Process.}, year = {2004}, volume = {3}, pages = {996999}, doi = {http://dx.doi.org/10.1109/ICASSP.2004.1326715} } 

Chandrasekaran, V., Wakin, M.B., Baron, D. & Baraniuk, R.G.  Representation and Compression of Multidimensional Piecewise Functions Using Surflets  2009  IEEE Trans. Inform. Theory Vol. 55(1), pp. 374400 
article  DOI 
Abstract: We study the representation, approximation, and compression of functions in M dimensions that consist of constant or smooth regions separated by smooth (M1)dimensional discontinuities. Examples include images containing edges, video sequences of moving objects, and seismic data containing geological horizons. For both function classes, we derive the optimal asymptotic approximation and compression rates based on Kolmogorov metric entropy. For piecewise constant functions, we develop a multiresolution predictive coder that achieves the optimal ratedistortion performance; for piecewise smooth functions, our coder has nearoptimal ratedistortion performance. Our coder for piecewise constant functions employs surflets, a new multiscale geometric tiling consisting of Mdimensional piecewise constant atoms containing polynomial discontinuities. Our coder for piecewise smooth functions uses surfprints, which wed surflets to wavelets for piecewise smooth approximation. Both of these schemes achieve the optimal asymptotic approximation performance. Key features of our algorithms are that they carefully control the potential growth in surflet parameters at higher smoothness and do not require explicit estimation of the discontinuity. We also extend our results to the corresponding discrete function spaces for sampled data. We provide asymptotic performance results for both discrete function spaces and relate this asymptotic performance to the sampling rate and smoothness orders of the underlying functions and discontinuities. For approximation of discrete data, we propose a new scaleadaptive dictionary that contains few elements at coarse and fine scales, but many elements at medium scales. Simulation results on synthetic signals provide a comparison between surfletbased coders and previously studied approximation schemes based on wedgelets and wavelets.  
BibTeX:
@article{Chandrasekaran_V_2009_jieeetit_rep_cmpfs, author = {Chandrasekaran, V. and Wakin, M. B. and Baron, D. and Baraniuk, R. G.}, title = {Representation and Compression of Multidimensional Piecewise Functions Using Surflets}, journal = {IEEE Trans. Inform. Theory}, year = {2009}, volume = {55}, number = {1}, pages = {374400}, doi = {http://dx.doi.org/10.1109/TIT.2008.2008153} } 

Chang, C.L. & Girod, B.  DirectionAdaptive Discrete Wavelet Transform for Image Compression  2007  IEEE Trans. Image Process. Vol. 16(5), pp. 12891302 
article  DOI 
Abstract: We propose a directionadaptive DWT (DADWT) that locally adapts the filtering directions to image content based on directional lifting. With the adaptive transform, energy compaction is improved for sharp image features. A mathematical analysis based on an anisotropic statistical image model is presented to quantify the theoretical gain achieved by adapting the filtering directions. The analysis indicates that the proposed DADWT is more effective than other liftingbased approaches. Experimental results report a gain of up to 2.5 dB in PSNR over the conventional DWT for typical test images. Subjectively, the reconstruction from the DADWT better represents the structure in the image and is visually more pleasing  
BibTeX:
@article{Chang_C_2007_tip_dir_adwtic, author = {C.L. Chang and Girod, B.}, title = {DirectionAdaptive Discrete Wavelet Transform for Image Compression}, journal = {IEEE Trans. Image Process.}, year = {2007}, volume = {16}, number = {5}, pages = {12891302}, doi = {http://dx.doi.org/10.1109/TIP.2007.894242} } 

Chappelier, V. & Guillemot, C.  Oriented Wavelet Transform for Image Compression and Denoising  2006  IEEE Trans. Image Process. Vol. 15(10), pp. 28922903 
article  DOI 
Abstract: In this paper, we introduce a new transform for image processing, based on wavelets and the lifting paradigm. The lifting steps of a unidimensional wavelet are applied along a local orientation defined on a quincunx sampling grid. To maximize energy compaction, the orientation minimizing the prediction error is chosen adaptively. A finegrained multiscale analysis is provided by iterating the decomposition on the lowfrequency band. In the context of image compression, the multiresolution orientation map is coded using a quad tree. The rate allocation between the orientation map and wavelet coefficients is jointly optimized in a ratedistortion sense. For image denoising, a Markov model is used to extract the orientations from the noisy image. As long as the map is sufficiently homogeneous, interesting properties of the original wavelet are preserved such as regularity and orthogonality. Perfect reconstruction is ensured by the reversibility of the lifting scheme. The mutual information between the wavelet coefficients is studied and compared to the one observed with a separable wavelet transform. The ratedistortion performance of this new transform is evaluated for image coding using stateoftheart subband coders. Its performance in a denoising application is also assessed against the performance obtained with other transforms or denoising methods  
BibTeX:
@article{Chappelier_V_2006_tip_ori_wticd, author = {Chappelier, V. and Guillemot, C.}, title = {Oriented Wavelet Transform for Image Compression and Denoising}, journal = {IEEE Trans. Image Process.}, year = {2006}, volume = {15}, number = {10}, pages = {28922903}, doi = {http://dx.doi.org/10.1109/TIP.2006.877526} } 

Chaudhury, K.N. & Unser, M.  On the Shiftability of DualTree Complex Wavelet Transforms  2010  IEEE Trans. Signal Process. Vol. 58(1), pp. 221232 
article  
Abstract: The dualtree complex wavelet transform (DTCWT) is known to exhibit better shiftinvariance than the conventional discrete wavelet transform. We propose an amplitudephase representation of the DTCWT which, among other things, offers a direct explanation for the improvement in the shiftinvariance. The representation is based on the shifting action of the group of fractional Hilbert transform (fHT) operators, which extends the notion of arbitrary phaseshifts from sinusoids to finiteenergy signals (wavelets in particular). In particular, we characterize the shiftability of the DTCWT in terms of the shifting property of the fHTs. At the heart of the representation are certain fundamental invariances of the fHT group, namely that of translation, dilation, and norm, which play a decisive role in establishing the key properties of the transform. It turns out that these fundamental invariances are exclusive to this group. Next, by introducing a generalization of the Bedrosian theorem for the fHT operator, we derive an explicitly understanding of the shifting action of the fHT for the particular family of wavelets obtained through the modulation of lowpass functions (e.g., the Shannon and Gabor wavelet). This, in effect, links the corresponding dualtree transform with the framework of windowedFourier analysis. Finally, we extend these ideas to the multidimensional setting by introducing a directional extension of the fHT, the fractional directional Hilbert transform. In particular, we derive a signal representation involving the superposition of directionselective wavelets with appropriate phaseshifts, which helps explain the improved shiftinvariance of the transform along certain preferential directions.  
BibTeX:
@article{Chaudhury_K_2010_jieeetsp_shi_dtcwt, author = {K. N. Chaudhury and M. Unser}, title = {On the Shiftability of DualTree Complex Wavelet Transforms}, journal = {IEEE Trans. Signal Process.}, year = {2010}, volume = {58}, number = {1}, pages = {221232} } 

Chaudhury, K.N. & Unser, M.  Construction of Hilbert Transform Pairs of Wavelet Bases and GaborLike Transforms  2009  IEEE Trans. Signal Process. Vol. 57(9), pp. 34113425 
article  
Abstract: We propose a novel method for constructing Hilbert transform (HT) pairs of wavelet bases based on a fundamental approximationtheoretic characterization of scaling functions?the Bspline factorization theorem. In particular, starting from welllocalized scaling functions, we construct HT pairs of biorthogonal wavelet bases of L2(?) by relating the corresponding wavelet filters via a discrete form of the continuous HT filter. As a concrete application of this methodology, we identify HT pairs of spline wavelets of a specific flavor, which are then combined to realize a family of complex wavelets that resemble the optimallylocalized Gabor function for sufficiently large orders. Analytic wavelets, derived from the complexification of HT wavelet pairs, exhibit a onesided spectrum. Based on the tensorproduct of such analytic wavelets, and, in effect, by appropriately combining four separable biorthogonal wavelet bases of L2(?2), we then discuss a methodology for constructing 2D directionalselective complex wavelets. In particular, analogous to the HT correspondence between the components of the 1D counterpart, we relate the real and imaginary components of these complex wavelets using a multidimensional extension of the HT?the directional HT. Next, we construct a family of complex spline wavelets that resemble the directional Gabor functions proposed by Daugman. Finally, we present an efficient FFTbased filterbank algorithm for implementing the associated complex wavelet transform.  
BibTeX:
@article{Chaudhury_K_2009_jieeetsp_con_htpwbglt, author = {Chaudhury, K. N. and Unser, M.}, title = {Construction of Hilbert Transform Pairs of Wavelet Bases and GaborLike Transforms}, journal = {IEEE Trans. Signal Process.}, year = {2009}, volume = {57}, number = {9}, pages = {34113425} } 

Chaudhury, K.N. & Unser, M.  Gabor Wavelet Analysis and the Fractional Hilbert Transform  2009  Vol. 7446Proc. SPIE, Wavelets: Appl. Signal Image Process., pp. 74460T174460T7 
inproceedings  
Abstract: We propose an amplitudephase representation of the dualtree complex wavelet transform (DTCWT) which provides an intuitive interpretation of the associated complex wavelet coefficients. The representation, in particular, is based on the shifting action of the group of fractional Hilbert transforms (fHT) which allow us to extend the notion of arbitrary phaseshifts beyond pure sinusoids. We explicitly characterize this shifting action for a particular family of Gaborlike wavelets which, in effect, links the corresponding dualtree transform with the framework of windowedFourier analysis. We then extend these ideas to the bivariate DTCWT based on certain directional extensions of the fHT. In particular, we derive a signal representation involving the superposition of directionselective wavelets affected with appropriate phaseshifts.  
BibTeX:
@inproceedings{Chaudhury_K_2009_pspiewasip_gab_wafht, author = {Chaudhury, K. N. and Unser, M.}, title = {Gabor Wavelet Analysis and the Fractional Hilbert Transform}, booktitle = {Proc. SPIE, Wavelets: Appl. Signal Image Process.}, year = {2009}, volume = {7446}, pages = {74460T174460T7} } 

Chaux, C., Duval, L., BenazzaBenyahia, A. & Pesquet, J.C.  A nonlinear Stein based estimator for multichannel image denoising  2008  IEEE Trans. Signal Process. Vol. 56(8), pp. 38553870 
article  DOI 
Abstract: The use of multicomponent images has become widespread with the improvement of multisensor systems having increased spatial and spectral resolutions. However, the observed images are often corrupted by an additive Gaussian noise. In this paper, we are interested in multichannel image denoising based on a multiscale representation of the images. A multivariate statistical approach is adopted to take into account both the spatial and the intercomponent correlations existing between the different wavelet subbands. More precisely, we propose a new parametric nonlinear estimator which generalizes many reported denoising methods. The derivation of the optimal parameters is achieved by applying Stein's principle in the multivariate case. Experiments performed on multispectral remote sensing images clearly indicate that our method outperforms conventional wavelet denoising techniques  
BibTeX:
@article{Chaux_C_2008_jieeetsp_non_sbemid, author = {Chaux, C. and Duval, L. and BenazzaBenyahia, A. and Pesquet, J.C.}, title = {A nonlinear Stein based estimator for multichannel image denoising}, journal = {IEEE Trans. Signal Process.}, year = {2008}, volume = {56}, number = {8}, pages = {38553870}, doi = {http://dx.doi.org/10.1109/TSP.2008.921757} } 

Chaux, C., Duval, L. & Pesquet, J.C.  Image analysis using a dualtree $M$band wavelet transform [BibTeX] 
2006  IEEE Trans. Image Process. Vol. 15(8), pp. 23972412 
article  
BibTeX:
@article{Chaux_C_2006_tip_ima_adtmbwt, author = {Chaux, C. and Duval, L. and Pesquet, J.C.}, title = {Image analysis using a dualtree $M$band wavelet transform}, journal = {IEEE Trans. Image Process.}, year = {2006}, volume = {15}, number = {8}, pages = {23972412} } 

Chen, S.S., Donoho, D.L. & Saunders, M.A.  Atomic Decomposition by Basis Pursuit  1998  SIAM J. Sci. Comput. Vol. 20(1), pp. 3361 
article  
Abstract: The timefrequency and timescale communities have recently developed a large number of overcomplete waveform dictionaries  stationary wavelets, wavelet packets, cosine packets, chirplets, and warplets, to name a few. Decomposition into overcomplete systems is not unique, and several methods for decomposition have been proposed, including the method of frames (MOF), Matching pursuit (MP), and, for special dictionaries, the best orthogonal basis (BOB). Basis Pursuit (BP) is a principle for decomposing a signal into an "optimal" superposition of dictionary elements, where optimal means having the smallest l1 norm of coefficients among all such decompositions. We give examples exhibiting several advantages over MOF, MP, and BOB, including better sparsity and superresolution. BP has interesting relations to ideas in areas as diverse as illposed problems, in abstract harmonic analysis, total variation denoising, and multiscale edge denoising. BP in highly overcomplete dictionaries leads to largescale optimization problems. With signals of length 8192 and a wavelet packet dictionary, one gets an equivalent linear program of size 8192 by 212,992. Such problems can be attacked successfully only because of recent advances in linear programming by interiorpoint methods. We obtain reasonable success with a primaldual logarithmic barrier method and conjugategradient solver.  
BibTeX:
@article{Chen_S_1998_jsiamscicomp_ato_dbp, author = {S. S. Chen and D. L. Donoho and M. A. Saunders}, title = {Atomic Decomposition by Basis Pursuit}, journal = {SIAM J. Sci. Comput.}, year = {1998}, volume = {20}, number = {1}, pages = {3361} } 

Christensen, O.  Frames, Riesz bases, and discrete Gabor/wavelet expansions  2001  Bull. Amer. Math. Soc. Vol. 38, pp. 273291 
article  
Abstract: This paper is a survey of research in discrete expansions over the last 10 years, mainly of functions in $L^2(mathbb R)$. The concept of an orthonormal basis $f_n$, allowing every function $f in L^2(mathbb R)$ to be written $f=sum c_nf_n$for suitable coefficients $c_n$, is well understood. In separable Hilbert spaces, a generalization known as frames exists, which still allows such a representation. However, the coefficients $c_n$ are not necessarily unique. We discuss the relationship between frames and Riesz bases, a subject where several new results have been proved over the last 10 years. Another central topic is the study of frames with additional structure, most important Gabor frames (consisting of modulated and translated versions of a single function) and wavelets (translated and dilated versions of one function). Along the way, we discuss some possible directions for future research.  
BibTeX:
@article{Christensen_O_2001_jbullamermathsoc_fra_rbdgwe, author = {O. Christensen}, title = {Frames, Riesz bases, and discrete Gabor/wavelet expansions}, journal = {Bull. Amer. Math. Soc.}, year = {2001}, volume = {38}, pages = {273291} } 

Chui, C.K., He, W. & Stöckler, J.  Compactly supported tight and sibling frames with maximum vanishing moments  2002  Appl. Comp. Harm. Analysis Vol. 13(3), pp. 224262 
article  DOI URL 
Abstract: The notion of vanishingmoment recovery (VMR) functions is introduced in this paper for the construction of compactly supported tight frames with two generators having the maximum order of vanishing moments as determined by the given refinable function, such as the mth order cardinal Bspline Nm. Tight frames are also extended to #sibling##frames# to allow additional properties, such as symmetry (or antisymmetry), minimum support, shiftinvariance, and interorthogonality. For Nm, it turns out that symmetry can be achieved for even m and antisymmetry for odd m, that minimum support and shiftinvariance can be attained by considering the frame generators with twoscale symbols 2m(1z)m and 2mz(1z)m, and that interorthogonality is always achievable, but sometimes at the sacrifice of symmetry. The results in this paper are valid for all compactly supported refinable functions that are reasonably smooth, such as piecewise Lip[alpha] for some [alpha]>0, as long as the corresponding twoscale Laurent polynomial symbols vanish at z=1. Furthermore, the methods developed here can be extended to the more general setting, such as arbitrary integer scaling factors, multiwavelets, and certainly biframes (i.e., allowing the dual frames to be associated with a different refinable function).  
BibTeX:
@article{Chui_C_2002_pacha_com_stsfmvm, author = {C. K. Chui and W. He and J. Stöckler}, title = {Compactly supported tight and sibling frames with maximum vanishing moments}, journal = {Appl. Comp. Harm. Analysis}, year = {2002}, volume = {13}, number = {3}, pages = {224262}, url = {http://www.sciencedirect.com/science/article/B6WB3474DMCF6/2/db9dc4c2133b1959e78efa4e3db881ff}, doi = {DOI: 10.1016/S10635203(02)005109} } 

Claypoole, R.L., Davis, G.M., Sweldens, W. & Baraniuk, R.G.  Nonlinear wavelet transforms for image coding via lifting  2003  IEEE Trans. Image Process. Vol. 12(12), pp. 14491459 
article  DOI 
Abstract: We investigate central issues such as invertibility, stability, synchronization, and frequency characteristics for nonlinear wavelet transforms built using the lifting framework. The nonlinearity comes from adaptively choosing between a class of linear predictors within the lifting framework. We also describe how earlier families of nonlinear filter banks can be extended through the use of prediction functions operating on a causal neighborhood of pixels. Preliminary compression results for model and realworld images demonstrate the promise of our techniques.  
BibTeX:
@article{Claypoole_R_2003_tip_non_wticl, author = {Claypoole, R. L. and Davis, G. M. and Sweldens, W. and Baraniuk, R. G.}, title = {Nonlinear wavelet transforms for image coding via lifting}, journal = {IEEE Trans. Image Process.}, year = {2003}, volume = {12}, number = {12}, pages = {14491459}, doi = {http://dx.doi.org/10.1109/TIP.2003.817237} } 

Clonda, D., Lina, J.M. & Goulard, B.  Complex Daubechies wavelets: properties and statistical image modelling  2004  Signal Process. Vol. 84(1), pp. 123 
article  
Abstract: This article presents the construction and various properties of complex Daubechies wavelets with a special emphasis on symmetric solutions. Such solutions exhibit interesting relationships between the real and imaginary components of the complex scaling function and the complex wavelet. We present those properties in the context of image processing. Within the framework of statistical modelling, we focus on the redundant description of real images given by the complex multiresolution representation. A hierarchical Markovian Graphical model is then explored. We present an Expectation Maximization algorithm for optimizing the model with observational complex wavelet data. This model is then applied to image estimation and texture classification. In both applications, we demonstrate the benefit brought by the Markovian hypothesis and the performance of the real images's complex multiscale representation.  
BibTeX:
@article{Clonda_D_2004_sp_com_dwpsim, author = {Clonda, D. and Lina, J.M. and Goulard, B.}, title = {Complex Daubechies wavelets: properties and statistical image modelling}, journal = {Signal Process.}, year = {2004}, volume = {84}, number = {1}, pages = {123} } 

Cohen, A., Daubechies, I. & Feauveau, J.C.  Biorthogonal bases of compactly supported wavelets  1992  Commun. ACM Vol. 45(5), pp. 485560 
article  DOI 
Abstract: Orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide. We show here that under fairly general conditions, exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual Riesz bases of compactly supported wavelets. We give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions, and we present a sufficient conditions for the decay of their Fourier transforms. We study the regularity of these biorthogonal bases. We provide several families of examples, all symmetric (corresponding to ldquolinear phaserdquo filters). In particular we can construct symmetric biorthogonal wavelet bases with arbitraily high preassigned regularity; we also show how to construct symmetric biorthogonal wavelet bases ldquocloserdquo to a (nonsymmetric) orthonormal basis.  
BibTeX:
@article{Cohen_A_1992_jcommacm_bio_bcsw, author = {Cohen, A. and Daubechies, I. and Feauveau, J.C.}, title = {Biorthogonal bases of compactly supported wavelets}, journal = {Commun. ACM}, year = {1992}, volume = {45}, number = {5}, pages = {485560}, doi = {http://dx.doi.org/10.1002/cpa.3160450502} } 

Cohen, A. & Dyn, N.  Nonstationary subdivision schemes, multiresolution analysis, and wavelet packets  1998  Vol. 7Signal and image representation in combined spaces, pp. 189200 
incollection  DOI URL 
Abstract: Nonstationary subdivision schemes consist of recursive refinements of an initial sparse sequence with the use of masks that may vary from one scale to the next finer one. We show that such schemes can be used to construct C[infinity] compactly supported orthonormal scaling functions, wavelets, and waveletpackets with better control on the frequency localization.  
BibTeX:
@incollection{Cohen_A_1998_incoll_non_ssmawp, author = {A. Cohen and N. Dyn}, title = {Nonstationary subdivision schemes, multiresolution analysis, and wavelet packets}, booktitle = {Signal and image representation in combined spaces}, publisher = {Academic Press}, year = {1998}, volume = {7}, pages = {189200}, url = {http://www.sciencedirect.com/science/article/B8H444NVH5KF8/2/5db5af84ac9c81a4c62f218ff48996e2}, doi = {DOI: 10.1016/S1874608X(98)800083} } 

Cohen, A., Dyn, N., Hecht, F. & Mirebeau, J.M.  Adaptive multiresolution analysis based on anisotropic triangulations  2011  Math. Comput.  article  
Abstract: A simple greedy refinement procedure for the generation of dataadapted triangulations is proposed and studied. Given a function of two variables, the algorithm produces a hierarchy of triangulations and piecewise polynomial approximations on these triangulations. The refinement procedure consists in bisecting a triangle T in a direction which is chosen so as to minimize the local approximation error in some prescribed norm between the approximated function and its piecewise polynomial approximation after T is bisected. The hierarchical structure allows us to derive various approximation tools such as multiresolution analysis, wavelet bases, adaptive triangulations based either on greedy or optimal CART trees, as well as a simple encoding of the corresponding triangulations. We give a general proof of convergence in the Lp norm of all these approximations. Numerical tests performed in the case of piecewise linear approximation of functions with analytic expressions or of numerical images illustrate the fact that the refinement procedure generates triangles with an optimal aspect ratio (which is dictated by the local Hessian of of the approximated function in case of C2 functions). 

BibTeX:
@article{Cohen_A_2011_PREPRINT_ada_mabat, author = {A. Cohen and N. Dyn and F. Hecht and J.M. Mirebeau}, title = {Adaptive multiresolution analysis based on anisotropic triangulations}, journal = {Math. Comput.}, year = {2011}, note = {Preprint, submitted, http://arxiv.org/abs/1101.1512} } 

Cohen, A. & Matei, B.  Nonlinear subdivision schemes: applications to image processing  2002  Tutorials on Multiresolution in Geometric Modelling, pp. 9397  incollection  
Abstract: The authors discuss some refinement rules for subdivision subschemes, which are of interest because of their relation to multiresolution analysis and wavelets bases, making them suitable for signal, hence image, processing. The refinements include linear refinement by polynomial reconstruction, nonlinear refinement by essentially nonoscillatory stencil selection, and nonlinear refinement by using stencil selection and subcell resolution.  
BibTeX:
@incollection{Cohen_A_2002_inbook_nonssaip, author = {A. Cohen and B. Matei}, title = {Nonlinear subdivision schemes: applications to image processing}, booktitle = {Tutorials on Multiresolution in Geometric Modelling}, publisher = {Springer Verlag}, year = {2002}, pages = {9397}, note = {Europ. summer school on principles of multiresolution in geometric modelling} } 

Cohen, A., de Vore, R., Petrushev, P. & Xu, H.  Non linear approximation and the space $BV(R^2)$ [BibTeX] 
1999  Am. J. Math. Vol. 121, pp. 587628 
article  
BibTeX:
@article{Cohen_A_1999_jamjmath_non_lasbvr2, author = {Cohen, A. and de Vore, R. and Petrushev, P. and Xu, H.}, title = {Non linear approximation and the space $BV(R^2)$}, journal = {Am. J. Math.}, year = {1999}, volume = {121}, pages = {587628} } 

Cohen, I., Raz, S. & Malah, D.  Orthonormal shiftinvariant adaptive local trigonometric decomposition  1997  Signal Process. Vol. 57(1), pp. 4364 
article  DOI URL 
Abstract: In this paper, an extended library of smooth local trigonometric bases is defined, and an appropriate fast #bestbasis# search algorithm is introduced. When compared with the standard local cosine decomposition (LCD), the proposed algorithm is advantageous in three respects. First, it leads to a bestbasis expansion that is shiftinvariant. Second, the resulting representation is characterized by a lower information cost. Third, the polarity of the folding operator is adapted to the parity properties of the segmented signal at the endpoints. The shift invariance stems from an adaptive relative shift of expansions in distinct resolution levels. We show that at any resolution level l it suffices to examine and select one of two relative shift options  a zero shift or a 2l1 shift. A variable folding operator, whose polarity is locally adapted to the parity properties of the signal, further enhances the representation. The computational complexity is manageable and comparable to that of the LCD.  
BibTeX:
@article{Cohen_I_1997_jsp_ort_sialtd, author = {I. Cohen and S. Raz and D. Malah}, title = {Orthonormal shiftinvariant adaptive local trigonometric decomposition}, journal = {Signal Process.}, year = {1997}, volume = {57}, number = {1}, pages = {4364}, url = {http://www.sciencedirect.com/science/article/B6V183SNV3JDJ/2/2e2fae13ff56a03e02d2aecad76a61b2}, doi = {DOI: 10.1016/S01651684(96)001855} } 

Coifman, R. & Donoho, D.  Translationinvariant denoising [BibTeX] 
1995  Vol. 103Wavelets and Statistics, pp. 125150 
incollection  
BibTeX:
@incollection{Coifman_R_1995_was_tra_id, author = {Coifman, R. and Donoho, D.}, title = {Translationinvariant denoising}, booktitle = {Wavelets and Statistics}, publisher = {Springer}, year = {1995}, volume = {103}, pages = {125150} } 

Coifman, R.R. & Maggioni, M.  Diffusion wavelets  2006  Appl. Comp. Harm. Analysis Vol. 21(1), pp. 5394 
article  DOI URL 
Abstract: Our goal in this paper is to show that many of the tools of signal processing, adapted Fourier and wavelet analysis can be naturally lifted to the setting of digital data clouds, graphs, and manifolds. We use diffusion as a smoothing and scaling tool to enable coarse graining and multiscale analysis. Given a diffusion operator T on a manifold or a graph, with large powers of low rank, we present a general multiresolution construction for efficiently computing, representing and compressing Tt. This allows a direct multiscale computation, to high precision, of functions of the operator, notably the associated Green's function, in compressed form, and their fast application. Classes of operators for which these computations are fast include certain diffusionlike operators, in any dimension, on manifolds, graphs, and in nonhomogeneous media. We use ideas related to the Fast Multipole Methods and to the wavelet analysis of CalderónZygmund and pseudodifferential operators, to numerically enforce the emergence of a natural hierarchical coarse graining of a manifold, graph or data set. For example for a body of text documents the construction leads to a directory structure at different levels of generalization. The dyadic powers of an operator can be used to induce a multiresolution analysis, as in classical LittlewoodPaley and wavelet theory: we construct, with efficient and stable algorithms, bases of orthonormal scaling functions and wavelets associated to this multiresolution analysis, together with the corresponding downsampling operators, and use them to compress the corresponding powers of the operator. While most of our discussion deals with symmetric operators and relates to localization to spectral bands, the symmetry of the operators and their spectral theory need not be considered, as the main assumption is reduction of the numerical ranks as we take powers of the operator.  
BibTeX:
@article{Coifman_R_2006_jacha_diff_w, author = {Coifman, R. R. and Maggioni, M.}, title = {Diffusion wavelets}, journal = {Appl. Comp. Harm. Analysis}, year = {2006}, volume = {21}, number = {1}, pages = {5394}, url = {http://www.sciencedirect.com/science/article/B6WB34K4PSX22/2/383131de381044772c31895fa7488ce3}, doi = {DOI: 10.1016/j.acha.2006.04.004} } 

Coifman, R.R. & Wickerhauser, M.V.  Entropybased algorithms for bestbasis selection [BibTeX] 
1992  IEEE Trans. Inform. Theory Vol. 38(2), pp. 713718 
article  
BibTeX:
@article{Coifman_R_1992_tit_ent_babbs, author = {Coifman, R. R. and Wickerhauser, M. V.}, title = {Entropybased algorithms for bestbasis selection}, journal = {IEEE Trans. Inform. Theory}, year = {1992}, volume = {38}, number = {2}, pages = {713718} } 

Combettes, P.L. & Pesquet, J.C.  Proximal splitting methods in signal processing  2010  Fixedpoint algorithms for inverse problems in science and engineering  incollection  
Abstract: The proximity operator of a convex function is a natural extension of the notion of a projection operator onto a convex set. This tool, which plays a central role in the analysis and the numerical solution of convex optimization problems, has recently been introduced in the arena of inverse problems and, especially, in signal processing, where it has become increasingly important. In this paper, we review the basic properties of proximity operators which are relevant to signal processing and present optimization methods based on these operators. These proximal splitting methods are shown to capture and extend several wellknown algorithms in a unify ing framework. Applications of proximal methods in signal recovery and synthesis are discussed.  
BibTeX:
@incollection{Combettes_P_2010_incoll_pro_smsp, author = {Combettes, P. L. and Pesquet, J.C.}, title = {Proximal splitting methods in signal processing}, booktitle = {Fixedpoint algorithms for inverse problems in science and engineering}, publisher = {Springer Verlag}, year = {2010} } 

Combettes, P.L. & Wajs, V.R.  Signal recovery by proximal forwardbackward splitting  2005  Multiscale Model. Simul. Vol. 4(4), pp. 11681200 
article  
Abstract: We show that various inverse problems in signal recovery can be formulated as the generic problem of minimizing the sum of two convex functions with certain regularity properties. This formulation makes it possible to derive existence, uniqueness, characterization, and stability results in a unified and standardized fashion for a large class of apparently disparate problems. Recent results on monotone operator splitting methods are applied to establish the convergence of a forwardbackward algorithm to solve the generic problem. In turn, we recover, extend, and provide a simplified analysis for a variety of existing iterative methods. Applications to geometry/texture image decomposition schemes are also discussed. A novelty of our framework is to use extensively the notion of a proximity operator, which was introduced by Moreau in the 1960s.  
BibTeX:
@article{Combettes_P_2005_siammms_sig_rpfbs, author = {Combettes, P. L. and Wajs, V. R.}, title = {Signal recovery by proximal forwardbackward splitting}, journal = {Multiscale Model. Simul.}, year = {2005}, volume = {4}, number = {4}, pages = {11681200} } 

Coulombe, S. & Dubois, E.  Multidimensional windows over arbitrary lattices and their application to FIR filter design  1996  Vol. 4Proc. Int. Conf. Acoust. Speech Signal Process., pp. 23832386 
inproceedings  DOI 
Abstract: This paper presents some applications to FIR filter design of multiD windows over arbitrary lattices and with arbitrary center of spatial symmetry. First, classic windows (such as Hamming, Blackman, etc.) are extended to windows over 1D and multiD lattices with arbitrary spatial symmetry centers (which multirate applications sometimes require). Then the problem of obtaining a target frequency response with a good transition band from an ideal frequency response (made of a passband having constant gain and a stopband for which the gain is zero) is studied. A method to obtain a described target response using multiD windows with application to the design of FIR filters is presented. Finally, a procedure for designing multiD FIR filters by windowing is explained  
BibTeX:
@inproceedings{Coulombe_S_1996_picassp_mul_walafirfd, author = {Coulombe, S. and Dubois, E.}, title = {Multidimensional windows over arbitrary lattices and their application to FIR filter design}, booktitle = {Proc. Int. Conf. Acoust. Speech Signal Process.}, year = {1996}, volume = {4}, pages = {23832386}, doi = {http://dx.doi.org/10.1109/ICASSP.1996.547762} } 

Cunha, A.L., Zhou, J. & Do, M.N.  The nonsubsampled contourlet transform: theory, design, and applications  2006  IEEE Trans. Image Process. Vol. 15(10), pp. 30893101 
article  DOI 
Abstract: In this paper, we develop the nonsubsampled contourlet transform (NSCT) and study its applications. The construction proposed in this paper is based on a nonsubsampled pyramid structure and nonsubsampled directional filter banks. The result is a flexible multiscale, multidirection, and shiftinvariant image decomposition that can be efficiently implemented via the à trous algorithm. At the core of the proposed scheme is the nonseparable twochannel nonsubsampled filter bank (NSFB). We exploit the less stringent design condition of the NSFB to design filters that lead to a NSCT with better frequency selectivity and regularity when compared to the contourlet transform. We propose a design framework based on the mapping approach, that allows for a fast implementation based on a lifting or ladder structure, and only uses onedimensional filtering in some cases. In addition, our design ensures that the corresponding frame elements are regular, symmetric, and the frame is close to a tight one. We assess the performance of the NSCT in image denoising and enhancement applications. In both applications the NSCT compares favorably to other existing methods in the literature.  
BibTeX:
@article{Cunha_A_2006_tip_non_cttda, author = {Cunha, A. L. and Zhou, J. and Do, M. N.}, title = {The nonsubsampled contourlet transform: theory, design, and applications}, journal = {IEEE Trans. Image Process.}, year = {2006}, volume = {15}, number = {10}, pages = {30893101}, doi = {http://dx.doi.org/10.1109/TIP.2006.877507} } 

Daragon, X., Couprie, M. & Bertrand, G.  Discrete Frontiers [BibTeX] 
2003  Vol. 2886Discrete geometry for computer imagery, pp. 236245 
incollection  
BibTeX:
@incollection{Daragon_X_2003_incoll_dis_f, author = {X. Daragon and M. Couprie and G. Bertrand}, title = {Discrete Frontiers}, booktitle = {Discrete geometry for computer imagery}, publisher = {Springer Verlag}, year = {2003}, volume = {2886}, pages = {236245} } 

Daubechies, I.  Ten Lectures on Wavelets [BibTeX] 
1992  book  
BibTeX:
@book{Daubechies_I_1992_book_ten_lw, author = {Daubechies, I.}, title = {Ten Lectures on Wavelets}, publisher = {CBMSNSF, SIAM Lecture Series}, year = {1992} } 

Daubechies, I., DeVore, R., Fornasier, M. & Güntürk, S.  Iteratively reweighted least squares minimization for sparse recovery [BibTeX] 
2010  Comm. Pure Applied Math. Vol. 63, pp. 138 
article  
BibTeX:
@article{Daubechies_I_2010_jcommpureapplmath_ite_rwlsmsr, author = {I. Daubechies and R. DeVore and Fornasier, M. and S. Güntürk}, title = {Iteratively reweighted least squares minimization for sparse recovery}, journal = {Comm. Pure Applied Math.}, year = {2010}, volume = {63}, pages = {138} } 

Daubechies, I., Han, B., Ron, A. & Shen, Z.  Framelets: MRAbased constructions of wavelet frames  2003  Appl. Comp. Harm. Analysis Vol. 14(1), pp. 146 
article  DOI URL 
Abstract: We discuss wavelet frames constructed via multiresolution analysis (MRA), with emphasis on tight wavelet frames. In particular, we establish general principles and specific algorithms for constructing framelets and tight framelets, and we show how they can be used for systematic constructions of spline, pseudospline tight frames, and symmetric biframes with short supports and high approximation orders. Several explicit examples are discussed. The connection of these frames with multiresolution analysis guarantees the existence of fast implementation algorithms, which we discuss briefly as well.  
BibTeX:
@article{Daubechies_I_2003_jacha_fra_mrabcwf, author = {Daubechies, I. and Han, B. and Ron, A. and Shen, Z.}, title = {Framelets: MRAbased constructions of wavelet frames}, journal = {Appl. Comp. Harm. Analysis}, year = {2003}, volume = {14}, number = {1}, pages = {146}, url = {http://www.sciencedirect.com/science/article/B6WB347G3R6F1/2/8f6f85efc54408c99a53768f71fb7c4b}, doi = {DOI: 10.1016/S10635203(02)005110} } 

Daubechies, I. & Sweldens, W.  Factoring Wavelet Transforms into Lifting Steps [BibTeX] 
1998  J. Fourier Anal. Appl. Vol. 4(3), pp. 245267 
article  
BibTeX:
@article{Daubechies_I_1998_jfouranalappl_fac_wtls, author = {I. Daubechies and W. Sweldens}, title = {Factoring Wavelet Transforms into Lifting Steps}, journal = {J. Fourier Anal. Appl.}, year = {1998}, volume = {4}, number = {3}, pages = {245267} } 

Daugman, J.  Twodimensional spectral analysis of cortical receptive field profile [BibTeX] 
1980  Vis. Res. Vol. 20, pp. 847856 
article  
BibTeX:
@article{Daugman_J_1980_jvisres_two_dsacrfp, author = {J. Daugman}, title = {Twodimensional spectral analysis of cortical receptive field profile}, journal = {Vis. Res.}, year = {1980}, volume = {20}, pages = {847856} } 

Daugman, J.G.  Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by twodimensional visual cortical filters [BibTeX] 
1985  J. Opt. Soc. Amer. A Vol. 2(7), pp. 11601169 
article  
BibTeX:
@article{Daugman_J_1985_joptsocama_unc_rrssfootvcf, author = {Daugman, J. G.}, title = {Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by twodimensional visual cortical filters}, journal = {J. Opt. Soc. Amer. A}, year = {1985}, volume = {2}, number = {7}, pages = {11601169} } 

Davis, G. & Nosratinia, A.  WaveletBased Image Coding: An Overview  1998  Vol. 1Applied and Computational Control, Signals, and Circuits, pp. 369434 
incollection  
Abstract: This paper presents an overview of waveletbased image coding. We develop the basics of image coding with a discussion of vector quantization. We motivate the use of transform coding in practical settings, and describe the properties of various decorrelating transforms. We motivate the use of the wavelet transform in coding using ratedistortion considerations as well as approximationtheoretic considerations. Finally, we give an overview of current coders in the literature.  
BibTeX:
@incollection{Davis_G_1998_incoll_wav_bico, author = {Davis, G. and Nosratinia, A.}, title = {WaveletBased Image Coding: An Overview}, booktitle = {Applied and Computational Control, Signals, and Circuits}, publisher = {Birkhäuser}, year = {1998}, volume = {1}, pages = {369434} } 

Deans, S.R.  The Radon transform and some of its applications [BibTeX] 
1983  book  
BibTeX:
@book{Deans_S_1983_book_rad_tsa, author = {Deans, S. R.}, title = {The Radon transform and some of its applications}, publisher = {John Wiley & Sons}, year = {1983} } 

Dekel, S. & Leviatan, D.  Adaptive Multivariate Approximation Using Binary Space Partitions and Geometric Wavelets [BibTeX] 
2005  SIAM J. Numer. Anal. Vol. 43(2), pp. 707732 
article  DOI 
BibTeX:
@article{Dekel_S_2005_jsiamjnumeranal_ada_mabspgw, author = {Dekel, S. and Leviatan, D.}, title = {Adaptive Multivariate Approximation Using Binary Space Partitions and Geometric Wavelets}, journal = {SIAM J. Numer. Anal.}, publisher = {Society for Industrial and Applied Mathematics}, year = {2005}, volume = {43}, number = {2}, pages = {707732}, doi = {http://dx.doi.org/10.1137/040604649} } 

Demanet, L. & Vandergheynst, P.  Gabor wavelets on the sphere  2003  Vol. 5207Proc. SPIE, Wavelets: Appl. Signal Image Process., pp. 208215 
inproceedings  
Abstract: We propose the construction of directional  or Gabor  continuous wavelets on the sphere. We provide a criterion to measure their angular selectivity. We finally discuss implementation issues and potential applications. The code for the spherical wavelet transform is available in the YAWTB Matlab Toolbox, //www.yawtb.be.tf  
BibTeX:
@inproceedings{Demanet_L_2003_pspiewasip_gab_ws, author = {Demanet, L. and Vandergheynst, P.}, title = {Gabor wavelets on the sphere}, booktitle = {Proc. SPIE, Wavelets: Appl. Signal Image Process.}, year = {2003}, volume = {5207}, pages = {208215} } 

Demanet, L. & Ying, L.  Wave atoms and sparsity of oscillatory patterns  2007  Appl. Comp. Harm. Analysis Vol. 23(3), pp. 368387 
article  DOI URL 
Abstract: We introduce #wave##atoms# as a variant of 2D wavelet packets obeying the parabolic scaling wavelength~(diameter)2. We prove that warped oscillatory functions, a toy model for texture, have a significantly sparser expansion in wave atoms than in other fixed standard representations like wavelets, Gabor atoms, or curvelets. We propose a novel algorithm for a tight frame of wave atoms with redundancy two, directly in the frequency plane, by the #wrapping# technique. We also propose variants of the basic transform for applications in image processing, including an orthonormal basis, and a shiftinvariant tight frame with redundancy four. Sparsity and denoising experiments on both seismic and fingerprint images demonstrate the potential of the tool introduced.  
BibTeX:
@article{Demanet_L_2007_jacha_wav_asop, author = {L. Demanet and L. Ying}, title = {Wave atoms and sparsity of oscillatory patterns}, journal = {Appl. Comp. Harm. Analysis}, year = {2007}, volume = {23}, number = {3}, pages = {368387}, url = {http://www.sciencedirect.com/science/article/B6WB34NG3TG42/2/f454d2df448c7f929cb1b55c1a9c4f8c}, doi = {DOI: 10.1016/j.acha.2007.03.003} } 

Demaret, L., Dyn, N. & Iske, A.  Image compression by linear splines over adaptive triangulations  2006  Signal Process. Vol. 86(7), pp. 16041616 
article  DOI URL 
Abstract: This paper proposes a new method for image compression. The method is based on the approximation of an image, regarded as a function, by a linear spline over an adapted triangulation, , which is the Delaunay triangulation of a small set Y of significant pixels. The linear spline minimizes the distance to the image, measured by the mean square error, among all linear splines over . The significant pixels in Y are selected by an adaptive thinning algorithm, which recursively removes less significant pixels in a greedy way, using a sophisticated criterion for measuring the significance of a pixel. The proposed compression method combines the approximation scheme with a customized scattered data coding scheme. We compare our compression method with JPEG2000 on two geometric images and on three popular test cases of real images.  
BibTeX:
@article{Demaret_L_2006_jsp_ima_clsat, author = {L. Demaret and N. Dyn and A. Iske}, title = {Image compression by linear splines over adaptive triangulations}, journal = {Signal Process.}, year = {2006}, volume = {86}, number = {7}, pages = {16041616}, url = {http://www.sciencedirect.com/science/article/B6V184H6XNBP1/2/9e038cbd48b235537548054efdbb1491}, doi = {DOI: 10.1016/j.sigpro.2005.09.003} } 

Deriche, R.  Recursively implementing the Gaussian and its derivative [BibTeX] 
1993  techreport  
BibTeX:
@techreport{Deriche_R_1993_tr_rec_igd, author = {Deriche, R.}, title = {Recursively implementing the Gaussian and its derivative}, year = {1993} } 

Distasi, R., Nappi, M. & Vitulano, S.  Image compression by Btree triangular coding  1997  IEEE Trans. Commun. Vol. 45(9), pp. 10951100 
article  DOI 
Abstract: This paper describes an algorithm for still image compression called Btree triangular coding (BTTC). The coding scheme is based on the recursive decomposition of the image domain into rightangled triangles arranged in a binary tree. The method is attractive because of its fast encoding, O(n log n), and decoding, Theta;(n), where n is the number of pixels, and because it is easy to implement and to parallelize. Experimental studies indicate that BTTC produces images of satisfactory quality from a subjective and objective point of view, One advantage of BTTC over JPEG is its shorter execution time  
BibTeX:
@article{Distasi_R_1997_jieeetcom_ima_cbttc, author = {Distasi, R. and Nappi, M. and Vitulano, S.}, title = {Image compression by Btree triangular coding}, journal = {IEEE Trans. Commun.}, year = {1997}, volume = {45}, number = {9}, pages = {10951100}, doi = {http://dx.doi.org/10.1109/26.623074} } 

Do, M.N. & Vetterli, M.  Contourlets [BibTeX] 
2003  Beyond Wavelets  incollection  
BibTeX:
@incollection{Do_M_2003_incoll_con, author = {Do, M. N. and Vetterli, M.}, title = {Contourlets}, booktitle = {Beyond Wavelets}, publisher = {Academic Press}, year = {2003} } 

Do, M.N. & Vetterli, M.  The contourlet transform: an efficient directional multiresolution image representation [BibTeX] 
2005  IEEE Trans. Image Process. Vol. 14(12), pp. 20912106 
article  
BibTeX:
@article{Do_M_2005_tip_con_tedmir, author = {Do, M. N. and Vetterli, M.}, title = {The contourlet transform: an efficient directional multiresolution image representation}, journal = {IEEE Trans. Image Process.}, year = {2005}, volume = {14}, number = {12}, pages = {20912106} } 

Do, M.N. & Vetterli, M.  Framing pyramids  2003  IEEE Trans. Signal Process. Vol. 51(9), pp. 23292342 
article  DOI 
Abstract: Burt and Adelson (1983) introduced the Laplacian pyramid (LP) as a multiresolution representation for images. We study the LP using the frame theory, and this reveals that the usual reconstruction is suboptimal. We show that the LP with orthogonal filters is a tight frame, and thus, the optimal linear reconstruction using the dual frame operator has a simple structure that is symmetric with the forward transform. In more general cases, we propose an efficient filterbank (FB) for the reconstruction of the LP using projection that leads to a proved improvement over the usual method in the presence of noise. Setting up the LP as an oversampled FB, we offer a complete parameterization of all synthesis FBs that provide perfect reconstruction for the LP. Finally, we consider the situation where the LP scheme is iterated and derive the continuousdomain frames associated with the LP.  
BibTeX:
@article{Do_M_2003_jieeetsp_fra_p, author = {Do, M. N. and Vetterli, M.}, title = {Framing pyramids}, journal = {IEEE Trans. Signal Process.}, year = {2003}, volume = {51}, number = {9}, pages = {23292342}, doi = {http://dx.doi.org/10.1109/TSP.2003.815389} } 

Do, M.N. & Vetterli, M.  The Finite Ridgelet Transform for Image Representation [BibTeX] 
2003  IEEE Trans. Image Process. Vol. 12(1), pp. 1628 
article  
BibTeX:
@article{Do_M_2003_tip_fin_rtir, author = {Do, M. N. and Vetterli, M.}, title = {The Finite Ridgelet Transform for Image Representation}, journal = {IEEE Trans. Image Process.}, year = {2003}, volume = {12}, number = {1}, pages = {1628} } 

Donoho, D.L.  Wedgelets: nearly minimax estimation of edges  1999  Ann. Stat. Vol. 27(3), pp. 859897 
article  
Abstract: We study a simple ?horizon model? for the problem of recovering an image from noisy data; in this model the image has an edge with $Hölder regularity. Adopting the viewpoint of computational harmonic analysis, we develop an overcomplete collection of atoms called wedgelets, dyadically organized indicator functions with a variety of locations, scales and orientations. The wedgelet representation provides nearly optimal representations of objects in the horizon model, as measured by minimax description length. We show how to rapidly compute a wedgelet approximation to noisy data by finding a special edgeletdecorated recursive partition which minimizes a complexitypenalized sum of squares. This estimate, using sufficient subpixel resolution, achieves nearly the minimax meansquared error in the horizon model. In fact, the method is adaptive in the sense that it achieves nearly the minimax risk for any value of the unknown degree of regularity of the horizon, $1 leq alpha leq 2$. Wedgelet analysis and denoising may be used successfully outside the horizon model. We study images modelled as indicators of starshaped sets with smooth boundaries and show that complexitypenalized wedgelet partitioning achieves nearly the minimax risk in that setting also.  
BibTeX:
@article{Donoho_D_1999_jannalsstatistics_wed_nmee, author = {Donoho, D. L.}, title = {Wedgelets: nearly minimax estimation of edges}, journal = {Ann. Stat.}, year = {1999}, volume = {27}, number = {3}, pages = {859897} } 

Donoho, D.L.  Tight frames of kplane ridgelets and the problem of representing objects that are smooth away from ddimensional singularities in $R^n$ [BibTeX] 
1999  Proc. Nat. Acad. Sci. U.S.A. Vol. 96(5), pp. 18281833 
article  URL 
BibTeX:
@article{Donoho_D_1999_jpnas_tig_fkprprosddsrn, author = {Donoho, D. L.}, title = {Tight frames of kplane ridgelets and the problem of representing objects that are smooth away from ddimensional singularities in $R^n$}, journal = {Proc. Nat. Acad. Sci. U.S.A.}, year = {1999}, volume = {96}, number = {5}, pages = {18281833}, url = {http://www.pnas.org/content/96/5/1828.abstract} } 

Donoho, D.L.  CART and bestorthobasis: A connection [BibTeX] 
1997  Ann. Stat. Vol. 25(5), pp. 18701911 
article  DOI URL 
BibTeX:
@article{Donoho_D_1997_jannalsstatisticscar_bobc, author = {D. L. Donoho}, title = {CART and bestorthobasis: A connection}, journal = {Ann. Stat.}, year = {1997}, volume = {25}, number = {5}, pages = {18701911}, url = {http://projecteuclid.org/euclid.aos/1069362377}, doi = {http://dx.doi.org/10.1214/aos/1069362377} } 

Donoho, D.L., Elad, M. & Temlyakov, V.N.  Stable recovery of sparse overcomplete representations in the presence of noise  2006  IEEE Trans. Inform. Theory Vol. 52(1), pp. 618 
article  DOI 
Abstract: Overcomplete representations are attracting interest in signal processing theory, particularly due to their potential to generate sparse representations of signals. However, in general, the problem of finding sparse representations must be unstable in the presence of noise. This paper establishes the possibility of stable recovery under a combination of sufficient sparsity and favorable structure of the overcomplete system. Considering an ideal underlying signal that has a sufficiently sparse representation, it is assumed that only a noisy version of it can be observed. Assuming further that the overcomplete system is incoherent, it is shown that the optimally sparse approximation to the noisy data differs from the optimally sparse decomposition of the ideal noiseless signal by at most a constant multiple of the noise level. As this optimalsparsity method requires heavy (combinatorial) computational effort, approximation algorithms are considered. It is shown that similar stability is also available using the basis and the matching pursuit algorithms. Furthermore, it is shown that these methods result in sparse approximation of the noisy data that contains only terms also appearing in the unique sparsest representation of the ideal noiseless sparse signal.  
BibTeX:
@article{Donoho_D_2006_jieeetit_sta_rsorpn, author = {D. L. Donoho and M. Elad and Temlyakov, V. N.}, title = {Stable recovery of sparse overcomplete representations in the presence of noise}, journal = {IEEE Trans. Inform. Theory}, year = {2006}, volume = {52}, number = {1}, pages = {618}, doi = {http://dx.doi.org/10.1109/TIT.2005.860430} } 

Donoho, D.L. & Flesia, A.G.  Digital ridgelet transform based on true ridge functions [BibTeX] 
2003  Vol. 10Beyond Wavelets, pp. 130 
incollection  DOI URL 
BibTeX:
@incollection{Donoho_D_2003_incoll_dig_rtbtrf, author = {Donoho, D. L. and Flesia, A. G.}, title = {Digital ridgelet transform based on true ridge functions}, booktitle = {Beyond Wavelets}, publisher = {Academic Press}, year = {2003}, volume = {10}, pages = {130}, url = {http://www.sciencedirect.com/science/article/B8GXW4NW0SWP2/2/c641f3555a4b704754addeae0cdf1dd0}, doi = {DOI: 10.1016/S1570579X(03)800290} } 

Driscoll, J.R. & Healy, D.M.  Computing Fourier Transforms and Convolutions on the 2Sphere  1994  Adv. Appl. Math. Vol. 15(2), pp. 202250 
article  DOI URL 
Abstract: This paper considers the problem of efficient computation of the spherical harmonic expansion, or Fourier transform, of functions defined on the two dimensional sphere, S2. The resulting algorithms are applied to the efficient computation of convolutions of functions on the sphere. We begin by proving convolution theorems generalizing well known and useful results from the abelian case. These convolution theorems are then used to develop a sampling theorem on the sphere. which reduces the calculation of Fourier transforms and convolutions of bandlimited functions to discrete computations. We show how to perform these efficiently, starting with an O(n(log n)2) time algorithm for computing the Legendre transform of a function defined on the interval [1,1] sampled at n points there. Theoretical and experimental results on the effects of finite precision arithmetic are presented. The Legendre transform algorithm is then generalized to obtain an algorithm for the Fourier transform, requiring O(n(log n)2) time, and an algorithm for its inverse in O(n1.5) time, where n is the number of points on the sphere at which the function is sampled. This improves the naive O(n2) bound, which is the best previously known. These transforms give an O(n1.5) algorithm for convolving two functions on the sphere.  
BibTeX:
@article{Driscoll_J_1994_jadvapplmath_com_ftc2s, author = {J. R. Driscoll and D. M. Healy}, title = {Computing Fourier Transforms and Convolutions on the 2Sphere}, journal = {Adv. Appl. Math.}, year = {1994}, volume = {15}, number = {2}, pages = {202250}, url = {http://www.sciencedirect.com/science/article/B6W9D45P0H0GB/2/44060d9d048f53ab56ccc47adf07d705}, doi = {DOI: 10.1006/aama.1994.1008} } 

Duffin, R. & Schaeffer, A.  A class of nonharmonic Fourier series [BibTeX] 
1952  Trans. Amer. Math. Soc. Vol. 72, pp. 341366 
article  
BibTeX:
@article{Duffin_R_1952_tams_cla_nhfs, author = {Duffin, R. and Schaeffer, A.}, title = {A class of nonharmonic Fourier series}, journal = {Trans. Amer. Math. Soc.}, year = {1952}, volume = {72}, pages = {341366} } 

Durand, S.  $M$band filtering and nonredundant directional wavelets [BibTeX] 
2007  Appl. Comp. Harm. Analysis Vol. 22, pp. 124139 
article  DOI 
BibTeX:
@article{Durand_S_2007_acha_m_bfndw, author = {Durand, S.}, title = {$M$band filtering and nonredundant directional wavelets}, journal = {Appl. Comp. Harm. Analysis}, year = {2007}, volume = {22}, pages = {124139}, doi = {http://dx.doi.org/10.1016/j.acha.2006.05.006} } 

Duval, L.  WITS: Where Is The Starlet? [BibTeX] 
misc  
BibTeX:
@misc{Duval_L_2005_url_wits, author = {Duval, L.}, title = {WITS: Where Is The Starlet?}, note = {http://www.laurentduval.eu/sivawitswhereisthestarlet.html} } 

Dyn, N., Gregory, J.A. & Levin, D.  A fourpoint interpolatory subdivision scheme for curve design [BibTeX] 
1987  Comput. Aided Geomet. Des. Vol. 4, pp. 257268 
article  
BibTeX:
@article{Dyn_N_1987_jcomputaidedgeometdesfou_pisccd, author = {N. Dyn and J. A. Gregory and D. Levin}, title = {A fourpoint interpolatory subdivision scheme for curve design}, journal = {Comput. Aided Geomet. Des.}, year = {1987}, volume = {4}, pages = {257268} } 

Efros, A.A. & Freeman, W.T.  Image Quilting for Texture Synthesis and Transfer  2001  Proc. SIGGRAPH Int. Conf. Comput. Graph. Interactive Tech., pp. 341346  inproceedings  
Abstract: We present a simple imagebased method of generating novel visual appearance in which a new image is synthesized by stitching together small patches of existing images. We call this process image quilting. First, we use quilting as a fast and very simple texture synthesis algorithm which produces surprisingly good results for a wide range of textures. Second, we extend the algorithm to perform texture transfer  rendering an object with a texture taken from a different object. More generally, we demonstrate how an image can be rerendered in the style of a different image. The method works directly on the images and does not require 3D information.  
BibTeX:
@inproceedings{Efros_A_2001_pacmsiggraph_ima_qtst, author = {A. A. Efros and W. T. Freeman}, title = {Image Quilting for Texture Synthesis and Transfer}, booktitle = {Proc. SIGGRAPH Int. Conf. Comput. Graph. Interactive Tech.}, year = {2001}, pages = {341346} } 

Egger, O., Li, W. & Kunt, M.  High compression image coding using an adaptive morphological subband decomposition  1995  Proc. IEEE Vol. 83(2), pp. 272287 
article  DOI 
Abstract: A morphological subband decomposition with perfect reconstruction is proposed. Critical subsampling is achieved. The reconstructed images using this decomposition do not suffer from any ringing effect. In order to avoid poor texture representation by the morphological filters an adaptive subband decomposition is introduced. It chooses linear filters on textured regions and morphological filters otherwise. A simple and efficient texture detection criterion is proposed and applied to the adaptive decomposition. Comparisons to other coding techniques such as JPEG and linear subband coding show that the proposed scheme performs significantly better both in terms of PSNR and visual quality  
BibTeX:
@article{Egger_1995_jprocieee_hig_cicamsd, author = {Egger, O. and Li, W. and Kunt, M.}, title = {High compression image coding using an adaptive morphological subband decomposition}, journal = {Proc. IEEE}, year = {1995}, volume = {83}, number = {2}, pages = {272287}, doi = {http://dx.doi.org/10.1109/5.364462} } 

Fadili, J.M. & Starck, J.L.  Curvelets and ridgelets [BibTeX] 
2009  Vol. 3Encyclopedia of Complexity and Systems Science, pp. 17181738 
incollection  
BibTeX:
@incollection{Fadili_J_2009_incoll_cur_r, author = {Fadili, J. M. and Starck, J.L.}, title = {Curvelets and ridgelets}, booktitle = {Encyclopedia of Complexity and Systems Science}, publisher = {Springer, New York}, year = {2009}, volume = {3}, pages = {17181738} } 

Faugère, J.C., Moreau de SaintMartin, F. & Rouillier, F.  Design of Regular Nonseparable Bidimensional Wavelets Using Gröbner Basis Techniques  1998  IEEE Trans. Signal Process. Vol. 46(4), pp. 845856 
article  DOI 
Abstract: The design of twodimensional (2D) filter banks yielding orthogonality and linearphase filters and generating regular wavelet bases is a difficult task involving the algebraic properties of multivariate polynomials. Using cascade forms implies dealing with nonlinear optimization. We turn the issue of optimizing the orthogonal linearphase cascade from Kovacevic and Vetterli (1992) into a polynomial problem and solve it using Grobner basis techniques and computer algebra. This leads to a complete description of maximally flat wavelets among the orthogonal linearphase family proposed by Kovacevic and Vetterli. We obtain up to five degrees of flatness for a $16times 16$ filter bank, whose Sobolev exponent is 2.11, making this wavelet the most regular orthogonal linearphase nonseparable wavelet to the authors' knowledge,  
BibTeX:
@article{Faugere_J_1998_tsp_des_rnbwgbt, author = {Faugère, J.C. and Moreau de SaintMartin, F. and Rouillier, F.}, title = {Design of Regular Nonseparable Bidimensional Wavelets Using Gröbner Basis Techniques}, journal = {IEEE Trans. Signal Process.}, year = {1998}, volume = {46}, number = {4}, pages = {845856}, doi = {http://dx.doi.org/10.1109/78.668541} } 

Feauveau, J.C.  Analyse multirésolution pour les images avec un facteur de résolution $2$  1990  Trait. Signal Vol. 7(2), pp. 117128 
article  
Abstract: Recently, an algorithm for multiresolution signal analysis, based on tvavelet theory las been proposed by S. Mallat. The basic idea is to capture signifïcant détails in the signal through the analysis ai successive scales . Mallat's algorithm uses a scaling factor of 2 between two successive scales . The aim of this paper is to develop a theory and an algorith n for image processing, with a scaling factor of $2)$ . This will allow for sinipler resulis interpretation and doubled sharpness of analysis . The theory is illustrated on both artificial and natural images, where our algorithm proves efficient for nonoriented contour detection . Récemment, un algorithme d'analyse multirésolution a été proposé par S. Mallat . Cet algorithme a pour but d'extraire les caractéristiques d'un signal en l'analysant à diverses échelles, un facteur de résolution 2 reliant deux échelles consécutives . Nous développons ici un cadre théorique et un algorithme dédié au traitement d'images, avec un facteur de résolution qui double la finesse d'analyse . Une autre caractéristique essentielle de cet algorithme est d'être non sélectif à l'orientation comme le montre les images sur lesquelles il a été testé .  
BibTeX:
@article{Feauveau_J_1990_ts_ana_mfrs2, author = {Feauveau, J. C.}, title = {Analyse multirésolution pour les images avec un facteur de résolution $2$}, journal = {Trait. Signal}, year = {1990}, volume = {7}, number = {2}, pages = {117128} } 

Felsberg, M.  LowLevel Image Processing with the Structure Multivector  2002  (Bericht Nr. 0203)  techreport  
Abstract: The present thesis deals with twodimensional signal processing for computer vision. The main topic is the development of a sophisticated generalization of the onedimensional analytic signal to two dimensions. Motivated by the fundamental property of the latter, the invariance ? equivariance constraint, and by its relation to complex analysis and potential theory, a twodimensional approach is derived. This method is called the monogenic signal and it is based on the Riesz transform instead of the Hilbert transform. By means of this linear approach it is possible to estimate the local orientation and the local phase of signals which are projections of onedimensional functions to two dimensions. For general twodimensional signals, however, the monogenic signal has to be further extended, yielding the structure multivector. The latter approach combines the ideas of the structure tensor and the quaternionic analytic signal. A rich feature set can be extracted from the structure multivector, which contains measures for local amplitudes, the local anisotropy, the local orientation, and two local phases. Both, the monogenic signal and the structure multivector are combined with an appropriate scalespace approach, resulting in generalized quadrature filters. Same as the monogenic signal, the applied scalespace approach is derived from the threedimensional Laplace equation instead of the diffusion equation. Hence, the twodimensional generalization of the analytic signal turns out to provide a whole new framework for lowlevel vision. Several applications are presented to show the efficiency and power of the theoretic considerations. Among these are methods for orientation estimation, edge and corner detection, stereo correspondence and disparity estimation, and adaptive smoothing.  
BibTeX:
@techreport{Felsberg_M_2002_tr_low_lipsm, author = {Felsberg, M.}, title = {LowLevel Image Processing with the Structure Multivector}, year = {2002}, number = {Bericht Nr. 0203} } 

Fernandes, F.C.A., van Spaendonck, R.L.C. & Burrus, C.S.  Multidimensional, MappingBased Complex Wavelet Transforms  2005  IEEE Trans. Image Process. Vol. 14(1), pp. 110124 
article  
Abstract: Although the discrete wavelet transform (DWT) is a powerful tool for signal and image processing, it has three serious disadvantages: shift sensitivity, poor directionality, and lack of phase information. To overcome these disadvantages, we introduce multidimensional, mappingbased, complex wavelet transforms that consist of a mapping onto a complex function space followed by a DWT of the complex mapping. Unlike other popular transforms that also mitigate DWT shortcomings, the decoupled implementation of our transforms has two important advantages. First, the controllable redundancy of the mapping stage offers a balance between degree of shift sensitivity and transform redundancy. This allows us to create a directional, nonredundant, complex wavelet transform with potential benefits for image coding systems. To the best of our knowledge, no other complex wavelet transform is simultaneously directional and nonredundant. The second advantage of our approach is the flexibility to use any DWT in the transform implementation. As an example, we exploit this flexibility to create the complex doubledensity DWT: a shiftinsensitive, directional, complex wavelet transform with a low redundancy of (3 1) (2 1) in dimensions. No other transform achieves all these properties at a lower redundancy, to the best of our knowledge. By exploiting the advantages of our multidimensional, mappingbased complex wavelet transforms in seismic signalprocessing applications, we have demonstrated stateoftheart results.  
BibTeX:
@article{Fernandes_F_2005_tip_mul_mbcwt, author = {Fernandes, F. C. A. and van Spaendonck, R. L. C. and Burrus, C. S.}, title = {Multidimensional, MappingBased Complex Wavelet Transforms}, journal = {IEEE Trans. Image Process.}, year = {2005}, volume = {14}, number = {1}, pages = {110124} } 

Fernandes, F.C.A., van Spaendonck, R.L.C. & Burrus, C.S.  A new framework for complex wavelet transforms  2003  IEEE Trans. Signal Process. Vol. 51(7), pp. 18251837 
article  DOI 
Abstract: Although the discrete wavelet transform (DWT) is a powerful tool for signal and image processing, it has three serious disadvantages: shift sensitivity, poor directionality, and lack of phase information. To overcome these disadvantages, we introduce twostage mappingbased complex wavelet transforms that consist of a mapping onto a complex function space followed by a DWT of the complex mapping. Unlike other popular transforms that also mitigate DWT shortcomings, the decoupled implementation of our transforms has two important advantages. First, the controllable redundancy of the mapping stage offers a balance between degree of shift sensitivity and transform redundancy. This allows us to create a directional, nonredundant, complex wavelet transform with potential benefits for image coding systems. To the best of our knowledge, no other complex wavelet transform is simultaneously directional and nonredundant. The second advantage of our approach is the flexibility to use any DWT in the transform implementation. As an example, we can exploit this flexibility to create the complex doubledensity DWT (CDDWT): a shiftinsensitive, directional, complex wavelet transform with a low redundancy of (3/sup m/1/2/sup m/1) in m dimensions. To the best of our knowledge, no other transform achieves all these properties at a lower redundancy.  
BibTeX:
@article{Fernandes_F_2003_tsp_new_fcwt, author = {Fernandes, F. C. A. and van Spaendonck, R. L. C. and Burrus, C. S.}, title = {A new framework for complex wavelet transforms}, journal = {IEEE Trans. Signal Process.}, year = {2003}, volume = {51}, number = {7}, pages = {18251837}, doi = {http://dx.doi.org/10.1109/TSP.2003.812841} } 

Fernandes, F.C.A., van Spaendonck, R.L.C. & Burrus, C.S.  A directional, shift insensitive, lowredundancy, wavelet transform  2001  Vol. 1Proc. Int. Conf. Image Process., pp. 618621 
inproceedings  DOI 
Abstract: Shift sensitivity and poor directionality, two major disadvantages of the discrete wavelet transform, have previously been circumvented either by using highly redundant, nonseparable wavelet transforms or by using restrictive designs to obtain a pair of wavelet trees with a transformdomain redundancy of 4.0 in 2D. We demonstrate that excellent shiftinvariance properties and directional selectivity may be obtained with a transformdomain redundancy of only 2.67 in 2D. We achieve this by projecting the wavelet coefficients from Selesnick's (see Wavelet Applications VII, Proceedings of SPIE, 2000) shift insensitive, doubledensity wavelet transform so as to separate approximately the positive and negative frequencies, thereby increasing directionality. Subsequent decimation and a novel inverse projection maintain the low redundancy while ensuring perfect reconstruction. Although our transform generates complexvalued coefficients that provide valuable phase information, it may be implemented with a fast algorithm that uses only real arithmetic. To demonstrate the efficacy of our new transform, we show that it achieves stateoftheart performance in a seismic imageprocessing application  
BibTeX:
@inproceedings{Fernandes_F_2001_picip_dir_silrwt, author = {Fernandes, F. C. A. and van Spaendonck, R. L. C. and Burrus, C. S.}, title = {A directional, shift insensitive, lowredundancy, wavelet transform}, booktitle = {Proc. Int. Conf. Image Process.}, year = {2001}, volume = {1}, pages = {618621}, doi = {http://dx.doi.org/10.1109/ICIP.2001.959121} } 

Fernandes, F.C.A., Wakin, M. & Baraniuk, R.  NonRedundant, LinearPhase, SemiOrthogonal, Directional Complex Wavelets [BibTeX] 
2004  Proc. Int. Conf. Acoust. Speech Signal Process.  inproceedings  
BibTeX:
@inproceedings{Fernandes_F_2004_icassp_non_rlpsodcw, author = {Fernandes, F. C. A. and Wakin, M. and Baraniuk, R.}, title = {NonRedundant, LinearPhase, SemiOrthogonal, Directional Complex Wavelets}, booktitle = {Proc. Int. Conf. Acoust. Speech Signal Process.}, year = {2004} } 

Figueras i Ventura, R., Vandergheynst, P. & Frossard, P.  Low rate and flexible image coding with redundant representations [BibTeX] 
2006  IEEE Trans. Image Process. Vol. 15(3), pp. 726739 
article  
BibTeX:
@article{FiguerasIVentura_R_2006_tip_low_rficrr, author = {Figueras i Ventura, R. and Vandergheynst, P. and Frossard, P.}, title = {Low rate and flexible image coding with redundant representations}, journal = {IEEE Trans. Image Process.}, year = {2006}, volume = {15}, number = {3}, pages = {726739} } 

Florack, L. & Kuijper, A.  The Topological Structure of ScaleSpace Images  1998  techreport  
Abstract: We investigate the “deep structure” of a scalespace image. The emphasis is on topology, i.e. we concentrate on critical points—points with vanishing gradient  and toppoints  critical points with degenerate Hessian  and monitor their displacements, respectively generic morsifications in scalespace. Relevant parts of catastrophe theory in the context of the scalespace paradigm are briefly reviewed, and subsequently rewritten into coordinate independent form. This enables one to implement topological descriptors using a conveniently defined coordinate system.  
BibTeX:
@techreport{Florack_L_1998_tr_top_sssi, author = {Florack, L. and Kuijper, A.}, title = {The Topological Structure of ScaleSpace Images}, year = {1998} } 

Forster, B., Blu, T., Van De Ville, D. & Unser, M.  Shiftinvariant spaces from rotationcovariant functions [BibTeX] 
2008  Appl. Comp. Harm. Analysis Vol. 25(2), pp. 240265 
article  DOI URL 
BibTeX:
@article{Forster_B_2008_jacha_shi_isrcf, author = {Forster, B. and Blu, T. and Van De Ville, D. and Unser, M.}, title = {Shiftinvariant spaces from rotationcovariant functions}, journal = {Appl. Comp. Harm. Analysis}, publisher = {Elsevier}, year = {2008}, volume = {25}, number = {2}, pages = {240265}, url = {http://bigwww.epfl.ch/publications/forster0801.ps, http://bigwww.epfl.ch/publications/forster0801.html}, doi = {http://dx.doi.org/10.1016/j.acha.2007.11.002} } 

Freeden, W., Maier, T. & Zimmermann, S.  A Survey on Wavelet Methods for (geo)applications  2003  Rev. Matemática Complutense Vol. 16(1), pp. 277310 
article  
Abstract: Wavelets originated in 1980's for the analysis of (seismic) signals and have seen an explosion of applications. However, almost all the material is based on wavelets over Euclidean spaces. This paper deals with an approach to the theory and algorithmic aspects of wavelets in a general separable Hilbert space framework. As examples Legendre wavelets on the interval $[1,1]$ and scalar and vector spherical wavelets on the unit sphere are discussed in more detail.  
BibTeX:
@article{Freeden_W_2003_jrevmatcomplutense_sur_wmga, author = {W. Freeden and T. Maier and S. Zimmermann}, title = {A Survey on Wavelet Methods for (geo)applications}, journal = {Rev. Matemática Complutense}, year = {2003}, volume = {16}, number = {1}, pages = {277310} } 

Freeden, W. & Windheuser, U.  Spherical Wavelet Transform and its Discretization  1996  Adv. Appl. Math. Vol. 5(1), pp. 5194 
article  
Abstract: A continuous version of spherical multiresolution is described, starting from continuous wavelet transform on the sphere. Scale discretization enables us to construct spherical counterparts to wavelet packets and scale discrete wavelets. The essential tool is the theory of singular integrals on the sphere. It is shown that singular integral operators forming a semigroup of contraction operators of class (C0) (like AbelPoisson or Gau Weierstra operators) lead in a canonical way to (pyramidal) algorithms.  
BibTeX:
@article{Freeden_W_1996_jadvapplmath_sph_wtd, author = {W. Freeden and U. Windheuser}, title = {Spherical Wavelet Transform and its Discretization}, journal = {Adv. Appl. Math.}, year = {1996}, volume = {5}, number = {1}, pages = {5194} } 

Freeman, W.T.  Steerable Filters and Local Analysis of Image Structure  1992  School: Massachusetts Institute of Technology  phdthesis  
Abstract: Two paradigms for visual analysis are topdown, starting from highlevel models or information about the image, and bottomup, where little is assumed about the image or objects in it. We explore a local, bottomup approach to image analysis. We develop operators to identify and classify image junctions, which contain important visual cues for identifying occlusion, transparency, and surface bends. Like the human visual system, we begin with the application of linear filters which are oriented in all possible directions. We develop an efficient way to create an oriented filter of arbitrary orientation by describing it as a linear combination of basis filters. This approach to oriented filtering, which we call steerable filters, offers advantages for analysis as well as computation. We design a variety of steerable filters, including steerable quadrature pairs, which measure local energy. We show applications of these filters in orientation and texture analysis, and image representation and enhancement. We develop methods based on steerable filters to study structures such as contours and junctions. We describe how to postfilter the energy measures in order to more efficiently analyze structures with multiple orientations. We introduce a new detector for contours, based on energy local maxima. We analyze contour phases at energy local maxima, and compare the results with the prediction of a simple model. Using these tools, we analyze junctions. Based on local oriented filters, we develop simple mechanisms which respond selectively to ``T'', ``L'', and ``X'' junctions. The T and X junctions may indicate occlusion and transparency, respectively. These mechanism show that detectors for important, lowlevel visual cues can be built out of oriented filters and energy measures, which resemble responses found in the visual cortex. We present a second approach to junction detection based on salient contours. We combine our contour detector with the structural saliency algorithm of Shashua and Ullman, which finds visually salient contours. To improve its descriptive power, we include a competitive mechanism in the algorithm. From the local configuration of saliencies, we form simple detectors which respond to cues for occlusion, transparency and surface bending. Using the saliency values and curve linking information, we can propagate this information along image contours. For both algorithms, we show successful results on simple synthetic and natural images. We show results for more complicated scenes and discuss the methods do not work, and why. Each algorithm uses only local calculations applied in parallel throughout the image, and assumes little prior information about the objects it expects to see.  
BibTeX:
@phdthesis{Freeman_W_1992_phd_ste_flais, author = {Freeman, W. T.}, title = {Steerable Filters and Local Analysis of Image Structure}, school = {Massachusetts Institute of Technology}, year = {1992} } 

Freeman, W.T. & Adelson, E.H.  The design and use of steerable filters  1991  IEEE Trans. Patt. Anal. Mach. Int. Vol. 13(9), pp. 891906 
article  
Abstract: Oriented filters are useful in many early vision and image processing tasks. One often needs to apply the same filter, rotated to different angles under adaptive control, or wishes to calculate the filter response at various orientations. We present an efficient architecture to synthesize filters of arbitrary orientations from linear combinations of basis filters, allowing one to adaptively ``steer'' a filter to any orientation, and to determine analytically the filter output as a function of orientation. Steerable filters may be designed in quadrature pairs to allow adaptive control over phase as well as orientation. We show how to design and steer the filters, and present examples of their use in several tasks: the analysis of orientation and phase, angularly adaptive filtering, edge detection, and shapefromshading. One can also build a selfsimilar steerable pyramid representation which may be used to implement a steerable ``wavelet'' decomposition. The same concepts can be generalized to the design of 3D steerable filters, which should be useful in the analysis of image sequences and volumetric data.  
BibTeX:
@article{Freeman_W_1991_tpami_des_usf, author = {Freeman, W. T. and Adelson, E. H.}, title = {The design and use of steerable filters}, journal = {IEEE Trans. Patt. Anal. Mach. Int.}, year = {1991}, volume = {13}, number = {9}, pages = {891906} } 

Freeman, W.T. & Adelson, E.H.  Steerable Filters for Early Vision, Image Analysis and Wavelet Decomposition [BibTeX] 
1990  Proc. IEEE Int. Conf. Comput. Vis., pp. 406415  inproceedings  
BibTeX:
@inproceedings{Freeman_W_1990_piccv_ste_feviawd, author = {Freeman, W. T. and Adelson, E. H.}, title = {Steerable Filters for Early Vision, Image Analysis and Wavelet Decomposition}, booktitle = {Proc. IEEE Int. Conf. Comput. Vis.}, year = {1990}, pages = {406415} } 

Friedrich, F., Demaret, L., Fuhr, H. & Wicker, K.  Efficient Moment Computation over Polygonal Domains with an Application to Rapid Wedgelet Approximation [BibTeX] 
2007  SIAM J. Sci. Comput. Vol. 29(2), pp. 842863 
article  
BibTeX:
@article{Friedrich_F_2007_jsiamscicomp_eff_mcpdarwa, author = {F. Friedrich and L. Demaret and H. Fuhr and K. Wicker}, title = {Efficient Moment Computation over Polygonal Domains with an Application to Rapid Wedgelet Approximation}, journal = {SIAM J. Sci. Comput.}, year = {2007}, volume = {29}, number = {2}, pages = {842863} } 

Führ, H., Demaret, L. & Friedrich, F.  Beyond wavelets: New image representation paradigms [BibTeX] 
2006  Document and Image Compression  incollection  
BibTeX:
@incollection{Fuhr_H_2006_incoll_bey_wnirp, author = {H. Führ and L. Demaret and F. Friedrich}, title = {Beyond wavelets: New image representation paradigms}, booktitle = {Document and Image Compression}, publisher = {CRC Press}, year = {2006} } 

Gabor, D.  Theory of Communication  1946  J. IEE Vol. 93(26), pp. 429457 
article  
Abstract: The purpose of these studies is an inquiry into the essence of the "information" conveyed by channels of communication, and the application of the result of this inquiry to the practical problem of optimum utilization of frequency bands. In Part 1, a new method of analysing signals is presented in which time and frequency play symmetrical parts, and which contains "time analysis" and "frequency analysis" as special cases. It is shown that the information conveyed by a frequency band in a given timeinterval can be analysed in various ways into the same number of elementary "quanta of information," each quantum conveying one numerical datum. In Part 2, this method is applied to the analysis of hearing sensations. It is shown on the basis of existing experimental material that in the band between 60 an 1000 c/s the human ear can discriminate very nearly every second datum of information, and that this efficiency of nearly 50 percent is independent of the duration of the signals in a remarkably wide interval. This fact, which cannot be explained by any mechanism in the inner ear, suggests a new phenomenon in nerve conduction. At frequencies above 1000 c/s the efficiency of discrimination falls off sharply, proving that sound reproductions which are far from faithful may be perceived by the ear as perfect, and that "condensed" methods of transmission and reproduction with improved waveband economy are possible in principle. In Part 3, suggestions are discussed for compresse transmission and reproduction of speech or music, and the first experimental results obtained with one of these methods are described.  
BibTeX:
@article{Gabor_D_1946_jiee_the_c, author = {Gabor, D.}, title = {Theory of Communication}, journal = {J. IEE}, year = {1946}, volume = {93}, number = {26}, pages = {429457}, note = {Part. III} } 

Gagnon, L., Lina, J.M. & Goulard, B.  Sharpening Enhancement of Digitized Mammograms with Complex Symmetric Daubechies Wavelets  1995  Proc. EMBS  inproceedings  
Abstract: Some complex symmetric Daubechies wavelets provide a natural way to calculate zero crossings because of a hidden "Laplacian operator" in the imaginary part of the scaling function. We propose a simple multiscale sharpening enhancement algorithm based on this property. The algorithm is tested on lowcontrast digitized mammograms.  
BibTeX:
@inproceedings{Gagnon_L_1995_pembs_sha_edmcsdw, author = {Gagnon, L. and Lina, J.M. and Goulard, B.}, title = {Sharpening Enhancement of Digitized Mammograms with Complex Symmetric Daubechies Wavelets}, booktitle = {Proc. EMBS}, year = {1995} } 

Gauthier, J., Duval, L. & Pesquet, J.C.  Optimization of Synthesis Oversampled Complex Filter Banks  2009  IEEE Trans. Signal Process. Vol. 57(10), pp. 38273843 
article  DOI 
Abstract: An important issue with oversampled FIR analysis filter banks (FBs) is to determine inverse synthesis FBs, when they exist. Given any complex oversampled FIR analysis FB, we first provide an algorithm to determine whether there exists an inverse FIR synthesis system. We also provide a method to ensure the Hermitian symmetry property on the synthesis side, which is serviceable to processing realvalued signals. As an invertible analysis scheme corresponds to a redundant decomposition, there is no unique inverse FB. Given a particular solution, we parameterize the whole family of inverses through a null space projection. The resulting reduced parameter set simplifies design procedures, since the perfect reconstruction constrained optimization problem is recast as an unconstrained optimization problem. The design of optimized synthesis FBs based on time or frequency localization criteria is then investigated, using a simple yet efficient gradient algorithm.  
BibTeX:
@article{Gauthier_J_2009_tsp_opt_socfb, author = {Gauthier, J. and Duval, L. and Pesquet, J.C.}, title = {Optimization of Synthesis Oversampled Complex Filter Banks}, journal = {IEEE Trans. Signal Process.}, year = {2009}, volume = {57}, number = {10}, pages = {38273843}, doi = {http://dx.doi.org/10.1109/TSP.2009.2023947} } 

Gerek, O.N. & Cetin, A.E.  Adaptive Polyphase Subband Decomposition Structures for Image Compression  2000  IEEE Trans. Image Process. Vol. 9(10), pp. 16491660 
article  DOI 
Abstract: Subband decomposition techniques have been extensively used for data coding and analysis. In most filter banks, the goal is to obtain subsampled signals corresponding to different spectral regions of the original data. However, this approach leads to various artifacts in images having spatially varying characteristics, such as images containing text, subtitles, or sharp edges. In this paper, adaptive filter banks with perfect reconstruction property are presented for such images. The filters of the decomposition structure which can be either linear or nonlinear vary according to the nature of the signal. This leads to improved image compression ratios. Simulation examples are presented  
BibTeX:
@article{Gerek_O_2000_jieeetip_ada_psdsic, author = {O. N. Gerek and A. E. Cetin}, title = {Adaptive Polyphase Subband Decomposition Structures for Image Compression}, journal = {IEEE Trans. Image Process.}, year = {2000}, volume = {9}, number = {10}, pages = {16491660}, doi = {http://dx.doi.org/10.1109/83.869176} } 

Golomb, S.  Polyominoes [BibTeX] 
1994  book  
BibTeX:
@book{Golomb_S_1994_book_pol, author = {Golomb, S.}, title = {Polyominoes}, publisher = {Princeton University Press}, year = {1994}, edition = {2nd} } 

Gopinath, R.A.  Phaselets of Framelets [BibTeX] 
2005  IEEE Trans. Signal Process. Vol. 53(5), pp. 17941806 
article  
BibTeX:
@article{Gopinath_R_2005_tsp_pha_f, author = {Gopinath, R. A.}, title = {Phaselets of Framelets}, journal = {IEEE Trans. Signal Process.}, year = {2005}, volume = {53}, number = {5}, pages = {17941806} } 

Gopinath, R.A.  The phaselet transform  an integral redundancy nearly shiftinvariant wavelet transform [BibTeX] 
2003  IEEE Trans. Signal Process. Vol. 51(7), pp. 17921805 
article  
BibTeX:
@article{Gopinath_R_2003_tsp_pha_tirnsiwt, author = {Gopinath, R. A.}, title = {The phaselet transform  an integral redundancy nearly shiftinvariant wavelet transform}, journal = {IEEE Trans. Signal Process.}, year = {2003}, volume = {51}, number = {7}, pages = {17921805} } 

Goutsias, J. & Heijmans, H.J.A.M.  Nonlinear multiresolution signal decomposition schemes. I. Morphological pyramids  2000  IEEE Trans. Image Process. Vol. 9(11), pp. 18621876 
article  DOI 
Abstract: Interest in multiresolution techniques for signal processing and analysis is increasing steadily. An important instance of such a technique is the socalled pyramid decomposition scheme. This paper presents a general theory for constructing linear as well as nonlinear pyramid decomposition schemes for signal analysis and synthesis. The proposed theory is based on the following ingredients: 1) the pyramid consists of a (finite or infinite) number of levels such that the information content decreases toward higher levels and 2) each step toward a higher level is implemented by an (informationreducing) analysis operator, whereas each step toward a lower level is implemented by an (informationpreserving) synthesis operator. One basic assumption is necessary: synthesis followed by analysis yields the identity operator, meaning that no information is lost by these two consecutive steps. Several examples of pyramid decomposition schemes are shown to be instances of the proposed theory: a particular class of linear pyramids, morphological skeleton decompositions, the morphological Haar pyramid, median pyramids, etc. Furthermore, the paper makes a distinction between singlescale and multiscale decomposition schemes, i.e., schemes without or with sample reduction. Finally, the proposed theory provides the foundation of a general approach to constructing nonlinear wavelet decomposition schemes and filter banks  
BibTeX:
@article{Goutsias_J_2000_jieeetip_non_msds1mp, author = {Goutsias, J. and Heijmans, H. J. A. M.}, title = {Nonlinear multiresolution signal decomposition schemes. I. Morphological pyramids}, journal = {IEEE Trans. Image Process.}, year = {2000}, volume = {9}, number = {11}, pages = {18621876}, doi = {http://dx.doi.org/10.1109/83.877209} } 

Gouze, A., Antonini, M., Barlaud, M. & Macq, B.  Design of signaladapted multidimensional lifting scheme for lossy coding  2004  IEEE Trans. Image Process. Vol. 13(12), pp. 15891603 
article  DOI 
Abstract: This paper proposes a new method for the design of lifting filters to compute a multidimensional nonseparable wavelet transform. Our approach is stated in the general case, and is illustrated for the 2D separable and for the quincunx images. Results are shown for the JPEG2000 database and for satellite images acquired on a quincunx sampling grid. The design of efficient quincunx filters is a difficult challenge which has already been addressed for specific cases. Our approach enables the design of less expensive filters adapted to the signal statistics to enhance the compression efficiency in a more general case. It is based on a twostep lifting scheme and joins the lifting theory with Wiener's optimization. The prediction step is designed in order to minimize the variance of the signal, and the update step is designed in order to minimize a reconstruction error. Application for lossy compression shows the performances of the method.  
BibTeX:
@article{Gouze_A_2004_jieeetip_des_samlslc, author = {Gouze, A. and Antonini, M. and Barlaud, M. and Macq, B.}, title = {Design of signaladapted multidimensional lifting scheme for lossy coding}, journal = {IEEE Trans. Image Process.}, year = {2004}, volume = {13}, number = {12}, pages = {15891603}, doi = {http://dx.doi.org/10.1109/TIP.2004.837556} } 

Grossman, A. & Morlet, J.  Decompositions of Functions into Wavelets of Constant Shape, and Related Transforms [BibTeX] 
1984  , pp. 135165  misc  
BibTeX:
@misc{Grossmann_A_1984_misc_dec_fwcsrt, author = {A. Grossman and J. Morlet}, title = {Decompositions of Functions into Wavelets of Constant Shape, and Related Transforms}, year = {1984}, pages = {135165}, note = {"Mathematics and Physics, Lectures on recent results", L. Streit, ed., World Scientific Publishing Co., Singapore} } 

Guilloux, F., Faÿ, G. & Cardoso, J.F.  Practical wavelet design on the sphere  2009  Appl. Comp. Harm. Analysis Vol. 26(2), pp. 143160 
article  DOI URL 
Abstract: This paper considers the design of isotropic analysis functions on the sphere which are perfectly limited in the spectral domain and optimally localized in the spatial domain. This is motivated by the need of localized analysis tools in domains where the data is lying on the sphere, e.g. the science of the Cosmic Microwave Background. Our construction is derived from the localized frames introduced by F. Narcowich et al. [F. Narcowich, P. Petrushev, J. Ward, Localized tight frames on spheres, SIAM J. Math. Anal. 38 (2) (2006) 574594]. The analysis frames are optimized for given applications and compared numerically using various criteria.  
BibTeX:
@article{Guilloux_F_2009_jacha_pra_wds, author = {F. Guilloux and G. Faÿ and J.F. Cardoso}, title = {Practical wavelet design on the sphere}, journal = {Appl. Comp. Harm. Analysis}, year = {2009}, volume = {26}, number = {2}, pages = {143160}, url = {http://www.sciencedirect.com/science/article/B6WB34S62RSR1/2/4efc6cf368a88710d74e68d414783415}, doi = {DOI: 10.1016/j.acha.2008.03.003} } 

Guo, K. & Labate, D.  Optimally Sparse Multidimensional Representation using Shearlets [BibTeX] 
2007  SIAM J. Math. Anal. Vol. 39, pp. 298318 
article  
BibTeX:
@article{Guo_K_2007_jsiammathanal_opt_smrs, author = {K. Guo and D. Labate}, title = {Optimally Sparse Multidimensional Representation using Shearlets}, journal = {SIAM J. Math. Anal.}, year = {2007}, volume = {39}, pages = {298318} } 

Haar, A.  Zur Theory der orthogalen Funktionen Systeme [BibTeX] 
1910  Math. Annalen Vol. 69, pp. 331371 
article  
BibTeX:
@article{Haar_A_1910_ma_zur_tofs, author = {Haar, A.}, title = {Zur Theory der orthogalen Funktionen Systeme}, journal = {Math. Annalen}, year = {1910}, volume = {69}, pages = {331371} } 

Hahn, S.L.  Multidimensional complex signals with singleorthant spectra  1992  Proc. IEEE Vol. 80(8), pp. 12871300 
article  DOI 
Abstract: An extension of the notion of the analytical signal to multidimensional signals is presented. The real and imaginary parts of this signal are a linear combination of the original signal and of its complete and partial Hilbert transforms. Its Fourier image exists only in one orthant of the multidimensional frequency space. The orthant is a halfaxis in one dimension, a quadrant in two dimensions, an octant in three dimensions, etc. A multidimensional complex signal makes it possible to introduce the definitions of instantaneous amplitude, instantaneous phase, and partial instantaneous complex frequencies (partial derivatives of the instantaneous phase) and to formulate a modulation theory of a multidimensional harmonic carrier. The 2D equivalent of 1D singlesideband modulation is defined and called single quadrant modulation. It is shown that the multidimensional complex signal with a signal orthant spectrum may be defined as a convolution of the real signal with the multidimensional complex delta distribution  
BibTeX:
@article{Hahn_S_1992_procieee_mul_cssos, author = {Hahn, S. L.}, title = {Multidimensional complex signals with singleorthant spectra}, journal = {Proc. IEEE}, year = {1992}, volume = {80}, number = {8}, pages = {12871300}, doi = {http://dx.doi.org/10.1109/5.158601} } 

Hammond, D.K., Vandergheynst, P. & Gribonval, R.  Wavelets on graphs via spectral graph theory  2011  Appl. Comp. Harm. Analysis Vol. 30(2), pp. 129150 
article  DOI 
Abstract: We propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. Our approach is based on defining scaling using the graph analogue of the Fourier domain, namely the spectral decomposition of the discrete graph Laplacian L. Given a wavelet generating kernel g and a scale parameter t, we define the scaled wavelet operator Ttg = g(tL). The spectral graph wavelets are then formed by localizing this operator by applying it to an indicator function. Subject to an admissibility condition on g, this procedure defines an invertible transform. We explore the localization properties of the wavelets in the limit of fine scales. Additionally, we present a fast Chebyshev polynomial approximation algorithm for computing the transform that avoids the need for diagonalizing L. We highlight potential applications of the transform through examples of wavelets on graphs corresponding to a variety of different problem domains.  
BibTeX:
@article{Hammond_2011_jacha_wav_gsgt, author = {D. K. Hammond and P. Vandergheynst and R. Gribonval}, title = {Wavelets on graphs via spectral graph theory}, journal = {Appl. Comp. Harm. Analysis}, year = {2011}, volume = {30}, number = {2}, pages = {129150}, doi = {http://dx.doi.org/10.1016/j.acha.2010.04.005} } 

Hampson, F.J. & Pesquet, J.C.  $M$band Nonlinear Subband Decompositions with Perfect Reconstruction  1998  IEEE Trans. Image Process. Vol. 7(11), pp. 15471560 
article  DOI 
Abstract: We investigate nonlinear multirate filterbanks with maximal decimation and perfect reconstruction. Definitions of the desired properties of such structures are given for general nonlinear filterbanks. We then consider a triangular representation of linear filterbanks and see that it may be easily extended to the nonlinear case. Furthermore, general nonlinear filterbanks are presented, for which perfect reconstruction is either inherently guaranteed or ensured subject to an easily verified condition. Extensions to bidimensional filters are also discussed and an application for nonlinear multiresolution schemes to feature sieves is shown  
BibTeX:
@article{Hampson_F_1998_jieeetip_m_bnsdpr, author = {F. J. Hampson and J.C. Pesquet}, title = {$M$band Nonlinear Subband Decompositions with Perfect Reconstruction}, journal = {IEEE Trans. Image Process.}, year = {1998}, volume = {7}, number = {11}, pages = {15471560}, doi = {http://dx.doi.org/10.1109/83.725362} } 

Healy, D.M., Rockmore, D.N., Kostelec, P.J. & Moore, S.  FFTs for the 2Sphere  Improvements and Variations [BibTeX] 
2003  J. Fourier Anal. Appl. Vol. 9(4), pp. 341385 
article  
BibTeX:
@article{Healy_D_2003_jfouranalappl_fft_2siv, author = {D. M. Healy and D. N. Rockmore and P. J. Kostelec and S. Moore}, title = {FFTs for the 2Sphere  Improvements and Variations}, journal = {J. Fourier Anal. Appl.}, year = {2003}, volume = {9}, number = {4}, pages = {341385} } 

Heeger, D.J. & Bergen, J.R.  PyramidBased Texture Analysis/Synthesis [BibTeX] 
1995  Proc. SIGGRAPH Int. Conf. Comput. Graph. Interactive Tech., pp. 229238  inproceedings  
BibTeX:
@inproceedings{Heeger_D_1995_pacmsiggraph_pyr_btas, author = {D. J. Heeger and J. R. Bergen}, title = {PyramidBased Texture Analysis/Synthesis}, booktitle = {Proc. SIGGRAPH Int. Conf. Comput. Graph. Interactive Tech.}, year = {1995}, pages = {229238} } 

Heijmans, H.J.A.M. & Goutsias, J.  Nonlinear multiresolution signal decomposition schemes. II. Morphological wavelets  2000  IEEE Trans. Image Process. Vol. 9(11), pp. 18971913 
article  DOI 
Abstract: For pt.I see ibid., vol.9, no.11, p.186276 (2000). In its original form, the wavelet transform is a linear tool. However, it has been increasingly recognized that nonlinear extensions are possible. A major impulse to the development of nonlinear wavelet transforms has been given by the introduction of the lifting scheme by Sweldens (1995, 1996, 1998). The aim of this paper, which is a sequel to a previous paper devoted exclusively to the pyramid transform, is to present an axiomatic framework encompassing most existing linear and nonlinear wavelet decompositions. Furthermore, it introduces some, thus far unknown, wavelets based on mathematical morphology, such as the morphological Haar wavelet, both in one and two dimensions. A general and flexible approach for the construction of nonlinear (morphological) wavelets is provided by the lifting scheme. This paper briefly discusses one example, the maxlifting scheme, which has the intriguing property that preserves local maxima in a signal over a range of scales, depending on how local or global these maxima are  
BibTeX:
@article{Heijmans_H_2000_jieeetip_non_msds2mw, author = {Heijmans, H. J. A. M. and Goutsias, J.}, title = {Nonlinear multiresolution signal decomposition schemes. II. Morphological wavelets}, journal = {IEEE Trans. Image Process.}, year = {2000}, volume = {9}, number = {11}, pages = {18971913}, doi = {http://dx.doi.org/10.1109/83.877211} } 

Heijmans, H.J.A.M., PesquetPopescu, B. & Piella, G.  Building nonredundant adaptive wavelets by update lifting [BibTeX] 
2005  Appl. Comp. Harm. Analysis Vol. 18(3), pp. 252281 
article  
BibTeX:
@article{Heijmans_H_2005_jacha_bui_nawul, author = {Heijmans, H. J. A. M. and B. PesquetPopescu and G. Piella}, title = {Building nonredundant adaptive wavelets by update lifting}, journal = {Appl. Comp. Harm. Analysis}, year = {2005}, volume = {18}, number = {3}, pages = {252281} } 

Helbert, D., Carré, P. & Andrès, É.  3D Discrete Analytical Ridgelet Transform  2006  IEEE Trans. Image Process. Vol. 15(12), pp. 37013714 
article  
Abstract: In this paper, we propose an implementation of the 3D Ridgelet transform: the 3D discrete analytical Ridgelet transform (3D DART). This transform uses the Fourier strategy for the computation of the associated 3D discrete Radon transform. The innovative step is the definition of a discrete 3D transform with the discrete analytical geometry theory by the construction of 3D discrete analytical lines in the Fourier domain. We propose two types of 3D discrete lines: 3D discrete radial lines going through the origin defined from their orthogonal projections and 3D planes covered with 2D discrete line segments. These discrete analytical lines have a parameter called arithmetical thickness, allowing us to define a 3D DART adapted to a specific application. Indeed, the 3D DART representation is not orthogonal, It is associated with a flexible redundancy factor. The 3D DART has a very simple forward/inverse algorithm that provides an exact reconstruction without any iterative method. In order to illustrate the potentiality of this new discrete transform, we apply the 3D DART and its extension to the LocalDART (with smooth windowing) to the denoising of 3D image and color video. These experimental results show that the simple thresholding of the 3D DART coefficients is efficient.  
BibTeX:
@article{Helbert_D_2006_tip_3d_dart, author = {D. Helbert and P. Carré and É. Andrès}, title = {3D Discrete Analytical Ridgelet Transform}, journal = {IEEE Trans. Image Process.}, year = {2006}, volume = {15}, number = {12}, pages = {37013714} } 

Held, S., Storath, M., Massopust, P. & Forster, B.  Steerable Wavelet Frames Based on the Riesz Transform  2010  IEEE Trans. Image Process. Vol. 19(3), pp. 653667 
article  DOI 
Abstract: We consider an extension of the 1D concept of analytical wavelet to $n$D which is by construction compatible with rotations. This extension, called a monogenic wavelet, yields a decomposition of the wavelet coefficients into amplitude, phase, and phase direction. The monogenic wavelet is based on the hypercomplex monogenic signal which is defined using Riesz transforms and perfectly isotropic wavelets frames. Employing the new concept of Clifford frames, we can show that the monogenic wavelet generates a wavelet frame. Furthermore, this approach yields wavelet frames that are steerable with respect to direction. Applications to descreening and contrast enhancement illustrate the versatility of this approach to image analysis and reconstruction.  
BibTeX:
@article{Held_S_2010_jieeetip_ste_wfrt, author = {Held, S. and Storath, M. and Massopust, P. and Forster, B.}, title = {Steerable Wavelet Frames Based on the Riesz Transform}, journal = {IEEE Trans. Image Process.}, year = {2010}, volume = {19}, number = {3}, pages = {653667}, doi = {http://dx.doi.org/10.1109/TIP.2009.2036713} } 

Hildreth, E.C.  Implementation of a Theory of Edge Detection  1980  (AITR579)  techreport  
Abstract: This report describes the implementation of a theory of edge detection, proposed by Marr and Hildreth (1979). According to this theory, the image is first processed independently through a set of different size filters, whose shape is the Laplacian of a Gaussian, ***. Zerocrossings in the output of these filters mark the positions of intensity changes at different resolutions. Information about these zerocrossings is then used for deriving a full symbolic description of changes in intensity in the image, called the raw primal sketch. The theory is closely tied with early processing in the human visual systems. In this report, we first examine the critical properties of the initial filters used in the edge detection process, both from a theoretical and practical standpoint. The implementation is then used as a test bed for exploring aspects of the human visual system; in particular, acuity and hyperacuity. Finally, we present some preliminary results concerning the relationship between zerocrossings detected at different resolutions, and some observations relevant to the process by which the human visual system integrates descriptions of intensity changes obtained at different resolutions  
BibTeX:
@techreport{Hildreth_E_1980_tr_imp_ted, author = {Hildreth, E. C.}, title = {Implementation of a Theory of Edge Detection}, year = {1980}, number = {AITR579} } 

Holschneider, M.  Wavelets, an analysis tool [BibTeX] 
1995  book  
BibTeX:
@book{Holscheider_M_1995_book_wav_at, author = {Holschneider, M.}, title = {Wavelets, an analysis tool}, publisher = {Oxford Science Publications}, year = {1995} } 

Jacques, L. & Antoine, J.P.  Multiselective Pyramidal Decomposition of Images: wavelets with Adaptive Angular Selectivity  2007  Int. J. Wavelets Multidim. Inform. Proc. Vol. 5(5), pp. 785814 
article  DOI 
Abstract: Many techniques have been devised these last ten years to add an appropriate directionality concept in decompositions of images from the specific transformations of a small set of atomic functions. Let us mention, for instance, works on directional wavelets, steerable filters, dualtree wavelet transform, curvelets, wave atoms, ridgelet packets, etc. In general, features that are best represented are straight lines or smooth curves as those defining contours of objects (e.g. in curvelets processing) or oriented textures (e.g. wave atoms, ridgelet packets). However, real images present also a set of details less oriented and more isotropic, like corners, spots, texture components, etc. This paper develops an adaptive representation for all these image elements, ranging from highly directional ones to fully isotropic ones. This new tool can indeed be tuned relatively to these image features by decomposing them into a Littlewood?Paley frame of directional wavelets with variable angular selectivity. Within such a decomposition, 2D wavelets inherit some particularities of the biorthogonal circular multiresolution framework in their angular behavior. Our method can therefore be seen as an angular multiselectivity analysis of images. Two applications of the proposed method are given at the end of the paper, namely, in the fields of image denoising and Nterm nonlinear approximation.  
BibTeX:
@article{Jacques_L_2007_jijwmip_mul_pdiwaas, author = {Jacques, L. and Antoine, J.P.}, title = {Multiselective Pyramidal Decomposition of Images: wavelets with Adaptive Angular Selectivity}, journal = {Int. J. Wavelets Multidim. Inform. Proc.}, year = {2007}, volume = {5}, number = {5}, pages = {785814}, doi = {http://dx.doi.org/10.1142/S0219691307002051} } 

Jacques, L., Duval, L., Chaux, C. & Peyré, G.  Addendum to ``A panorama on multiscale geometric representations, intertwining spatial, directional and frequency selectivity'' [BibTeX] 
2011  misc  
BibTeX:
@misc{Jacques_L_2011_url_panorama_addendum, author = {L. Jacques and L. Duval and C. Chaux and G. Peyré}, title = {Addendum to ``A panorama on multiscale geometric representations, intertwining spatial, directional and frequency selectivity''}, year = {2011}, note = {http://www.laurentduval.eu/sivapanoramamultiscalegeometricrepresentations.html} } 

Jacques, L. & Vleeschouwer, C.D.  A Geometrical Study of Matching Pursuit Parametrization [BibTeX] 
2008  IEEE Trans. Image Process. Vol. 56(7), pp. 28352848 
article  
BibTeX:
@article{Jacques_L_2008_jieeetip_geo_smpp, author = {Jacques, L. and Vleeschouwer, C. D.}, title = {A Geometrical Study of Matching Pursuit Parametrization}, journal = {IEEE Trans. Image Process.}, publisher = {New York, NY: Institute of Electrical and Electronics Engineers, c1991}, year = {2008}, volume = {56}, number = {7}, pages = {28352848} } 

Jalobeanu, A., Kingsbury, N. & Zerubia, J.  Image deconvolution using hidden Markov tree modeling of complex wavelet packets [BibTeX] 
2001  Vol. 1Proc. Int. Conf. Image Process., pp. 201204 
inproceedings  
BibTeX:
@inproceedings{Jalobeanu_A_2001_picip_ima_deconv, author = {Jalobeanu, A. and Kingsbury, N. and Zerubia, J.}, title = {Image deconvolution using hidden Markov tree modeling of complex wavelet packets}, booktitle = {Proc. Int. Conf. Image Process.}, year = {2001}, volume = {1}, pages = {201204} } 

Jansen, M., Baraniuk, R.G. & Lavu, S.  Multiscale Approximation of Piecewise Smooth TwoDimensional Function using Normal Triangulated Meshes  2005  Appl. Comp. Harm. Analysis Vol. 19(1), pp. 92130 
article  
Abstract: Multiresolution triangulation meshes are widely used in computer graphics for representing threedimensional(3d) shapes. We propose to use these tools to represent 2d piecewise smooth functions such as grayscale images,because triangles have potential to more efficiently approximate the discontinuities between the smooth pieces than other standard tools like wavelets. We show that normal mesh subdivision is an efficient triangulation, thanks to its local adaptivity to the discontinuities. Indeed, we prove that, within a certain function class, the normal mesh representation has an optimal asymptotic error decay rate as the number of terms in the representation grows. This function class is the socalled horizon class comprising constant regions separated by smooth discontinuities,where the line of discontinuity is C2 continuous. This optimal decay rate is possible because normal meshes automatically generate a polyline (piecewise linear) approximation of each discontinuity, unlike the blocky piecewise constant approximation of tensor product wavelets. In this way, the proposed nonlinear multiscale normal mesh decomposition is an anisotropic representation of the 2d function. The same idea of anisotropic representations lies at the basis of decompositions such as wedgelet and curvelet transforms, but the proposed normal mesh approach has a unique construction.  
BibTeX:
@article{Jansen_M_2005_jacha_mul_apstdfntm, author = {M. Jansen and R. G. Baraniuk and S. Lavu}, title = {Multiscale Approximation of Piecewise Smooth TwoDimensional Function using Normal Triangulated Meshes}, journal = {Appl. Comp. Harm. Analysis}, year = {2005}, volume = {19}, number = {1}, pages = {92130} } 

Kâaniche, M., BenazzaBenyahia, A., PesquetPopescu, B. & Pesquet, J.C.  Non separable lifting scheme with adaptive update step for still and stereo image coding [BibTeX] 
2011  Signal Process.  article  
BibTeX:
@article{Kaaniche_M_2011_PREPRINT_non_slsausssic, author = {Kâaniche, M. and BenazzaBenyahia, A. and PesquetPopescu, B. and Pesquet, J.C.}, title = {Non separable lifting scheme with adaptive update step for still and stereo image coding}, journal = {Signal Process.}, year = {2011}, note = {In press} } 

Kâaniche, M., BenazzaBenyahia, A., PesquetPopescu, B. & Pesquet, J.C.  Vector Lifting Schemes for Stereo Image Coding  2009  IEEE Trans. Image Process. Vol. 18(11), pp. 24632475 
article  DOI 
Abstract: Many research efforts have been devoted to the improvement of stereo image coding techniques for storage or transmission. In this paper, we are mainly interested in lossytolossless coding schemes for stereo images allowing progressive reconstruction. The most commonly used approaches for stereo compression are based on disparity compensation techniques. The basic principle involved in this technique first consists of estimating the disparity map. Then, one image is considered as a reference and the other is predicted in order to generate a residual image. In this paper, we propose a novel approach, based on vector lifting schemes (VLS), which offers the advantage of generating two compact multiresolution representations of the left and the right views. We present two versions of this new scheme. A theoretical analysis of the performance of the considered VLS is also conducted. Experimental results indicate a significant improvement using the proposed structures compared with conventional methods.  
BibTeX:
@article{Kaaniche_M_2009_jieeetip_vec_lssic, author = {Kâaniche, M. and BenazzaBenyahia, A. and PesquetPopescu, B. and Pesquet, J.C.}, title = {Vector Lifting Schemes for Stereo Image Coding}, journal = {IEEE Trans. Image Process.}, year = {2009}, volume = {18}, number = {11}, pages = {24632475}, doi = {http://dx.doi.org/10.1109/TIP.2009.2026672} } 

Kassim, A.A., Lee, W.S. & Zonoobi, D.  Hierarchical SegmentationBased Image Coding Using Hybrid QuadBinary Trees  2009  IEEE Trans. Image Process. Vol. 18(6), pp. 1284291 
article  DOI 
Abstract: A novel segmentationbased image approximation and coding technique is proposed. A hybrid quadbinary (QB) tree structure is utilized to efficiently model and code geometrical information within images. Compared to other treebased representation such as wedgelets, the proposed QBtree based method is more efficient for a wide range of contour features such as junctions, corners and ridges, especially at low bit rates.  
BibTeX:
@article{Kassim_A_2009_jieeetip_hie_sbichqbt, author = {Kassim, A. A. and W. S. Lee and Zonoobi, D.}, title = {Hierarchical SegmentationBased Image Coding Using Hybrid QuadBinary Trees}, journal = {IEEE Trans. Image Process.}, year = {2009}, volume = {18}, number = {6}, pages = {1284291}, doi = {http://dx.doi.org/10.1109/TIP.2009.2017339} } 

Kerkyacharian, G., Petrushev, P., Picard, D. & Willer, T.  Needlet algorithms for estimation in inverse problems  2007  Electron. J. Stat. Vol. 1, pp. 3076 
article  URL 
Abstract: We provide a new algorithm for the treatment of inverse problems which combines the traditional SVD inversion with an appropriate thresholding technique in a well chosen new basis. Our goal is to devise an inversion procedure which has the advantages of localization and multiscale analysis of wavelet representations without losing the stability and computability of the SVD decompositions. To this end we utilize the construction of localized frames (termed "needlets") built upon the SVD bases. We consider two different situations: the "wavelet" scenario, where the needlets are assumed to behave similarly to true wavelets, and the "Jacobitype" scenario, where we assume that the properties of the frame truly depend on the SVD basis at hand (hence on the operator). To illustrate each situation, we apply the estimation algorithm respectively to the deconvolution problem and to the Wicksell problem. In the latter case, where the SVD basis is a Jacobi polynomial basis, we show that our scheme is capable of achieving rates of convergence which are optimal in the $L_2$ case, we obtain interesting rates of convergence for other $L_p$ norms which are new (to the best of our knowledge) in the literature, and we also give a simulation study showing that the NEEDD estimator outperforms other standard algorithms in almost all situations.  
BibTeX:
@article{Kerkyacharian_G_2007_jelecjstat_nee_aeip, author = {G. Kerkyacharian and P. Petrushev and D. Picard and T. Willer}, title = {Needlet algorithms for estimation in inverse problems}, journal = {Electron. J. Stat.}, year = {2007}, volume = {1}, pages = {3076}, url = {doi:10.1214/07EJS014} } 

King, F.W.  Hilbert Transforms  2009  Vol. 125 
book  
Abstract: The Hilbert transform has many uses, including solving problems in aerodynamics, condensed matter physics, optics, fluids, and engineering. Written in a style that will suit a wide audience (including the physical sciences), this book will become the reference of choice on the topic, whatever the subject background of the reader. It explains all the common Hilbert transforms, mathematical techniques for evaluating them, and has detailed discussions of their application. Especially useful for researchers are the tabulation of analytically evaluated Hilbert transforms, and an atlas that immediately illustrates how the Hilbert transform alters a function. A collection of exercises helps the reader to test their understanding of the material in each chapter. The bibliography is a wideranging collection of references both to the classical mathematical papers, and to a diverse array of applications.  
BibTeX:
@book{King_F_2009_book_hil_t, author = {King, F. W.}, title = {Hilbert Transforms}, publisher = {Cambridge University Press}, year = {2009}, volume = {125} } 

Kingsbury, N.G.  Image Processing with Complex Wavelets  1999  Phil. Trans. Roy. Soc. Lond. A Vol. 357, pp. 25432560 
article  
Abstract: We first review how wavelets may be used for multiresolution image processing, describing the filterbank implementation of the discrete wavelet transform (DWT) and how it may be extended via separable filtering for processing images and other multidimensional signals. We then show that the condition for inversion of the DWT (perfect reconstruction) forces many commonly used wavelets to be similar in shape, and that this shape produces severe shift dependence (variation of DWTco efficient energy at any given scale with shift of the input signal). It is also shown that separable filtering with the DWTprev ents the transform from providing directionally selective filters for diagonal image features. Complex wavelets can provide both shift invariance and good directional selectivity, with only modest increases in signal redundancy and computation load. However, development of a complex wavelet transform (CWT) with perfect reconstruction and good filter characteristics has proved difficult until recently. We now propose the dualtree CWTas a solution to this problem, yielding a transform with attractive properties for a range of signal and image processing applications, including motion estimation, denoising, texture analysis and synthesis, and object segmentation.  
BibTeX:
@article{Kingsbury_N_1999_jphiltransroysoclonda_ima_pcw, author = {Kingsbury, N. G.}, title = {Image Processing with Complex Wavelets}, journal = {Phil. Trans. Roy. Soc. Lond. A}, year = {1999}, volume = {357}, pages = {25432560} } 

Kingsbury, N.G.  The dualtree complex wavelet transform: a new technique for shift invariance and directional filters [BibTeX] 
1998  Proc. IEEE Digital Signal Process. Workshop  inproceedings  
BibTeX:
@inproceedings{Kingsbury_N_1998_pieeedspw_dua_tcwtntsidf, author = {Kingsbury, N. G.}, title = {The dualtree complex wavelet transform: a new technique for shift invariance and directional filters}, booktitle = {Proc. IEEE Digital Signal Process. Workshop}, year = {1998} } 

Kittipoom, P., Kutyniok, G. & Lim, W.Q.  Irregular Shearlet Frames: Geometry and Approximation Properties  2010  J. Fourier Anal. Appl., pp. 136  article  DOI URL 
Abstract: Recently, shearlet systems were introduced as a means to derive efficient encoding methodologies for anisotropic features in 2dimensional data with a unified treatment of the continuum and digital setting. However, only very few construction strategies for discrete shearlet systems are known so far. In this paper, we take a geometric approach to this problem. Utilizing the close connection with group representations, we first introduce and analyze an upper and lower weighted shearlet density based on the shearlet group. We then apply this geometric measure to provide necessary conditions on the geometry of the sets of parameters for the associated shearlet systems to form a frame for $L^2(R^2)$, either when using all possible generators or a large class exhibiting some decay conditions. While introducing such a feasible class of shearlet generators, we analyze approximation properties of the associated shearlet systems, which themselves lead to interesting insights into homogeneous approximation abilities of shearlet frames. We also present examples, such as oversampled shearlet systems and coshearlet systems, to illustrate the usefulness of our geometric approach to the construction of shearlet frames.  
BibTeX:
@article{Kittipoom_P_2010_jfouranalappl_irr_sfgap, author = {Kittipoom, P. and Kutyniok, G. and Lim, W.Q.}, title = {Irregular Shearlet Frames: Geometry and Approximation Properties}, journal = {J. Fourier Anal. Appl.}, publisher = {Birkhäuser Boston}, year = {2010}, pages = {136}, url = {http://dx.doi.org/10.1007/s0004101091630}, doi = {http://dx.doi.org/10.1007/s0004101091630} } 

Knutsson, H. & Andersson, M.  Implications of invariance and uncertainty for local structure analysis filter sets  2005  Signal Process. Image Comm. Vol. 20, pp. 569581 
article  
Abstract: The paper discusses which properties of filter sets used in local structure estimation that are the most important. Answers are provided via the introduction of a number of fundamental invariances. Mathematical formulations corresponding to the required invariances leads up to the introduction of a new class of filter sets termed loglets. Loglets are polar separable and have excellent uncertaintyproperties. The directional part uses a spherical harmonics basis. Using loglets it is shown how the concepts of quadrature and phase can be defined in ndimensions. It is also shown how a reliable measure of the certaintyof the estimate can be obtained byfinding the deviation from the signal model manifold. Local structure analysis algorithms are quite complex and involve a lot more than the filters used. This makes comparisons difficult to interpret from a filter point of view. To reduce the number ?free? parameters and target the filter design aspects a number of simple 2D experiments have been carried out. The evaluation supports the claim that loglets are preferable to other designs. In particular it is demonstrated that the loglet approach outperforms a Gaussian derivative approach in resolution and robustness.  
BibTeX:
@article{Knutsson_H_2005_jspic_imp_iulsafs, author = {Knutsson, H. and Andersson, M.}, title = {Implications of invariance and uncertainty for local structure analysis filter sets}, journal = {Signal Process. Image Comm.}, year = {2005}, volume = {20}, pages = {569581} } 

Kovačević, J. & Chebira, A.  Life beyond bases: The advent of frames (Part I) [BibTeX] 
2007  IEEE Signal Process. Mag., pp. 86104  article  
BibTeX:
@article{Kovacevic_J_2007_ieeespm_lif_bbaf1, author = {Kovačević, J. and Chebira, A.}, title = {Life beyond bases: The advent of frames (Part I)}, journal = {IEEE Signal Process. Mag.}, year = {2007}, pages = {86104} } 

Kovačević, J. & Chebira, A.  Life beyond bases: The advent of frames (Part II) [BibTeX] 
2007  IEEE Signal Process. Mag., pp. 115125  article  
BibTeX:
@article{Kovacevic_J_2007_ieeespm_lif_bbaf2, author = {Kovačević, J. and Chebira, A.}, title = {Life beyond bases: The advent of frames (Part II)}, journal = {IEEE Signal Process. Mag.}, year = {2007}, pages = {115125} } 

Kovačević, J. & Vetterli, M.  Nonseparable two and threedimensional wavelets [BibTeX] 
1995  IEEE Trans. Signal Process. Vol. 43(5), pp. 12691273 
article  
BibTeX:
@article{Kovacevic_J_1995_tsp_non_ttdw, author = {Kovačević, J. and Vetterli, M.}, title = {Nonseparable two and threedimensional wavelets}, journal = {IEEE Trans. Signal Process.}, year = {1995}, volume = {43}, number = {5}, pages = {12691273} } 

Kovačević, J. & Vetterli, M.  Nonseparable Multidimensional Perfect Reconstruction Filters Banks and Wavelets Bases for $R^n$ [BibTeX] 
1992  IEEE Trans. Inform. Theory Vol. 38(2), pp. 533555 
article  
BibTeX:
@article{Kovacevic_J_1992_tit_non_mprfbwbrn, author = {Kovačević, J. and Vetterli, M.}, title = {Nonseparable Multidimensional Perfect Reconstruction Filters Banks and Wavelets Bases for $R^n$}, journal = {IEEE Trans. Inform. Theory}, year = {1992}, volume = {38}, number = {2}, pages = {533555} } 

Krommweh, J. & Plonka, G.  Directional Haar wavelet frames on triangles  2009  Appl. Comp. Harm. Analysis Vol. 27(2), pp. 215234 
article  DOI URL 
Abstract: Traditional wavelets are not very effective in dealing with images that contain orientated discontinuities (edges). To achieve a more efficient representation one has to use basis elements with much higher directional sensitivity. In recent years several approaches like curvelets and shearlets have been studied providing essentially optimal approximation properties for images that are piecewise smooth and have discontinuities along C2curves. While curvelets and shearlets have compact support in frequency domain, we construct directional wavelet frames generated by functions with compact support in time domain. Our Haar wavelet constructions can be seen as special composite dilation wavelets, being based on a generalized multiresolution analysis (MRA) associated with a dilation matrix and a finite collection of [`]shear' matrices. The complete system of constructed wavelet functions forms a Parseval frame. Based on this MRA structure we provide an efficient filter bank algorithm. The freedom obtained by the redundancy of the applied Haar functions will be used for an efficient sparse representation of piecewise constant images as well as for image denoising.  
BibTeX:
@article{Krommweh_J_2009_jacha_dir_hwft, author = {J. Krommweh and G. Plonka}, title = {Directional Haar wavelet frames on triangles}, journal = {Appl. Comp. Harm. Analysis}, year = {2009}, volume = {27}, number = {2}, pages = {215234}, url = {http://www.sciencedirect.com/science/article/B6WB34VXMPVH1/2/164801f99390b93b7838b6eabdce65ce}, doi = {DOI: 10.1016/j.acha.2009.03.002} } 

Kutyniok, G. & Labate, D.  The construction of regular and irregular shearlet frames [BibTeX] 
2007  J. Wavelet Theory Appl. Vol. 1, pp. 110 
article  
BibTeX:
@article{Kutyniok_G_2007_jwavelettheoryappl_con_risf, author = {G. Kutyniok and D. Labate}, title = {The construction of regular and irregular shearlet frames}, journal = {J. Wavelet Theory Appl.}, year = {2007}, volume = {1}, pages = {110} } 

Le Pennec, E. & Mallat, S.  Bandelet image approximation and compression  2005  Multiscale Model. Simul. Vol. 4(3), pp. 9921039 
article  DOI 
Abstract: Finding efficient geometric representations of images is a central issue to improving image compression and noise removal algorithms. We introduce bandelet orthogonal bases and frames that are adapted to the geometric regularity of an image. Images are approximated by finding a best bandelet basis or frame that produces a sparse representation. For functions that are uniformly regular outside a set of edge curves that are geometrically regular, the main theorem proves that bandelet approximations satisfy an optimal asymptotic error decay rate. A bandelet image compression scheme is derived. For computational applications, a fast discrete bandelet transform algorithm is introduced, with a fast best basis search which preserves asymptotic approximation and coding error decay rates.  
BibTeX:
@article{LePennec_E_2005_jsiammms_ban_ac, author = {Le Pennec, E. and Mallat, S.}, title = {Bandelet image approximation and compression}, journal = {Multiscale Model. Simul.}, year = {2005}, volume = {4}, number = {3}, pages = {9921039}, doi = {http://dx.doi.org/10.1137/040619454} } 

Lee, T.S.  Image Representation Using 2D Gabor Wavelets  1996  IEEE Trans. Patt. Anal. Mach. Int. Vol. 18(10), pp. 959971 
article  DOI 
Abstract: This paper extends to two dimensions the frame criterion developed by Daubechies for onedimensional wavelets, and it computes the frame bounds for the particular case of 2D Gabor wavelets. Completeness criteria for 2D Gabor image representations are important because of their increasing role in many computer vision applications and also in modeling biological vision, since recent neurophysiological evidence from the visual cortex of mammalian brains suggests that the filter response profiles of the main class of linearlyresponding cortical neurons (called simple cells) are best modeled as a family of selfsimilar 2D Gabor wavelets. We therefore derive the conditions under which a set of continuous 2D Gabor wavelets will provide a complete representation of any image, and we also find selfsimilar wavelet parametrization which allow stable reconstruction by summation as though the wavelets formed an orthonormal basis. Approximating a ldquo;tight frame rdquo; generates redundancy which allows lowresolution neural responses to represent highresolution images  
BibTeX:
@article{Lee_T_1996_jieeetpami_ima_r2dgw, author = {T. S. Lee}, title = {Image Representation Using 2D Gabor Wavelets}, journal = {IEEE Trans. Patt. Anal. Mach. Int.}, publisher = {IEEE Computer Society}, year = {1996}, volume = {18}, number = {10}, pages = {959971}, doi = {http://doi.ieeecomputersociety.org/10.1109/34.541406} } 

Lessig, C. & E., F.  SOHO: Orthogonal and symmetric Haar wavelets on the sphere  2008  ACM Trans. Graph. Vol. 27(1), pp. 4:14:11 
article  DOI URL 
Abstract: We propose the SOHO wavelet basis?the first spherical Haar wavelet basis that is both orthogonal and symmetric, making it particularly well suited for the approximation and processing of allfrequency signals on the sphere. We obtain the basis with a novel spherical subdivision scheme that defines a partition acting as the domain of the basis functions. Our construction refutes earlier claims doubting the existence of a basis that is both orthogonal and symmetric. Experimental results for the representation of spherical signals verify that the superior theoretical properties of the SOHO wavelet basis are also relevant in practice.  
BibTeX:
@article{Lessig_C_2008_jacmtog_soh_oshws, author = {Lessig, C. and Fiu E.}, title = {SOHO: Orthogonal and symmetric Haar wavelets on the sphere}, journal = {ACM Trans. Graph.}, publisher = {ACM}, year = {2008}, volume = {27}, number = {1}, pages = {4:14:11}, url = {http://doi.acm.org/10.1145/1330511.1330515}, doi = {http://dx.doi.org/10.1145/1330511.1330515} } 

Lim, W.Q.  The Discrete Shearlet Transform: A New Directional Transform and Compactly Supported Shearlet Frames  2010  IEEE Trans. Image Process. Vol. 19(5), pp. 11661180 
article  DOI 
Abstract: It is now widely acknowledged that analyzing the intrinsic geometrical features of the underlying image is essential in many applications including image processing. In order to achieve this, several directional image representation schemes have been proposed. In this paper, we develop the discrete shearlet transform (DST) which provides efficient multiscale directional representation and show that the implementation of the transform is built in the discrete framework based on a multiresolution analysis (MRA). We assess the performance of the DST in image denoising and approximation applications. In image approximations, our approximation scheme using the DST outperforms the discrete wavelet transform (DWT) while the computational cost of our scheme is comparable to the DWT. Also, in image denoising, the DST compares favorably with other existing transforms in the literature.  
BibTeX:
@article{Lim_W_2010_jieeetip_dis_stndtcssf, author = {Lim, W.Q.}, title = {The Discrete Shearlet Transform: A New Directional Transform and Compactly Supported Shearlet Frames}, journal = {IEEE Trans. Image Process.}, year = {2010}, volume = {19}, number = {5}, pages = {11661180}, doi = {http://dx.doi.org/10.1109/TIP.2010.2041410} } 

Lindeberg, T.  Generalized Gaussian scalespace axiomatics comprising linear scalespace, affine scalespace and spatiotemporal scalespace [BibTeX] 
2011  J. Math. Imaging Vis. Vol. 40, pp. 3681 
article  DOI URL 
BibTeX:
@article{Lindeberg_T_2011_jmathimagingvis_gen_gssaclssassstss, author = {Lindeberg, T.}, title = {Generalized Gaussian scalespace axiomatics comprising linear scalespace, affine scalespace and spatiotemporal scalespace}, journal = {J. Math. Imaging Vis.}, year = {2011}, volume = {40}, pages = {3681}, url = {http://www.csc.kth.se/~tony/earlyvision.html}, doi = {http://dx.doi.org/10.1007/s1085101002422} } 

Lindeberg, T.  Discrete Derivative Approximations with ScaleSpace Properties: A Basis for LowLevel Feature Extraction  1993  J. Math. Imaging Vis. Vol. 3(4), pp. 349379 
article  
Abstract: It is developed how discrete derivative approximations can be dened so that scalespace properties hold exactly also in the discrete domain. Starting from a set of natural requirements on the rst processing stages of a visual system, the visual front end, an axiomatic derivation is given of how a multiscale representation of derivative approximations can be constructed from a discrete signal, so that it possesses an algebraic structure similar to that possessed by the derivatives of the traditional scalespace representation in the continuous domain. A family of kernels is derived which constitute discrete analogues to the continuous Gaussian derivatives. The representation has theoretical advantages to other discretizations of the scalespace theory in the sense that operators which commute before discretization commute after discretization. Some computational implications of this are that derivative approximations can be computed directly from smoothed data, and that this will give exactly the same result as convolution with the corresponding derivative approximation kernel. Moreover, a number of normalization conditions are automatically satised. The proposed methodology leads to a conceptually very simple scheme of computations for multiscale lowlevel feature extraction, consisting of four basic steps; (i) large support convolution smoothing, (ii) small support dierence computations, (iii) point operations for computing dierential geometric entities, and (iv) nearest neighbour op erations for feature detection. Applications are given demonstrating how the proposed scheme can be used for edge detection and junction detection based on derivatives up to order three.  
BibTeX:
@article{Lindeberg_T_1993_jmathimagingvis_dis_dasspbllfe, author = {Lindeberg, T.}, title = {Discrete Derivative Approximations with ScaleSpace Properties: A Basis for LowLevel Feature Extraction}, journal = {J. Math. Imaging Vis.}, year = {1993}, volume = {3}, number = {4}, pages = {349379} } 

Lisowska, A.  Geometrical wavelets and their generalizations in digital image coding and processing  2005  School: Univ. Silesia, Sosnowiec, Poland  phdthesis  
Abstract: In recent years the problem of efficient image coding and processing has gained popularity and is of great interest both for computer scientists and mathematicians. Image coding which tends to obtain efficient compression, especially progressive one, allows to save time when sending images in a network and disc space during storage. On the other hand image processing may be used for image quality improvement as well as extraction of specific features. So, efficient representation of an image plays a crucial role in computer graphics because it forms the foundation for image coding and processing. Recently, it has become evident that separable transforms, as for example wavelet ones, are not the best ones in image representation due to their disability of catching line discontinuities present in images in the form of edges. What follows they are blind for image geometry. To overcome that problem the competitive theory of geometrical wavelets has arisen recently. As shown in literature, the use of geometrical wavelets, thanks to better approximations, is superior to nearly all of the classical wavelet based applications of image coding (including compression) and processing. Investigations of images may not be carried out without the relation to Human Visual System. Recent researches in psychology of vision have proven that the amount of information which is gathered by receptors of the retina in the eye is far larger than dozens of bits per second which are transmitted to the brain from the eye. Additionally, recent investigations in neuropsychology give us information what kinds of signals are perceived by brain in first order and which ones are less important. Two main observations follow from the researches. The first one is that, basing on better image approximations, it is possible to reduce the amount of data used in representation. The second one makes us realize that less important information, which does not reach the brain, may be removed from an image without corruption of the visual quality of an image. So two practical questions arise. How images can be approximated better, which will lead to improving its coding and processing properties? And how the most important information, from the Human Visual System point of view, may be extracted from an image in an automatic way? In this dissertation it has tried to answer both these questions. Thus firstly, the generalization of wedgelets (the class of geometrical wavelets) has been proposed and it has been shown that thanks to better approximations of images they improve the properties of image coding and processing in comparison with classical wedgelets. Especially, the use of such generalized wedgelets tends to produce more sparse representation of an image. It has been additionally shown that generalized wedgelets give better results in noisy image processing in comparison with other standard methods. Secondly, the new application of geometrical wavelets in extraction of different classes of signals with different importance for perception by the human brain has been proposed. With the help of such wavelets (especially beamlets) an operator has been defined, which extracts such signals quite automatically. Thanks to the geometrical approach, such solution is competitive in comparison with the other ones described in literature. The results presented in the dissertation appear to improve the results presented so far in literature, which has been confirmed both theoretically and experimentally.  
BibTeX:
@phdthesis{Lisowska_A_2005_phd_geo_wgdicp, author = {Lisowska, A.}, title = {Geometrical wavelets and their generalizations in digital image coding and processing}, school = {Univ. Silesia, Sosnowiec, Poland}, year = {2005} } 

Lounsbery, M., DeRose, T.D. & Warren, J.  Multiresolution analysis for surfaces of arbitrary topological type  1997  ACM Trans. Graph. Vol. 16(1), pp. 3473 
article  
Abstract: Multiresolution analysis and wavelets provide useful and efficient tools for representing functions at multiple levels of detail. Wavelet representations have been used in a broad range of applications, including image compression, physical simulation, and numerical analysis. In this article, we present a new class of wavelets, based on subdivision surfaces, that radically extends the class of representable functions. Whereas previous twodimensional methods were restricted to functions defined on $R^2$, the subdivision wavelets developed here may be applied to functions defined on compact surfaces of arbitrary topological type. We envision many applications of this work, including continuous levelofdetail control for graphics rendering, compression of geometric models, and acceleration of global illumination algorithms. Levelof detail control for spherical domains is illustrated using two examples: shape approximation of a polyhedral model, and color approximation of global terrain data.  
BibTeX:
@article{Lounsbery_M_1997_jacmtog_mul_asatt, author = {M. Lounsbery and T. D. DeRose and J. Warren}, title = {Multiresolution analysis for surfaces of arbitrary topological type}, journal = {ACM Trans. Graph.}, publisher = {ACM}, year = {1997}, volume = {16}, number = {1}, pages = {3473} } 

Lu, Y. & Do, M.N.  CRISP contourlets: a critically sampled directional multiresolution image representation  2003  Vol. 5207Proc. SPIE, Wavelets: Appl. Signal Image Process., pp. 655665 
inproceedings  
Abstract: Directional multiresolution image representations have lately attracted much attention. A number of new systems, such as the curvelet transform and the more recent contourlet transform, have been proposed. A common issue of these transforms is the redundancy in representation, an undesirable feature for certain applications (e.g. compression). Though some critically sampled transforms have also been proposed in the past, they can only provide limited directionality or limited flexibility in the frequency decomposition. In this paper, we propose a filter bank structure achieving a nonredundant multiresolution and multidirectional expansion of images. It can be seen as a critically sampled version of the original contourlet transform (hence the name CRISPcontourets) in the sense that the corresponding frequency decomposition is similar to that of contourlets, which divides the whole spectrum both angularly and radially. However, instead of performing the multiscale and directional decomposition steps separately as is done in contourlets, the key idea here is to use a combined iterated nonseparable filter bank for both steps. Aside from critical sampling, the proposed transform possesses other useful properties including perfect reconstruction, flexible configuration of the number of directions at each scale, and an efficient treestructured implementation.  
BibTeX:
@inproceedings{Lu_Y_2003_pspiewasip_cri_ccsdmir, author = {Lu, Y. and Do, M. N.}, title = {CRISP contourlets: a critically sampled directional multiresolution image representation}, booktitle = {Proc. SPIE, Wavelets: Appl. Signal Image Process.}, year = {2003}, volume = {5207}, pages = {655665} } 

Lu, Y.M. & Do, M.N.  Multidimensional Directional Filter Banks and Surfacelets  2007  IEEE Trans. Image Process. Vol. 16(4), pp. 918931 
article  DOI 
Abstract: In 1992, Bamberger and Smith proposed the directional filter bank (DFB) for an efficient directional decomposition of 2D signals. Due to the nonseparable nature of the system, extending the DFB to higher dimensions while still retaining its attractive features is a challenging and previously unsolved problem. We propose a new family of filter banks, named NDFB, that can achieve the directional decomposition of arbitrary Ndimensional (Nges2) signals with a simple and efficient treestructured construction. In 3D, the ideal passbands of the proposed NDFB are rectangularbased pyramids radiating out from the origin at different orientations and tiling the entire frequency space. The proposed NDFB achieves perfect reconstruction via an iterated filter bank with a redundancy factor of N in ND. The angular resolution of the proposed NDFB can be iteratively refined by invoking more levels of decomposition through a simple expansion rule. By combining the NDFB with a new multiscale pyramid, we propose the surfacelet transform, which can be used to efficiently capture and represent surfacelike singularities in multidimensional data  
BibTeX:
@article{Lu_Y_2007_tip_mul_dfbs, author = {Lu, Y. M. and Do, M. N.}, title = {Multidimensional Directional Filter Banks and Surfacelets}, journal = {IEEE Trans. Image Process.}, year = {2007}, volume = {16}, number = {4}, pages = {918931}, doi = {http://dx.doi.org/10.1109/TIP.2007.891785} } 

Ma, J. & Plonka, G.  The Curvelet Transform  a review of recent applications  2010  IEEE Signal Process. Mag. Vol. 27(2), pp. 118133 
article  DOI 
Abstract: Multiresolution methods are deeply related to image processing, biological and computer vision, and scientific computing. The curvelet transform is a multiscale directional transform that allows an almost optimal nonadaptive sparse representation of objects with edges. It has generated increasing interest in the community of applied mathematics and signal processing over the years. In this article, we present a review on the curvelet transform, including its history beginning from wavelets, its logical relationship to other multiresolution multidirectional methods like contourlets and shearlets, its basic theory and discrete algorithm. Further, we consider recent applications in image/video processing, seismic exploration, fluid mechanics, simulation of partial different equations, and compressed sensing.  
BibTeX:
@article{Ma_J_2010_jieeespm_cur_t, author = {J. Ma and Plonka, G.}, title = {The Curvelet Transform  a review of recent applications}, journal = {IEEE Signal Process. Mag.}, year = {2010}, volume = {27}, number = {2}, pages = {118133}, doi = {http://dx.doi.org/10.1109/MSP.2009.935453} } 

Mallat, S.  Geometrical grouplets  2009  Appl. Comp. Harm. Analysis Vol. 26(2), pp. 161180 
article  DOI 
Abstract: Grouplet orthogonal bases and tight frames are constructed with association fields that group points to take advantage of geometrical image regularities in space or time. These association fields have a multiscale geometry that can incorporate multiple junctions. A fast grouplet transform is computed with orthogonal multiscale hierarchical groupings. A grouplet transform applied to wavelet image coefficients defines an orthogonal basis or a tight frame of grouping bandlets. Applications to noise removal and image zooming are described.  
BibTeX:
@article{Mallat_S_2009_acha_geo_g, author = {Mallat, S.}, title = {Geometrical grouplets}, journal = {Appl. Comp. Harm. Analysis}, year = {2009}, volume = {26}, number = {2}, pages = {161180}, doi = {http://dx.doi.org/10.1016/j.acha.2008.03.004} } 

Mallat, S.  A wavelet tour of signal processing: the sparse way [BibTeX] 
2009  book  
BibTeX:
@book{Mallat_S_2009_book_wav_tspsw, author = {Mallat, S.}, title = {A wavelet tour of signal processing: the sparse way}, publisher = {Academic Press}, year = {2009}, edition = {3rd} } 

Mallat, S. & Zhang, Z.  Matching pursuits with timefrequency dictionaries  1993  IEEE Trans. Signal Process. Vol. 41(12), pp. 33973415 
article  
Abstract: We introduce an algorithm, called matching pursuit, that decomposes any signal into a linear expansion of waveforms that are selected from a redundant dictionary of functions. These waveforms are chosen in order to best match the signal structures. Matching pursuits are general procedures to compute adaptive signal representations. With a dictionary of Gabor functions a matching pursuit defines an adaptive timefrequency transform. We derive a signal energy distribution in the timefrequency plane, which does not include interference terms, unlike Wigner and Cohen class distributions. A matching pursuit isolates the signal structures that are coherent with respect to a given dictionary. An application to pattern extraction from noisy signals is described. We compare a matching pursuit decomposition with a signal expansion over an optimized wavepacket orthonormal basis, selected with the algorithm of Coifman and Wickerhauser.  
BibTeX:
@article{Mallat_S_1993_tsp_mat_ptfd, author = {Mallat, S. and Zhang, Z.}, title = {Matching pursuits with timefrequency dictionaries}, journal = {IEEE Trans. Signal Process.}, year = {1993}, volume = {41}, number = {12}, pages = {33973415} } 

Mallat, S.G.  A theory for multiresolution signal decomposition: the wavelet representation  1989  IEEE Trans. Patt. Anal. Mach. Int. Vol. 11(7), pp. 674693 
article  DOI 
Abstract: Multiresolution representations are effective for analyzing the information content of images. The properties of the operator which approximates a signal at a given resolution were studied. It is shown that the difference of information between the approximation of a signal at the resolutions 2j+1 and 2j (where j is an integer) can be extracted by decomposing this signal on a wavelet orthonormal basis of L2(Rn), the vector space of measurable, squareintegrable ndimensional functions. In L2(R), a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function ψ(x). This decomposition defines an orthogonal multiresolution representation called a wavelet representation. It is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters. Wavelet representation lies between the spatial and Fourier domains. For images, the wavelet representation differentiates several spatial orientations. The application of this representation to data compression in image coding, texture discrimination and fractal analysis is discussed  
BibTeX:
@article{Mallat_S_1989_tpami_the_msdwr, author = {Mallat, S. G.}, title = {A theory for multiresolution signal decomposition: the wavelet representation}, journal = {IEEE Trans. Patt. Anal. Mach. Int.}, year = {1989}, volume = {11}, number = {7}, pages = {674693}, doi = {http://dx.doi.org/10.1109/34.192463} } 

Malvar, H.S.  Fast progressive image coding without wavelets  2000  Proc. Data Compression Conf., pp. 243252  inproceedings  DOI 
Abstract: We introduce a new image compression algorithm that allows progressive image reconstruction ? both in resolution and in fidelity, with a fully embedded bitstream. The algorithm is based on bitplane entropy coding of reordered transform coefficients, similar to the progressive wavelet codec (PWC) previously introduced. Unlike PWC, however, our new progressive transform coder (PTC) does not use wavelets; it performs the spacefrequency decomposition step via a new lapped biorthogonal transform (LBT). PTC achieves a rate vs. distortion performance that is comparable (within 2%) to that of the stateoftheart SPIHT (set partitioning in hierarchical trees) codec. However, thanks to the use of the LBT, the spacefrequency decomposition step in PTC reduces the number of multiplications per pixel by a factor of 2.7, and the number of additions by about 15%, when compared to the fastest possible implementation of the ?9/7? wavelet transform via lifting. Furthermore, since most of the computation in the LBT is in fact performed by a DCT, our PTC codec can make full use of fast software and hardware modules for 1D and 2D DCTs. 

BibTeX:
@inproceedings{Malvar_H_2000_pdcc_fas_picww, author = {Malvar, H. S.}, title = {Fast progressive image coding without wavelets}, booktitle = {Proc. Data Compression Conf.}, year = {2000}, pages = {243252}, doi = {http://dx.doi.org/10.1109/DCC.2000.838164} } 

Malvar, H.S., Hallapuro, A., Karczewicz, M. & Kerofsky, L.  Lowcomplexity transform and quantization in H.264/AVC  2003  IEEE Trans. Circ. Syst. Video Tech. Vol. 13(7), pp. 598603 
article  DOI 
Abstract: This paper presents an overview of the transform and quantization designs in H.264. Unlike the popular 8 times;8 discrete cosine transform used in previous standards, the 4 times;4 transforms in H.264 can be computed exactly in integer arithmetic, thus avoiding inverse transform mismatch problems. The new transforms can also be computed without multiplications, just additions and shifts, in 16bit arithmetic, thus minimizing computational complexity, especially for lowend processors. By using short tables, the new quantization formulas use multiplications but avoid divisions.  
BibTeX:
@article{Malvar_H_2003_jieeetcsvt_low_ctqh264avc, author = {Malvar, H. S. and Hallapuro, A. and Karczewicz, M. and Kerofsky, L.}, title = {Lowcomplexity transform and quantization in H.264/AVC}, journal = {IEEE Trans. Circ. Syst. Video Tech.}, year = {2003}, volume = {13}, number = {7}, pages = {598603}, doi = {http://dx.doi.org/10.1109/TCSVT.2003.814964} } 

Manduchi, R., Perona, P. & Shy, D.  Efficient deformable filter banks  1998  IEEE Trans. Signal Process. Vol. 46(4), pp. 11681173 
article  DOI 
Abstract: This article describes efficient schemes for the computation of a large number of differently scaled/oriented filtered versions of an image. We generalize the wellknown steerable/scalable (?deformable?) filter bank structure by imposing XY separability on the basis filters. The resulting systems, designed by an iterative projections technique, achieve substantial reduction of the computational cost. To reduce the memory requirement, we adopt a multirate implementation. Due to the inner sampling rate alteration, the resulting structure is not shift invariant. We introduce a design criterion for multirate deformable structures that jointly controls the approximation error and the shift variance  
BibTeX:
@article{Manduchi_R_1998_tsp_eff_dfb, author = {Manduchi, R. and Perona, P. and Shy, D.}, title = {Efficient deformable filter banks}, journal = {IEEE Trans. Signal Process.}, year = {1998}, volume = {46}, number = {4}, pages = {11681173}, doi = {http://dx.doi.org/10.1109/78.668570} } 

Marr, D.  Vision: A Computational Investigation into the Human Representation and Processing of Visual Information [BibTeX] 
1982  book  
BibTeX:
@book{Marr_D_1982_book_vis_cihrpvi, author = {Marr, D.}, title = {Vision: A Computational Investigation into the Human Representation and Processing of Visual Information}, publisher = {W. H. Freeman}, year = {1982} } 

Marr, D. & Hildreth, E.  Theory of Edge Detection  1980  Proc. Roy. Soc. Lond. B Biol. Sci. Vol. 207(1167), pp. 187217 
article  DOI 
Abstract: A theory of edge detection is presented. The analysis proceeds in two parts. (1) Intensity changes, which occur in a natural image over a wide range of scales, are detected separately at different scales. An appropriate filter for this purpose at a given scale is found to be the second derivative of a Gaussian, and it is shown that, provided some simple conditions are satisfied, these primary filters need not be orientationdependent. Thus, intensity changes at a given scale are best detected by finding the zero values of $nabla ^2$G(x, y)* I(x, y)$ for image I, where $G(x, y)$ is a twodimensional Gaussian distribution and $nabla ^2$ is the Laplacian. The intensity changes thus discovered in each of the channels are then represented by oriented primitives called zerocrossing segments, and evidence is given that this representation is complete. (2) Intensity changes in images arise from surface discontinuities or from reflectance or illumination boundaries, and these all have the property that they are spatially localized. Because of this, the zerocrossing segments from the different channels are not independent, and rules are deduced for combining them into a description of the image. This description is called the raw primal sketch. The theory explains several basic psychophysical findings, and the operation of forming oriented zerocrossing segments from the output of centresurround $nabla ^2$G filters acting on the image forms the basis for a physiological model of simple cells (see Marr & Ullman 1979).  
BibTeX:
@article{Marr_D_1980_jprocroysocb_the_ed, author = {Marr, D. and Hildreth, E.}, title = {Theory of Edge Detection}, journal = {Proc. Roy. Soc. Lond. B Biol. Sci.}, year = {1980}, volume = {207}, number = {1167}, pages = {187217}, doi = {http://dx.doi.org/10.1098/rspb.1980.0020} } 

Marr, D. & Poggio, T.  A Computational Theory of Human Stereo Vision [BibTeX] 
1979  Proc.Roy. Soc. Lond. B Biol. Sci. Vol. 204(1156), pp. 301328 
article  
BibTeX:
@article{Marr_D_1979_jroystatsocb_com_thsv, author = {D. Marr and T. Poggio}, title = {A Computational Theory of Human Stereo Vision}, journal = {Proc.Roy. Soc. Lond. B Biol. Sci.}, year = {1979}, volume = {204}, number = {1156}, pages = {301328} } 

Massopust, P.  Fractal Functions, Fractal Surfaces, and Wavelets [BibTeX] 
1994  book  
BibTeX:
@book{Massopust_P_1994_book_fra_ffsw, author = {Massopust, P.}, title = {Fractal Functions, Fractal Surfaces, and Wavelets}, publisher = {Academic Press}, year = {1994} } 

Meyer, F.G. & Coifman, R.R.  Brushlets: A Tool for Directional Image Analysis and Image Compression  1997  Appl. Comp. Harm. Analysis Vol. 4(2), pp. 147187 
article  DOI URL 
Abstract: We construct a new adaptive basis of functions which is reasonably well localized with only one peak in frequency. We develop a compression algorithm that exploits this basis to obtain the most economical representation of the image in terms of textured patterns with different orientations, frequencies, sizes, and positions. The technique directly works in the Fourier domain and has potential applications for compression of highly textured images, texture analysis, etc.  
BibTeX:
@article{Meyer_F_1997_jacha_bru_tdiaic, author = {F. G. Meyer and R. R. Coifman}, title = {Brushlets: A Tool for Directional Image Analysis and Image Compression}, journal = {Appl. Comp. Harm. Analysis}, year = {1997}, volume = {4}, number = {2}, pages = {147187}, url = {http://www.sciencedirect.com/science/article/B6WB345MH2TF8/2/26f109da3fd9f249aa4f21ceb5482349}, doi = {DOI: 10.1006/acha.1997.0208} } 

Meyer, Y.  Oscillating Patterns in Image Processing and Nonlinear Evolution Equations [BibTeX] 
2001  The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures  incollection  
BibTeX:
@incollection{Meyer_Y_2001_incoll_osc_pipnee, author = {Meyer, Y.}, title = {Oscillating Patterns in Image Processing and Nonlinear Evolution Equations}, booktitle = {The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures}, publisher = {Amer. Math. Soc.}, year = {2001} } 

Monaci, G., Escoda, Ò.D. & Vandergheynst, P.  Analysis of multimodal sequences using geometric video representations  2006  Signal Process. Vol. 86(12), pp. 35343548 
article  DOI 
Abstract: This paper presents a novel method to correlate audio and visual data generated by the same physical phenomenon, based on sparse geometric representation of video sequences. The video signal is modeled as a sum of geometric primitives evolving through time, that jointly describe the geometric and motion content of the scene. The displacement through time of relevant visual features, like the mouth of a speaker, can thus be compared with the evolution of an audio feature to assess the correspondence between acoustic and visual signals. Experiments show that the proposed approach allows to localize and track the speaker's mouth when several persons are present on the scene, in presence of distracting motion, and without prior face or mouth detection.  
BibTeX:
@article{Monaci_G_2006_jsp_ana_msgvr, author = {Monaci, G. and Escoda, Ò. D. and Vandergheynst, P.}, title = {Analysis of multimodal sequences using geometric video representations}, journal = {Signal Process.}, publisher = {Elsevier}, year = {2006}, volume = {86}, number = {12}, pages = {35343548}, doi = {http://dx.doi.org/10.1016/j.sigpro.2006.02.044} } 

Narcowich, F.J., Petrushev, P. & Ward, J.D.  Localized tight frames on spheres [BibTeX] 
2006  SIAM J. Math. Anal. Vol. 38(2), pp. 574594 
article  
BibTeX:
@article{Narcowich_F_2006_jsiammathanal_loc_tfs, author = {F. J. Narcowich and P. Petrushev and J. D. Ward}, title = {Localized tight frames on spheres}, journal = {SIAM J. Math. Anal.}, year = {2006}, volume = {38}, number = {2}, pages = {574594} } 

Nason, G.P. & Silverman, B.W.  The stationary wavelet transform and some statistical applications [BibTeX] 
1995  Vol. 103Wavelets and Statistics, pp. 281300 
incollection  
BibTeX:
@incollection{Nason_G_1995_incollwas_sta_wtsa, author = {Nason, G. P. and Silverman, B. W.}, title = {The stationary wavelet transform and some statistical applications}, booktitle = {Wavelets and Statistics}, publisher = {Springer Verlag}, year = {1995}, volume = {103}, pages = {281300} } 

Natarajan, B.K.  Sparse Approximate Solutions to Linear Systems  1995  SIAM J. Comp. Vol. 24(2), pp. 227234 
article  DOI URL 
Abstract: The following problem is considered: given a matrix $A$ in $bf R^mtimes n$, ($m$ rows and $n$ columns), a vector $b$ in $bf R^m$, and $epsilon > 0$, compute a vector $x$ satisfying $Ax  b_2 leq epsilon $ if such exists, such that $x$ has the fewest number of nonzero entries over all such vectors. It is shown that the problem is NPhard, but that the wellknown greedy heuristic is good in that it computes a solution with at most $lceil 18 Opt(2)bf A^+_2^2 ln ( b _2/epsilon) nonzero entries, where $Opt(2)$ is the optimum number of nonzero entries at error $2$, $A$ is the matrix obtained by normalizing each column of $A$ with respect to the $L_2$ norm, and $A^+$ is its pseudoinverse.  
BibTeX:
@article{Natarajan_B_1995_jsiamcomp_spa_asls, author = {B. K. Natarajan}, title = {Sparse Approximate Solutions to Linear Systems}, journal = {SIAM J. Comp.}, publisher = {SIAM}, year = {1995}, volume = {24}, number = {2}, pages = {227234}, url = {http://link.aip.org/link/?SMJ/24/227/1}, doi = {http://dx.doi.org/10.1137/S0097539792240406} } 

Neff, R. & Zakhor, A.  VeryLow BitRate Video Coding Based on Matching Pursuits [BibTeX] 
1997  IEEE Trans. Circ. Syst. Video Tech. Vol. 7(1), pp. 158171 
article  
BibTeX:
@article{Neff_R_1997_jieeetcsvt_ver_lbrvcbmp, author = {Neff, R. and Zakhor, A.}, title = {VeryLow BitRate Video Coding Based on Matching Pursuits}, journal = {IEEE Trans. Circ. Syst. Video Tech.}, year = {1997}, volume = {7}, number = {1}, pages = {158171} } 

Nestares, O., Navarro, R., Portilla, J. & Tabernero, A.  Efficient Spatial Domain Implementation of a Multiscale Image Representation Based on Gabor Functions [BibTeX] 
1998  J. Electronic Imaging Vol. 7(1), pp. 166173 
article  
BibTeX:
@article{Nestares_O_1998_jelecim_eff_sdimirbgf, author = {Nestares, O. and Navarro, R. and Portilla, J. and Tabernero, A.}, title = {Efficient Spatial Domain Implementation of a Multiscale Image Representation Based on Gabor Functions}, journal = {J. Electronic Imaging}, year = {1998}, volume = {7}, number = {1}, pages = {166173} } 

Nguyen, T.T. & Oraintara, S.  A Class of Multiresolution Directional Filter Banks  2007  IEEE Trans. Signal Process. Vol. 55(3), pp. 949961 
article  DOI 
Abstract: In this paper, we introduced a class of directional filter banks (DFBs) having the previously proposed uniform DFB (uDFB) as a special case. Except for the uDFB, each DFB in this class can be used to decompose an image yielding up to 12 directions while maintaining perfect reconstruction and maximal decimation. A multiresolution representation can be obtained by repeating the same decomposition at the lowpass band. The permissible property of the filter banks in cases of being implemented by a tree structure and by direct implementation is discussed. The result shows that only one DFB in the class, called the uniform quincunx DFB (uqDFB), satisfies the permissible property when being implemented directly without using the tree structure. The nonuniform quincunx DFB (nuqDFB) is then constructed from the uqDFB by merging its two lowpass subbands. An alternative structure for constructing the nuqDFB is presented. The new structure, while yielding the same frequency partitioning, allows the DFB to be realized with complexity comparable to that of the separable wavelet filter bank. The connection between the discrete filter bank and the continuous directional wavelet is also established. Numerical experiments on directional feature extractions, image denoising and nonlinear approximation are presented at the end of the paper to demonstrate the potential of the nuqDFB  
BibTeX:
@article{Nguyen_T_2007_jieeetsp_cla_mdfb, author = {T. T. Nguyen and S. Oraintara}, title = {A Class of Multiresolution Directional Filter Banks}, journal = {IEEE Trans. Signal Process.}, year = {2007}, volume = {55}, number = {3}, pages = {949961}, doi = {http://dx.doi.org/10.1109/TSP.2006.887140} } 

Ogden, J.M., Adelson, E.H., Bergen, J.R. & Burt, P.J.  PyramidBased Computer Graphics  1985  RCA Eng. Vol. 30(5), pp. 415 
article  
Abstract: This paper describes pyramid solutions to graphics problems that have proven difficult in other image representations. The "physics simulation" approach grows more out of the physics and mathematical modelling traditions. Greater realism can be achieved by using the ph  
BibTeX:
@article{Ogden_J_1985_jrcaeng_pyr_bcg, author = {Ogden, J. M. and Adelson, E. H. and Bergen, J. R. and Burt, P. J.}, title = {PyramidBased Computer Graphics}, journal = {RCA Eng.}, year = {1985}, volume = {30}, number = {5}, pages = {415} } 

Olhede, S.C. & Metikas, G.  The Monogenic Wavelet Transform  2009  IEEE Trans. Signal Process. Vol. 57(9), pp. 34263441 
article  DOI 
Abstract: This paper extends the 1D analytic wavelet transform to the 2D monogenic wavelet transform. The transformation requires care in its specification to ensure suitable transform coefficients are calculated, and it is constructed so that the wavelet transform may be considered as both local and monogenic. This is consistent with defining the transform as a real wavelet transform of a monogenic signal in analogy with the analytic wavelet transform. Classes of monogenic wavelets are proposed with suitable local properties. It is shown that the monogenic wavelet annihilates antimonogenic signals, that the monogenic wavelet transform is phaseshift covariant and that the transform magnitude is phaseshift invariant. A simple form for the magnitude and orientation of the isotropic transform coefficients of a unidirectional signal when observed in a rotated frame of reference is derived. The monogenic wavelet ridges of local plane waves are given.  
BibTeX:
@article{Olhede_S_2009_jieeetsp_mon_wt, author = {Olhede, S. C. and Metikas, G.}, title = {The Monogenic Wavelet Transform}, journal = {IEEE Trans. Signal Process.}, year = {2009}, volume = {57}, number = {9}, pages = {34263441}, doi = {http://dx.doi.org/10.1109/TSP.2009.2023397} } 

Olshausen, B.A. & Field, D.J.  Sparse coding with an overcomplete basis set: A strategy employed by V1?  1997  Vis. Res. Vol. 37(23), pp. 33113325 
article  DOI URL 
Abstract: The spatial receptive fields of simple cells in mammalian striate cortex have been reasonably well described physiologically and can be characterized as being localized, oriented, and bandpass, comparable with the basis functions of wavelet transforms. Previously, we have shown that these receptive field properties may be accounted for in terms of a strategy for producing a sparse distribution of output activity in response to natural images. Here, in addition to describing this work in a more expansive fashion, we examine the neurobiological implications of sparse coding. Of particular interest is the case when the code is overcompletei.e., when the number of code elements is greater than the effective dimensionality of the input space. Because the basis functions are nonorthogonal and not linearly independent of each other, sparsifying the code will recruit only those basis functions necessary for representing a given input, and so the inputoutput function will deviate from being purely linear. These deviations from linearity provide a potential explanation for the weak forms of nonlinearity observed in the response properties of cortical simple cells, and they further make predictions about the expected interactions among units in response to naturalistic stimuli.  
BibTeX:
@article{Olshausen_B_1997_jvisres_spa_cobssev1, author = {B. A. Olshausen and D. J. Field}, title = {Sparse coding with an overcomplete basis set: A strategy employed by V1?}, journal = {Vis. Res.}, year = {1997}, volume = {37}, number = {23}, pages = {33113325}, url = {http://www.sciencedirect.com/science/article/B6T0W494SR7019/2/b4c138506b06df6f332ced73e8501a3e}, doi = {DOI: 10.1016/S00426989(97)001697} } 

Ouarti, N. & Peyré, G.  Best Basis Search in a Nonstationary Wavelet Packets Dictionary [BibTeX] 
2009  Proc. Int. Conf. Image Process.  inproceedings  
BibTeX:
@inproceedings{Ouarti_N_2009_picip_bes_bsnswpd, author = {N. Ouarti and G. Peyré}, title = {Best Basis Search in a Nonstationary Wavelet Packets Dictionary}, booktitle = {Proc. Int. Conf. Image Process.}, year = {2009} } 

Pati, Y.C., Rezaifar, R. & Krishnaprasa, P.S.  Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition [BibTeX] 
1993  Proc. Asilomar Conf. Signal, Syst. Comput.  inproceedings  
BibTeX:
@inproceedings{Pati_Y_1993_pasilomar_ort_mprfaawd, author = {Y. C. Pati and R. Rezaifar and P. S. Krishnaprasa}, title = {Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition}, booktitle = {Proc. Asilomar Conf. Signal, Syst. Comput.}, year = {1993} } 

Pesquet, J.C., Krim, H. & Carfantan, H.  Timeinvariant orthogonal wavelet representations  1996  IEEE Trans. Signal Process. Vol. 44(8), pp. 19641970 
article  
Abstract: A simple construction of an orthonormal basis starting with a socalled mother wavelet, together with an efficient implementation gained the wavelet decomposition easy acceptance and generated a great research interest in its applications. An orthonormal basis may not, however, always be a suitable representation of a signal, particularly when time (or space) invariance is a required property. The conventional way around this problem is to use a redundant decomposition. We address the timeinvariance problem for orthonormal wavelet transforms and propose an extension to wavelet packet decompositions. We show that it,is possible to achieve time invariance and preserve the orthonormality. We subsequently propose an efficient approach to obtain such a decomposition. We demonstrate the importance of our method by considering some application examples in signal reconstruction and time delay estimation  
BibTeX:
@article{Pesquet_J_1996_tsp_tim_iowr, author = {Pesquet, J.C. and Krim, H. and Carfantan, H.}, title = {Timeinvariant orthogonal wavelet representations}, journal = {IEEE Trans. Signal Process.}, year = {1996}, volume = {44}, number = {8}, pages = {19641970} } 

PesquetPopescu, B. & Bottreau, V.  Threedimensional lifting schemes for motion compensated video compression  2001  Vol. 3Proc. Int. Conf. Acoust. Speech Signal Process., pp. 17931796 
inproceedings  DOI 
Abstract: Threedimensional wavelet decompositions are efficient tools for scalable video coding. We show a lifting formulation for these decompositions. The temporal wavelet transform is inherently nonlinear, due to the motion estimation step, and the lifting formalism allows us to provide several improvements to the scheme initially proposed by Choi and Woods: a better processing of the uncovered areas is proposed and an overlapped motioncompensated temporal filtering method is introduced in the multiresolution decomposition. As shown by simulations, the proposed method results in higher coding efficiency, while keeping the scalability functionalities  
BibTeX:
@inproceedings{PesquetPopescu_B_2001_picassp_thr_dlsmcvc, author = {PesquetPopescu, B. and Bottreau, V.}, title = {Threedimensional lifting schemes for motion compensated video compression}, booktitle = {Proc. Int. Conf. Acoust. Speech Signal Process.}, year = {2001}, volume = {3}, pages = {17931796}, doi = {http://dx.doi.org/10.1109/ICASSP.2001.941289} } 

Peyré, G.  Best Basis Compressed Sensing  2010  IEEE Trans. Signal Process. Vol. 58(5), pp. 26132622 
article  DOI 
Abstract: This paper proposes a best basis extension of compressed sensing recovery. Instead of regularizing the compressed sensing inverse problem with a sparsity prior in a fixed basis, our framework makes use of sparsity in a treestructured dictionary of orthogonal bases. A new iterative thresholding algorithm performs both the recovery of the signal and the estimation of the best basis. The resulting reconstruction from compressive measurements optimizes the basis to the structure of the sensed signal. Adaptivity is crucial to capture the regularity of complex natural signals. Numerical experiments on sounds and geometrical images indeed show that this best basis search improves the recovery with respect to fixed sparsity priors.  
BibTeX:
@article{Peyre_G_2010_jieeetsp_bes_bcs, author = {G. Peyré}, title = {Best Basis Compressed Sensing}, journal = {IEEE Trans. Signal Process.}, year = {2010}, volume = {58}, number = {5}, pages = {26132622}, doi = {http://dx.doi.org/10.1109/TSP.2010.2042490} } 

Peyré, G.  Texture Processing with Grouplets  2009  IEEE Trans. Patt. Anal. Mach. Int. Vol. 32(4), pp. 733746 
article  DOI 
Abstract: This paper proposes a new method to synthesize and inpaint geometric textures. The texture model is composed of a geometric layer that drives the computation of a new grouplet transform. The geometry is an orientation flow that follows the patterns of the texture to analyze or synthesize. The grouplet transform extends the original construction of Mallat and is adapted to the modeling of natural textures. Each grouplet atoms is an elongated stroke located along the geometric flow. These atoms exhibit a wide range of lengths and widths, which is important to match the variety of structures present in natural images. Statistical modeling and sparsity optimization over these grouplet coefficients enable the synthesis of texture patterns along the flow. This paper explores texture inpainting and texture synthesis, which both require the joint optimization of the geometric flow and the grouplet coefficients.  
BibTeX:
@article{Peyre_G_2009_jieeetpami_tex_pg, author = {G. Peyré}, title = {Texture Processing with Grouplets}, journal = {IEEE Trans. Patt. Anal. Mach. Int.}, year = {2009}, volume = {32}, number = {4}, pages = {733746}, doi = {http://dx.doi.org/10.1109/TPAMI.2009.54} } 

Peyré, G. & Mallat, S.  Orthogonal Bandlet Bases for Geometric Images Approximation  2008  Comm. Pure Applied Math. Vol. 61(9), pp. 11731212 
article  
Abstract: This paper introduces orthogonal bandelet bases to approximate images having some geometrical regularity. These bandelet bases are computed by applying parametrized Alpert transform operators over an orthogonal wavelet basis. These bandeletization operators depend upon a multiscale geometric ?ow that is adapted to the image at each wavelet scale. This bandelet construction has a hierarchical structure over wavelet coef?cients taking advantage of existing regularity among these coef?cients. It is proved that C? images having singularities along Calphacurves are approximated in a best orthogonal bandelet basis with an optimal asymptotic error decay. Fast algorithms and compression applications are described.  
BibTeX:
@article{Peyre_G_2008_jcommpureapplmath_ort_bbgia, author = {G. Peyré and S. Mallat}, title = {Orthogonal Bandlet Bases for Geometric Images Approximation}, journal = {Comm. Pure Applied Math.}, year = {2008}, volume = {61}, number = {9}, pages = {11731212} } 

Plonka, G.  The easy path wavelet transform: A new adaptive wavelet transform for sparse representation of twodimensional data [BibTeX] 
2009  Multiscale Model. Simul. Vol. 7(3), pp. 14741496 
article  
BibTeX:
@article{Plonka_G_2009_jsiammms_eas_pwt_nawtsrtdd, author = {Plonka, G.}, title = {The easy path wavelet transform: A new adaptive wavelet transform for sparse representation of twodimensional data}, journal = {Multiscale Model. Simul.}, year = {2009}, volume = {7}, number = {3}, pages = {14741496} } 

Portilla, J. & Simoncelli, E.P.  A Parametric Texture Model based on Joint Statistics of Complex Wavelet Coefficients [BibTeX] 
2000  Int. J. Comp. Vis. Vol. 40, pp. 4971 
article  
BibTeX:
@article{Portilla_J_2000_jijcv_par_tmbjscwc, author = {Portilla, J. and Simoncelli, E. P.}, title = {A Parametric Texture Model based on Joint Statistics of Complex Wavelet Coefficients}, journal = {Int. J. Comp. Vis.}, year = {2000}, volume = {40}, pages = {4971} } 

Portilla, J., Strela, V., Wainwright, M.J. & Simoncelli, E.P.  Image denoising using scale mixtures of Gaussians in the wavelet domain [BibTeX] 
2003  IEEE Trans. Image Process. Vol. 12(11), pp. 13381351 
article  
BibTeX:
@article{Portilla_J_2003_tip_ima_dsmgwd, author = {Portilla, J. and Strela, V. and Wainwright, M. J. and Simoncelli, E. P.}, title = {Image denoising using scale mixtures of Gaussians in the wavelet domain}, journal = {IEEE Trans. Image Process.}, year = {2003}, volume = {12}, number = {11}, pages = {13381351} } 

Quellec, G., Lamard, M., Cazuguel, G., Cochener, B. & Roux, C.  Adaptive Nonseparable Wavelet Transform via Lifting and its Application to ContentBased Image Retrieval  2010  IEEE Trans. Image Process. Vol. 19(1), pp. 2535 
article  DOI 
Abstract: We present in this paper a novel way to adapt a multidimensional wavelet filter bank, based on the nonseparable lifting scheme framework, to any specific problem. It allows the design of filter banks with a desired number of degrees of freedom, while controlling the number of vanishing moments of the primal wavelet (mathtilde NÂ¿ moments) and of the dual wavelet ( NÂ¿ moments). The prediction and update filters, in the lifting scheme based filter banks, are defined as Neville filters of order mathtilde NÂ¿ and NÂ¿ , respectively. However, in order to introduce some degrees of freedom in the design, these filters are not defined as the simplest Neville filters. The proposed method is convenient: the same algorithm is used whatever the dimensionality of the signal, and whatever the lattice used. The method is applied to contentbased image retrieval (CBIR): an image signature is derived from this new adaptive nonseparable wavelet transform. The method is evaluated on four image databases and compared to a similar CBIR system, based on an adaptive separable wavelet transform. The mean precision at five of the nonseparable wavelet based system is notably higher on three out of the four databases, and comparable on the other one. The proposed method also compares favorably with the dualtree complex wavelet transform, an overcomplete nonseparable wavelet transform.  
BibTeX:
@article{Quellec_G_2010_jieeetip_ada_nwtlacbir, author = {Quellec, G. and Lamard, M. and Cazuguel, G. and Cochener, B. and Roux, C.}, title = {Adaptive Nonseparable Wavelet Transform via Lifting and its Application to ContentBased Image Retrieval}, journal = {IEEE Trans. Image Process.}, year = {2010}, volume = {19}, number = {1}, pages = {2535}, doi = {http://dx.doi.org/10.1109/TIP.2009.2030479} } 

Reissell, L.M.  Wavelet multiresolution representation of curves and surfaces  1996  Graph. Model. Image Process. Vol. 58(3), pp. 198217 
article  
Abstract: We develop wavelet methods for the multiresolution representation of parametric curves and surfaces. To support the representation, we construct a new family of compactly supported symmetric biorthogonal wavelets with interpolating scaling functions. The wavelets in these biorthogonal pairs have properties better suited for curves and surfaces than many commonly used filters. We also give examples of the applications of the wavelet approach: these include the derivation of compact hierarchical curve and surface representations using modified wavelet compression, the identification of smooth sections of surfaces, and a subdivisionlike intersection algorithm for discrete plane curves.  
BibTeX:
@article{Reissell_L_1996_jgraphmodelimageprocess_wav_mrcs, author = {Reissell, L.M.}, title = {Wavelet multiresolution representation of curves and surfaces}, journal = {Graph. Model. Image Process.}, year = {1996}, volume = {58}, number = {3}, pages = {198217} } 

Rioul, O. & Duhamel, P.  Fast algorithms for discrete and continuous wavelet transforms  1992  IEEE Trans. Inform. Theory Vol. 38(2), pp. 569586 
article  
Abstract: Several algorithms are reviewed for computing various types of wavelet transforms: the Mallat algorithm (1989), the `a trous' algorithm, and their generalizations by Shensa. The goal of this work is to develop guidelines for implementing discrete and continuous wavelet transforms efficiently, and to compare the various algorithms obtained and give an idea of possible gains by providing operation counts. Most wavelet transform algorithms compute sampled coefficients of the continuous wavelet transform using the filter bank structure of the discrete wavelet transform. Although this general method is already efficient, it is shown that noticeable computational savings can be obtained by applying known fast convolution techniques, such as the FFT (fast Fourier transform), in a suitable manner. The modified algorithms are termed `fast' because of their ability to reduce the computational complexity per computed coefficient from L to log L (within a small constant factor) for large filter lengths L. For short filters, smaller gains are obtained: `fast running FIR (finite impulse response) filtering' techniques allow one to achieve typically 30% savings in computations  
BibTeX:
@article{Rioul_O_1992_tit_fas_adcwt, author = {Rioul, O. and Duhamel, P.}, title = {Fast algorithms for discrete and continuous wavelet transforms}, journal = {IEEE Trans. Inform. Theory}, year = {1992}, volume = {38}, number = {2}, pages = {569586} } 

Romberg, J.  Multiscale geometric image processing  2003  School: Rice university  phdthesis  
Abstract: Since their introduction a little more than 10 years ago, wavelets have revolutionized image processing. Wavelet based algorithms define the stateoftheart for applications including image coding (JPEG2000), restoration, and segmentation. Despite their success, wavelets have significant shortcomings in their treatment of edges. Wavelets do not parsimoniously capture even the simplest geometrical structure in images, and wavelet based processing algorithms often produce images with ringing around the edges. As a first step towards accounting for this structure, we will show how to explicitly capture the geometric regularity of contours in cartoon images using the wedgelet representation and a multiscale geometry model. The wedgelet representation builds up an image out of simple piecewise constant functions with linear discontinuities. We will show how the geometry model, by putting a joint distribution on the orientations of the linear discontinuities, allows us to weigh several factors when choosing the wedgelet representation: the error between the representation and the original image, the parsimony of the representation, and whether the wedgelets in the representation form "natural" geometrical structures. We will analyze a simple wedgelet coder based on these principles, and show that it has optimal asymptotic performance for simple cartoon images. Next, we turn our attention to piecewise smooth images; images that are smooth away from a smooth contour. Using a representation composed of wavelets and wedgeprints (wedgelets projected into the wavelet domain), we develop a quadtree based prototype coder whose ratedistortion performance is asymptotically nearoptimal. We use these ideas to implement a fullscale image coder that outperforms JPEG2000 both in peak signal to noise ratio (by 11.5dB at low bitrates) and visually. Finally, we shift our focus to building a statistical image model directly in the wavelet domain. For applications other than compression, the approximate shiftinvariance and directional selectivity of the slightly redundant complex wavelet transform make it particularly wellsuited for modeling singularity structure. Around edges in images, complex wavelet coefficients behave very predictably, exhibiting dependencies that we will exploit using a hidden Markov tree model. We demonstrate the effectiveness of the complex wavelet model with several applications: image denoising, multiscale segmentation, and feature extraction.  
BibTeX:
@phdthesis{Romberg_J_2003_phd_mul_gip, author = {Romberg, J.}, title = {Multiscale geometric image processing}, school = {Rice university}, year = {2003} } 

Rosenfeld, A. & Klette, R.  Digital Straightness  2001  Electron. Notes Theor. Comput. Sci. Vol. 46, pp. 132 
article  DOI URL 
Abstract: A digital arc is called [`]straight' if it is the digitization of a straight line segment. Since the concept of digital straightness was introduced in the mid1970's, dozens of papers on the subject have appeared; many characterizations of digital straight lines have been formulated, and many algorithms for determining whether a digital arc is straight have been defined. This paper reviews the literature on digital straightness and discusses its relationship to other concepts of geometry, the theory of words, and number theory.  
BibTeX:
@article{Rosenfeld_A_2001_jentcs_dig_s, author = {A. Rosenfeld and R. Klette}, title = {Digital Straightness}, journal = {Electron. Notes Theor. Comput. Sci.}, year = {2001}, volume = {46}, pages = {132}, note = {8th Int. Workshop on Combinatorial Image Analysis (IWCIA)}, url = {http://www.sciencedirect.com/science/article/B75H14DDWJGP7M/2/203cd8c8e11100fdc0fca32495f50dbe}, doi = {DOI: 10.1016/S15710661(04)809769} } 

Rosiene, C.P. & Nguyen, T.Q.  Tensorproduct wavelet vs. Mallat decomposition: a comparative analysis  1999  Vol. 3Proc. Int. Symp. Circuits Syst., pp. 431434 
inproceedings  DOI 
Abstract: The twodimensional tensor product wavelet transform is compared to the Mallat representation for the purpose of data compression. It is shown that the tensor product wavelet transform will always provide a coding gain greater than or equal to that of the Mallat representation. Further, the costs of obtaining the tensor product wavelet transform are outlined  
BibTeX:
@inproceedings{Rosiene_C_1999_piscas_ten_pwmdca, author = {Rosiene, C. P. and Nguyen, T. Q.}, title = {Tensorproduct wavelet vs. Mallat decomposition: a comparative analysis}, booktitle = {Proc. Int. Symp. Circuits Syst.}, year = {1999}, volume = {3}, pages = {431434}, doi = {http://dx.doi.org/10.1109/ISCAS.1999.778877} } 

Roşca, D.  Wavelet bases on the sphere obtained by radial projection [BibTeX] 
2007  J. Fourier Anal. Appl. Vol. 13(4), pp. 421434 
article  
BibTeX:
@article{Rosca_D_2007_jfouranalappl_wav_bsorp, author = {Roşca, D.}, title = {Wavelet bases on the sphere obtained by radial projection}, journal = {J. Fourier Anal. Appl.}, publisher = {Springer}, year = {2007}, volume = {13}, number = {4}, pages = {421434} } 

Rubinstein, R., Bruckstein, A.M. & Elad, M.  Dictionaries for Sparse Representation Modeling  2010  Proc. IEEE Vol. 98(6), pp. 10451057 
article  DOI 
Abstract: Sparse and redundant representation modeling of data assumes an ability to describe signals as linear combinations of a few atoms from a prespecified dictionary. As such, the choice of the dictionary that sparsifies the signals is crucial for the success of this model. In general, the choice of a proper dictionary can be done using one of two ways: i) building a sparsifying dictionary based on a mathematical model of the data, or ii) learning a dictionary to perform best on a training set. In this paper we describe the evolution of these two paradigms. As manifestations of the first approach, we cover topics such as wavelets, wavelet packets, contourlets, and curvelets, all aiming to exploit 1D and 2D mathematical models for constructing effective dictionaries for signals and images. Dictionary learning takes a different route, attaching the dictionary to a set of examples it is supposed to serve. From the seminal work of Field and Olshausen, through the MOD, the KSVD, the Generalized PCA and others, this paper surveys the various options such training has to offer, up to the most recent contributions and structures.  
BibTeX:
@article{Rubinstein_R_2010_jprocieee_dic_srm, author = {Rubinstein, R. and Bruckstein, A. M. and Elad, M.}, title = {Dictionaries for Sparse Representation Modeling}, journal = {Proc. IEEE}, year = {2010}, volume = {98}, number = {6}, pages = {10451057}, doi = {http://dx.doi.org/10.1109/JPROC.2010.2040551} } 

Rudin, L.I., Osher, S. & Fatemi, E.  Nonlinear total variation based noise removal algorithms  1992  Physica D Vol. 60(14), pp. 259268 
article  DOI 
Abstract: A constrained optimization type of numerical algorithm for removing noise from images is presented. The total variation of the image is minimized subject to constraints involving the statistics of the noise. The constraints are imposed using Lagrange multipliers. The solution is obtained using the gradientprojection method. This amounts to solving a time dependent partial differential equation on a manifold determined by the constraints. As t > infty the solution converges to a steady state which is the denoised image. The numerical algorithm is simple and relatively fast. The results appear to be stateoftheart for very noisy images. The method is noninvasive, yielding sharp edges in the image. The technique could be interpreted as a first step of moving each level set of the image normal to itself with velocity equal to the curvature of the level set divided by the magnitude of the gradient of the image, and a second step which projects the image back onto the constraint set.  
BibTeX:
@article{Rudin_L_1992_jphysd_non_tvbnra, author = {Rudin, L. I. and Osher, S. and Fatemi, E.}, title = {Nonlinear total variation based noise removal algorithms}, journal = {Physica D}, year = {1992}, volume = {60}, number = {14}, pages = {259268}, doi = {http://dx.doi.org/10.1016/01672789(92)90242F} } 

Said, M., Lachaud, J.O. & Feschet, F.  Multiscale discrete geometry  2009  Proc. IAPR Int. Conf. Discrete Geom. Comput. Imagery (DGCI), pp. 118131  inproceedings  URL 
Abstract: This paper presents a first step in analyzing how digital shapes behave with respect to multiresolution. We first present an analysis of the covering of a standard digital straight line by a multiresolution grid. We then study the multiresolution of Digital Straight Segments (DSS): we provide a sublinear algorithm computing the exact characteristics of a DSS whenever it is a subset of a known standard line. We finally deduce an algorithm for computing a multiscale representation of a digital shape, based only on a DSS decomposition of its boundary.  
BibTeX:
@inproceedings{Said_M_2009_pdgci_mul_dg, author = {Said, M. and Lachaud, J.O. and Feschet, F.}, title = {Multiscale discrete geometry}, booktitle = {Proc. IAPR Int. Conf. Discrete Geom. Comput. Imagery (DGCI)}, publisher = {Springer }, year = {2009}, pages = {118131}, url = {http://hal.archivesouvertes.fr/hal00413681/en/} } 

Sala Llonch, R., Kokiopoulou, E., Tošić, I. & Frossard, P.  3D face recognition with sparse spherical representations  2010  Pattern Recogn. Vol. 43(3), pp. 824834 
article  DOI URL 
Abstract: This paper addresses the problem of 3D face recognition using simultaneous sparse approximations on the sphere. The 3D face point clouds are first aligned with a fully automated registration process. They are then represented as signals on the 2sphere in order to preserve depth and geometry information. Next, we implement a dimensionality reduction process with simultaneous sparse approximations and subspace projection. It permits to represent each 3D face by only a few spherical functions that are able to capture the salient facial characteristics, and hence to preserve the discriminant facial information. We eventually perform recognition by effective matching in the reduced space, where linear discriminant analysis can be further activated for improved recognition performance. The 3D face recognition algorithm is evaluated on the FRGC v.1.0 data set, where it is shown to outperform classical stateoftheart solutions that work with depth images.  
BibTeX:
@article{Llonch_S_2010_jpattrec_3d_frssr, author = {Sala Llonch, R. and Kokiopoulou, E. and Tošić, I. and Frossard, P.}, title = {3D face recognition with sparse spherical representations}, journal = {Pattern Recogn.}, year = {2010}, volume = {43}, number = {3}, pages = {824834}, url = {http://www.sciencedirect.com/science/article/B6V144WT3WG71/2/90b71f122fe206e818fc9f2a8633dad9}, doi = {DOI: 10.1016/j.patcog.2009.07.005} } 

Sampat, M.P., Wang, Z., Gupta, S., Bovik, A.C. & Markey, M.K.  Complex Wavelet Structural Similarity: A New Image Similarity Index  2009  IEEE Trans. Image Process. Vol. 18(11), pp. 24022418 
article  DOI 
Abstract: The monogenic signal is the natural 2D counterpart of the 1D analytic signal. We propose to transpose the concept to the wavelet domain by considering a complexified version of the Riesz transform which has the remarkable property of mapping a realvalued (primary) wavelet basis of $L_2(BBR^2)$ into a complex one. The Riesz operator is also steerable in the sense that it give access to the Hilbert transform of the signal along any orientation. Having set those foundations, we specify a primary polyharmonic spline wavelet basis of $L_2(BBR^2)$ that involves a single Mexicanhatlike mother wavelet (Laplacian of a Bspline). The important point is that our primary wavelets are quasiisotropic: they behave like multiscale versions of the fractional Laplace operator from which they are derived, which ensures steerability. We propose to pair these realvalued basis functions with their complex Riesz counterparts to specify a multiresolution monogenic signal analysis. This yields a representation where each wavelet index is associated with a local orientation, an amplitude and a phase. We give a corresponding waveletdomain method for estimating the underlying instantaneous frequency. We also provide a mechanism for improving the shift and rotationinvariance of the wavelet decomposition and show how to implement the transform efficiently using perfect reconstruction filterbanks. We illustrate the specific feature extraction capabilities of the representation and present novel examples of waveletdomain processing; in particular, a robust, tensorbased analysis of directional image patterns, the demodulation of interferograms, and the reconstruction of digital holograms.  
BibTeX:
@article{Sampat_M_2009_tip_com_wssnisi, author = {Sampat, M. P. and Wang, Z. and Gupta, S. and Bovik, A. C. and Markey, M. K.}, title = {Complex Wavelet Structural Similarity: A New Image Similarity Index}, journal = {IEEE Trans. Image Process.}, year = {2009}, volume = {18}, number = {11}, pages = {24022418}, doi = {http://dx.doi.org/10.1109/TIP.2009.2027628} } 

Schröder, P. & Sweldens, W.  Spherical wavelets: efficiently representing functions on the sphere  1995  Proc. SIGGRAPH Int. Conf. Comput. Graph. Interactive Tech., pp. 161172  inproceedings  DOI 
Abstract: Wavelets have proven to be powerful bases for use in numerical analysis and signal processing. Their power lies in the fact that they only require a small number of coefficients to represent general functions and large data sets accurately. This allows compression and efficient computations. Classical constructions have been limited to simple domains such as intervals and rectangles. In this paper we present a wavelet construction for scalar functions defined on the sphere. We show how biorthogonal wavelets with custom properties can be constructed with the lifting scheme. The bases are extremely easy to implement and allow fully adaptive subdivisions. We give examples of functions defined on the sphere, such as topographic data, bidirectional reflection distribution functions, and illumination, and show how they can be efficiently represented with spherical wavelets. 

BibTeX:
@inproceedings{Schroder_P_1995_pacmsiggraph_sph_werfs, author = {P. Schröder and W. Sweldens}, title = {Spherical wavelets: efficiently representing functions on the sphere}, booktitle = {Proc. SIGGRAPH Int. Conf. Comput. Graph. Interactive Tech.}, year = {1995}, pages = {161172}, doi = {http://dx.doi.org/10.1145/218380.218439} } 

Secker, A. & Taubman, D.  Liftingbased invertible motion adaptive transform (LIMAT) framework for highly scalable video compression [BibTeX] 
2003  IEEE Trans. Image Process. Vol. 12(12), pp. 15301542 
article  
BibTeX:
@article{Secker_A_2003_jieeetip_lif_bimatlimatfhsvc, author = {A. Secker and D. Taubman}, title = {Liftingbased invertible motion adaptive transform (LIMAT) framework for highly scalable video compression}, journal = {IEEE Trans. Image Process.}, year = {2003}, volume = {12}, number = {12}, pages = {15301542} } 

Selesnick, I.W.  The Characterization and Design of Hilbert Transform Pairs of Wavelet Bases [BibTeX] 
2001  Proc. Conf. Inform. Sciences Syst.  inproceedings  
BibTeX:
@inproceedings{Selesnick_I_2001_pciss_cha_dhtpwb, author = {Selesnick, I. W.}, title = {The Characterization and Design of Hilbert Transform Pairs of Wavelet Bases}, booktitle = {Proc. Conf. Inform. Sciences Syst.}, year = {2001} } 

Selesnick, I.W.  Hilbert transform pairs of wavelet bases  2001  #spl# Vol. 8(6)IEEE Trans. on Signal Processing, pp. 170173 
article  DOI 
Abstract: This paper considers the design of pairs of wavelet bases where the wavelets form a Hilbert transform pair. The derivation is based on the limit functions defined by the infinite product formula. It is found that the scaling filters should be offset from one another by a half sample. This gives an alternative derivation and explanation for the result by Kingsbury (1999), that the dualtree DWT is (nearly) shiftinvariant when the scaling filters satisfy the same offset  
BibTeX:
@article{Selesnick_I_2001_spl_hil_tpwb, author = {Selesnick, I. W. }, title = {Hilbert transform pairs of wavelet bases}, booktitle = {IEEE Trans. on Signal Processing}, journal = {#spl#}, year = {2001}, volume = {8}, number = {6}, pages = {170173}, doi = {http://dx.doi.org/10.1109/97.923042} } 

Selesnick, I.W., Baraniuk, R.G. & Kingsbury, N.G.  The dualtree complex wavelet transform [BibTeX] 
2005  IEEE Signal Process. Mag. Vol. 22(6), pp. 123151 
article  
BibTeX:
@article{Selesnick_I_2005_spm_dua_tcwt, author = {Selesnick, I. W. and Baraniuk, R. G. and Kingsbury, N. G.}, title = {The dualtree complex wavelet transform}, journal = {IEEE Signal Process. Mag.}, year = {2005}, volume = {22}, number = {6}, pages = {123151} } 

Shapiro, J.M.  Embedded image coding using zerotrees of wavelet coefficients  1993  IEEE Trans. Signal Process. Vol. 41, pp. 34453462 
article  DOI 
Abstract: The embedded zerotree wavelet algorithm (EZW) is a simple, yet remarkably effective, image compression algorithm, having the property that the bits in the bit stream are generated in order of importance, yielding a fully embedded code. The embedded code represents a sequence of binary decisions that distinguish an image from the "null" image. Using an embedded coding algorithm, an encoder can terminate the encoding at any point thereby allowing a target rate or target distortion metric to be met exactly. Also, given a bit stream, the decoder can cease decoding at any point in the bit stream and still produce exactly the same image that would have been encoded at the bit rate corresponding to the truncated bit stream. In addition to producing a fully embedded bit stream, the EZW consistently produces compression results that are competitive with virtually all known compression algorithms on standard test images. Yet this performance is achieved with a technique that requires absolutely no training, no prestored tables or codebooks, and requires no prior knowledge of the image source. The EZW algorithm is based on four key concepts: (1) a discrete wavelet transform or hierarchical subband decomposition, (2) prediction of the absence of significant information across scales by exploiting the selfsimilarity inherent in images, (3) entropycoded successiveapproximation quantization, and (4) universal lossless data compression which is achieved via adaptive arithmetic coding  
BibTeX:
@article{Shapiro_J_1993_jieeetsp_emb_iczwc, author = {Shapiro, J. M.}, title = {Embedded image coding using zerotrees of wavelet coefficients}, journal = {IEEE Trans. Signal Process.}, year = {1993}, volume = {41}, pages = {34453462}, doi = {http://dx.doi.org/10.1109/78.258085} } 

Shen, L., Papadakis, M., Kakadiaris, I.A., Konstantinidis, I., Kouri, D. & Hoffman, D.  Image denoising using a tight frame  2006  IEEE Trans. Image Process. Vol. 15(5), pp. 12541263 
article  DOI 
Abstract: We present a general mathematical theory for lifting frames that allows us to modify existing filters to construct new ones that form Parseval frames. We apply our theory to design nonseparable Parseval frames from separable (tensor) products of a piecewise linear spline tight frame. These new frame systems incorporate the weighted average operator, the Sobel operator, and the Laplacian operator in directions that are integer multiples of 45°. A new image denoising algorithm is then proposed, tailored to the specific properties of these new frame filters. We demonstrate the performance of our algorithm on a diverse set of images with very encouraging results.  
BibTeX:
@article{Shen_L_2006_tsp_ima_dtf, author = {Shen, L. and Papadakis, M. and Kakadiaris, I. A. and Konstantinidis, I. and Kouri, D. and Hoffman, D.}, title = {Image denoising using a tight frame}, journal = {IEEE Trans. Image Process.}, year = {2006}, volume = {15}, number = {5}, pages = {12541263}, doi = {http://dx.doi.org/10.1109/TIP.2005.864240} } 

Shensa, M.J.  The discrete wavelet transform: wedding the à trous and Mallat algorithms  1992  IEEE Trans. Signal Process. Vol. 40(10), pp. 24642482 
article  DOI 
Abstract: In a general sense this paper represents an effort to clarify the relationship of discrete and continuous wavelet transforms. More narrowly, it focuses on bringing together two separately motivated implementations of the wavelet transform, the algorithme U trous and Mallat?s multiresolution decomposition. It is observed that these algorithms are both special cases of a single filter bank structure, the discrete wavelet transform, the behavior of which is governed by one?s choice of filters. In fact, the h trow algorithm, originally devised as a computationally efficient implementation, is more properly viewed as a nonorthonormal multiresolution algorithm for which the discrete wavelet transform is exact. Moreover, it is shown that the commonly used Lagrange i~ trous filters are in onetoone correspondence with the convolutional squares of the Daubechies filters for orthonormal wavelets of compact support. A systematic framework for the discrete wavelet transform is provided, and conditions are derived under which it computes the continuous wavelet transform exactly. Suitable filter constraints for finite energy and boundedness of the discrete transform are also derived. Finally, relevant signal processing parameters are examined, and it is remarked that orthonormality is balanced by restrictions on resolution. 

BibTeX:
@article{Shensa_M_1992_jieeetsp_dis_wtwatma, author = {Shensa, M. J.}, title = {The discrete wavelet transform: wedding the à trous and Mallat algorithms}, journal = {IEEE Trans. Signal Process.}, year = {1992}, volume = {40}, number = {10}, pages = {24642482}, doi = {http://dx.doi.org/10.1109/78.157290} } 

Shi, X., Castro, A.L.R., Manduchi, R. & Montgomery, R.  Rotational Invariant Operators based on Steerable Filter Banks [BibTeX] 
2006  #spl# Vol. 13(11) 
article  
BibTeX:
@article{Shi_X_2006_spl_rot_iosfb, author = {X. Shi and A. L. Ribeiro Castro and R. Manduchi and R. Montgomery}, title = {Rotational Invariant Operators based on Steerable Filter Banks}, journal = {#spl#}, year = {2006}, volume = {13}, number = {11} } 

Shukla, R., Dragotti, P.L., Do, M.N. & Vetterli, M.  RateDistorsion optimized treestructured compression algorithms for piecewise polynomial images [BibTeX] 
2005  IEEE Trans. Image Process. Vol. 14(3), pp. 343359 
article  
BibTeX:
@article{Shukla_R_2005_tip_rat_dotscappi, author = {Shukla, R. and Dragotti, P. L. and Do, M. N. and Vetterli, M.}, title = {RateDistorsion optimized treestructured compression algorithms for piecewise polynomial images}, journal = {IEEE Trans. Image Process.}, year = {2005}, volume = {14}, number = {3}, pages = {343359} } 

Simoncelli, E.P. & Farid, H.  Steerable wedge filters for local orientation analysis  1996  IEEE Trans. Image Process. Vol. 5(9), pp. 13771382 
article  DOI 
Abstract: Steerable filters have been used to analyze local orientation patterns in imagery. Such filters are typically based on directional derivatives, whose symmetry produces orientation responses that are periodic with period π, independent of image structure. We present a more general set of steerable filters that alleviate this problem  
BibTeX:
@article{Simoncelli_E_1996_tip_ste_wfloa, author = {E. P. Simoncelli and H. Farid}, title = {Steerable wedge filters for local orientation analysis}, journal = {IEEE Trans. Image Process.}, year = {1996}, volume = {5}, number = {9}, pages = {13771382}, doi = {http://dx.doi.org/10.1109/83.535851} } 

Simoncelli, E.P. & Freeman, W.T.  The steerable pyramid: a flexible architecture for multiscale derivative computation [BibTeX] 
1995  Proc. Int. Conf. Image Process., pp. 444447  inproceedings  
BibTeX:
@inproceedings{Simoncelli_E_1995_icip_ste_pfamdc, author = {Simoncelli, E. P. and Freeman, W. T.}, title = {The steerable pyramid: a flexible architecture for multiscale derivative computation}, booktitle = {Proc. Int. Conf. Image Process.}, year = {1995}, pages = {444447} } 

Simoncelli, E.P., Freeman, W.T., Adelson, E.H. & Heeger, D.J.  Shiftable Multiscale Transforms  1992  IEEE Trans. Inform. Theory Vol. 38(2), pp. 587607 
article  
Abstract: Orthogonal wavelet transforms have recently become a popular representation for multiscale signal and image analysis. One of the major drawbacks of these representations is their lack of translation invariance: the content of wavelet subbands is unstable under translations of the input signal. Wavelet transforms are also unstable with respect to dilations of the input signal, and in two dimensions, rotations of the input signal. We formalize these problems by defining a type of translation invariance that we call "shiftability". In the spatial domain, shiftability corresponds to a lack of aliasing; thus, the conditions under which the property holds are specified by the sampling theorem. Shiftability may also be considered in the context of other domains, particularly orientation and scale. We explore ``jointly shiftable'' transforms that are simultaneously shiftable in more than one domain. Two examples of jointly shiftable transforms are designed and implemented: a onedimensional transform that is jointly shiftable in position and scale, and a twodimensional transform that is jointly shiftable in position and orientation. We demonstrate the usefulness of these image representations for scalespace analysis, stereo disparity measurement, and image enhancement.  
BibTeX:
@article{Simoncelli_E_1992_tit_shi_mst, author = {Simoncelli, E. P. and Freeman, W. T. and Adelson, E. H. and Heeger, D. J.}, title = {Shiftable Multiscale Transforms}, journal = {IEEE Trans. Inform. Theory}, year = {1992}, volume = {38}, number = {2}, pages = {587607}, note = {Special Issue on Wavelets} } 

Smith, M.J.T. & Barnwell, T.P.  A procedure for designing exact reconstruction filter banks for tree structured subband coders  1984  Vol. 9Proc. Int. Conf. Acoust. Speech Signal Process., pp. 421424 
inproceedings  
Abstract: In recent years, treestructured analysis/reconstruction systems have been extensively studied for use in subband coders for speech. In such systems, it is important that the individual channel signals be decimated in such a way that the number of samples coded and transmitted does not exceed the number of samples in the original speech signal. Under this constraint, the systems presented in the past have sought to remove the aliasing distortion while minimizing the overall analysis/reconstruction distortion. In this paper, it is shown that it is possible to design treestructured analysis/reconstruction systems which meet the sampling rate condition and which also result in exact reconstruction of the input signal. This paper develops the conditions for exact reconstruction and presents a general method for designing the corresponding high quality analysis and reconstruction filters.  
BibTeX:
@inproceedings{Smith_M_1984_picassp_pro_derfbtssc, author = {Smith, M. J. T. and Barnwell, T. P.}, title = {A procedure for designing exact reconstruction filter banks for tree structured subband coders}, booktitle = {Proc. Int. Conf. Acoust. Speech Signal Process.}, year = {1984}, volume = {9}, pages = {421424} } 

Smith, M.J.T. & Chung, W.C.L.  Recursive timevarying filter banks for subband image coding  1995  IEEE Trans. Image Process. Vol. 4(7), pp. 885895 
article  DOI 
Abstract: Filter banks, subband/wavelets, and multiresolution decompositions that employ recursive filters have been considered previously and are recognized for their efficiency in partitioning the frequency spectrum. This paper presents an analysis of a new infinite impulse response (IIR) filter bank in which these computationally efficient filters may be changed adaptively in response to the input. The new filter bank framework is presented and discussed in the context of subband image coding. In the absence of quantization errors, exact reconstruction can be achieved. By the proper choice of an adaptation scheme, it is shown that recursive linear timevarying (LTV) filter banks can yield improvement over conventional ones  
BibTeX:
@article{Smith_M_1995_jieeetip_rec_tvfbsic, author = {Smith, M. J. T. and Chung, W. C.L.}, title = {Recursive timevarying filter banks for subband image coding}, journal = {IEEE Trans. Image Process.}, year = {1995}, volume = {4}, number = {7}, pages = {885895}, doi = {http://dx.doi.org/10.1109/83.392331} } 

van Spaendonck, R., Fernandes, F., Coates, M. & Burrus, C.  Nonredundant, Directionally selective, complex wavelets  2000  Vol. 2Proc. Int. Conf. Image Process., pp. 379382 
inproceedings  DOI 
Abstract: Poor directional selectivity, a major disadvantage of the 2D separable discrete wavelet transform (DWT), has heretofore been circumvented either by using highly redundant, nonseparable wavelet transforms or by using restrictive designs to obtain a pair of wavelet trees. In this paper, we demonstrate that superior directional selectivity may be obtained with no redundancy in any separable wavelet transform. We achieve this by projecting the wavelet transform coefficients onto the Softy space of signals and decimating before processing. A novel reconstruction step guarantees perfect reconstruction within this criticallysampled framework.  
BibTeX:
@inproceedings{VanSpaendonck_R_2000_picip_non_rdscw, author = {van Spaendonck, R. and Fernandes, F. and Coates, M. and Burrus, C.}, title = {Nonredundant, Directionally selective, complex wavelets}, booktitle = {Proc. Int. Conf. Image Process.}, year = {2000}, volume = {2}, pages = {379382}, doi = {http://dx.doi.org/10.1109/ICIP.2000.899399} } 

Starck, J.L., Candès, E.J. & Donoho, D.L.  The curvelet transform for image denoising  2002  IEEE Trans. Image Process. Vol. 11(6), pp. 670685 
article  DOI 
Abstract: We describe approximate digital implementations of two new mathematical transforms, namely, the ridgelet transform and the curvelet transform. Our implementations offer exact reconstruction, stability against perturbations, ease of implementation, and low computational complexity. A central tool is Fourierdomain computation of an approximate digital Radon transform. We introduce a very simple interpolation in the Fourier space which takes Cartesian samples and yields samples on a rectopolar grid, which is a pseudopolar sampling set based on a concentric squares geometry. Despite the crudeness of our interpolation, the visual performance is surprisingly good. Our ridgelet transform applies to the Radon transform a special overcomplete wavelet pyramid whose wavelets have compact support in the frequency domain. Our curvelet transform uses our ridgelet transform as a component step, and implements curvelet subbands using a filter bank of a` trous wavelet filters. Our philosophy throughout is that transforms should be overcomplete, rather than critically sampled. We apply these digital transforms to the denoising of some standard images embedded in white noise. In the tests reported here, simple thresholding of the curvelet coefficients is very competitive with "state of the art" techniques based on wavelets, including thresholding of decimated or undecimated wavelet transforms and also including treebased Bayesian posterior mean methods. Moreover, the curvelet reconstructions exhibit higher perceptual quality than waveletbased reconstructions, offering visually sharper images and, in particular, higher quality recovery of edges and of faint linear and curvilinear features. Existing theory for curvelet and ridgelet transforms suggests that these new approaches can outperform wavelet methods in certain image reconstruction problems. The empirical results reported here are in encouraging agreement  
BibTeX:
@article{Starck_J_2002_tip_cur_tid, author = {Starck, J.L. and Candès, E. J. and Donoho, D. L.}, title = {The curvelet transform for image denoising}, journal = {IEEE Trans. Image Process.}, year = {2002}, volume = {11}, number = {6}, pages = {670685}, doi = {http://dx.doi.org/10.1109/TIP.2002.1014998} } 

Starck, J.L., Elad, M. & Donoho, D.L.  Redundant Multiscale Transforms and their Application for Morphological Component Analysis [BibTeX] 
2004  Adv. Imag. Electron Phys. Vol. 132, pp. 287348 
article  
BibTeX:
@article{Starck_J_2004_jadvimagelectronphys_red_tamca, author = {J.L. Starck and M. Elad and D. L. Donoho}, title = {Redundant Multiscale Transforms and their Application for Morphological Component Analysis}, journal = {Adv. Imag. Electron Phys.}, year = {2004}, volume = {132}, pages = {287348} } 

Starck, J.L., Moudden, Y., Abrial, P. & Nguyen, M.  Wavelets, ridgelets and curvelets on the sphere  2006  Astron. Astrophys. Vol. 446, pp. 11911204 
article  
Abstract: We present in this paper new multiscale transforms on the sphere, namely the isotropic undecimated wavelet transform, the pyramidal wavelet transform, the ridgelet transform and the curvelet transform. All of these transforms can be inverted i.e. we can exactly reconstruct the original data from its coefficients in either representation. Several applications are described. We show how these transforms can be used in denoising and especially in a Combined Filtering Method, which uses both the wavelet and the curvelet transforms, thus benefiting from the advantages of both transforms. An application to component separation from multichannel data mapped to the sphere is also described in which we take advantage of moving to a wavelet representation.  
BibTeX:
@article{Starck_J_2006_jastronastrophys_wav_rcs, author = {Starck, J.L. and Moudden, Y. and Abrial, P. and Nguyen, M.}, title = {Wavelets, ridgelets and curvelets on the sphere}, journal = {Astron. Astrophys.}, year = {2006}, volume = {446}, pages = {11911204} } 

Starck, J.L., Murtagh, F. & Fadili, J.M.  Sparse Image and Signal Processing: Wavelets, Curvelets, Morphological Diversity  2010  book  
Abstract: This book presents the state of the art in sparse and multiscale image and signal processing, covering linear multiscale transforms, such as wavelet, ridgelet, or curvelet transforms, and nonlinear multiscale transforms based on the median and mathematical morphology operators. Recent concepts of sparsity and morphological diversity are described and exploited for various problems such as denoising, inverse problem regularization, sparse signal decomposition, blind source separation, and compressed sensing. This book weds theory and practice in examining applications in areas such as astronomy, biology, physics, digital media, and forensics. A final chapter explores a paradigm shift in signal processing, showing that previous limits to information sampling and extraction can be overcome in very significant ways. Matlab and IDL code accompany these methods and applications to reproduce the experiments and illustrate the reasoning and methodology of the research available for download at the associated Web site.  
BibTeX:
@book{Starck_J_2010_book_spa_ispwcmd, author = {Starck, J.L. and Murtagh, F. and Fadili, J. M.}, title = {Sparse Image and Signal Processing: Wavelets, Curvelets, Morphological Diversity}, publisher = {Cambridge University Press}, year = {2010} } 

Steffen, P., Heller, P.N., Gopinath, R.A. & Burrus, C.S.  Theory of regular $M$band wavelet bases [BibTeX] 
1993  IEEE Trans. Signal Process. Vol. 41(12), pp. 34973511 
article  
BibTeX:
@article{Steffen_P_1993_tsp_the_rmbwb, author = {Steffen, P. and Heller, P. N. and Gopinath, R. A. and Burrus, C. S.}, title = {Theory of regular $M$band wavelet bases}, journal = {IEEE Trans. Signal Process.}, year = {1993}, volume = {41}, number = {12}, pages = {34973511} } 

Storath, M.  Directional Multiscale Amplitude and Phase Decomposition by the Monogenic Curvelet Transform  2011  SIAM J. Imaging Sci. Vol. 4(1), pp. 5778 
article  DOI 
Abstract: We reconsider the continuous curvelet transform from a signal processing point of view. We show that the analyzing elements of the curvelet transform, the curvelets, can be understood as analytic signals in the sense of the partial Hilbert transform. We then generalize the usual curvelets by the monogenic curvelets, which are analytic signals in the sense of the Riesz transform. They yield a new transform, called the monogenic curvelet transform. This transform has the useful property that it behaves at the fine scales like the usual curvelet transform and at the coarse scales like the monogenic wavelet transform. In particular, the new transform is highly anisotropic at the fine scales and yields a wellinterpretable amplitude/phase decomposition of the transform coefficients over all scales. We illustrate the advantage of this new directional multiscale amplitude/phase decomposition for the estimation of directional regularity.  
BibTeX:
@article{Storath_M_2011_jsiamjimagingsci_dir_mapdmct, author = {Storath, M.}, title = {Directional Multiscale Amplitude and Phase Decomposition by the Monogenic Curvelet Transform}, journal = {SIAM J. Imaging Sci. }, year = {2011}, volume = {4}, number = {1}, pages = {5778}, doi = {http://dx.doi.org/10.1137/100803924} } 

Sweldens, W.  The lifting scheme: a construction of second generation wavelets [BibTeX] 
1997  SIAM J. Math. Anal. Vol. 29(2), pp. 511546 
article  
BibTeX:
@article{Sweldens_W_1997_jsiammathanal_lif_scsgw, author = {W. Sweldens}, title = {The lifting scheme: a construction of second generation wavelets}, journal = {SIAM J. Math. Anal.}, year = {1997}, volume = {29}, number = {2}, pages = {511546} } 

Sweldens, W.  The lifting scheme: a customdesign construction of biorthogonal wavelets [BibTeX] 
1996  Appl. Comp. Harm. Analysis Vol. 3(2), pp. 186200 
article  
BibTeX:
@article{Sweldens_W_1996_jacha_lif_scdcbw, author = {W. Sweldens}, title = {The lifting scheme: a customdesign construction of biorthogonal wavelets}, journal = {Appl. Comp. Harm. Analysis}, year = {1996}, volume = {3}, number = {2}, pages = {186200} } 

Szatmáry, K. & Vinkó, J.  Periodicities of the light curve of the semiregular variable star Y Lyncis [BibTeX] 
1992  Mon. Not. Roy. Astron. Soc. Vol. 256, pp. 321328 
article  
BibTeX:
@article{Szatmary_K_1992_jmonnotroyastronsoc_per_lcsvsyl, author = {Szatmáry, K. and Vinkó, J.}, title = {Periodicities of the light curve of the semiregular variable star Y Lyncis}, journal = {Mon. Not. Roy. Astron. Soc.}, year = {1992}, volume = {256}, pages = {321328} } 

Tanaka, T.  A direct design of oversampled perfect reconstruction FIR filter banks of 50overlapping filters [BibTeX] 
2006  IEEE Trans. Signal Process. Vol. 54(8), pp. 30113022 
article  DOI 
BibTeX:
@article{Tanaka_T_2006_tsp_dir_doprfirfbfof, author = {Tanaka, T.}, title = {A direct design of oversampled perfect reconstruction FIR filter banks of 50overlapping filters}, journal = {IEEE Trans. Signal Process.}, year = {2006}, volume = {54}, number = {8}, pages = {30113022}, doi = {http://dx.doi.org/10.1109/TSP.2006.875384} } 

Tanaka, T. & Yamashita, Y.  The generalized lapped pseudobiorthogonal transform: Oversampled linearphase perfect reconstruction filter banks with lattice structures [BibTeX] 
2004  IEEE Trans. Signal Process. Vol. 52(2), pp. 434446 
article  
BibTeX:
@article{Tanaka_T_2004_tsp_gen_lpbtolpprfbls, author = {Tanaka, T. and Yamashita, Y.}, title = {The generalized lapped pseudobiorthogonal transform: Oversampled linearphase perfect reconstruction filter banks with lattice structures}, journal = {IEEE Trans. Signal Process.}, year = {2004}, volume = {52}, number = {2}, pages = {434446} } 

Tanaka, Y., Hasegawa, M., Kato, S., Ikehara, M. & Nguyen, T.Q.  Adaptive Directional Wavelet Transform Based on Directional Prefiltering  2010  IEEE Trans. Image Process. Vol. 19(4), pp. 934945 
article  DOI 
Abstract: This paper proposes an efficient approach for adaptive directional wavelet transform (WT) based on directional prefiltering. Although the adaptive directional WT is able to transform an image along diagonal orientations as well as traditional horizontal and vertical directions, it sacrifices computation speed for good image coding performance. We present two efficient methods to find the best transform directions by prefiltering using 2D filter bank or 1D directional WT along two fixed directions. The proposed direction calculation methods achieve comparable image coding performance comparing to the conventional one with less complexity. Furthermore, transform direction data of the proposed method can be used for contentbased image retrieval to increase retrieval ratio.  
BibTeX:
@article{Tanaka_Y_2010_jieeetip_ada_dwtbdp, author = {Tanaka, Y. and Hasegawa, M. and Kato, S. and Ikehara, M. and Nguyen, T. Q.}, title = {Adaptive Directional Wavelet Transform Based on Directional Prefiltering}, journal = {IEEE Trans. Image Process.}, year = {2010}, volume = {19}, number = {4}, pages = {934945}, doi = {http://dx.doi.org/10.1109/TIP.2009.2038820} } 

Tanaka, Y., Ikehara, M. & Nguyen, T.Q.  Multiresolution Image Representation Using Combined 2D and 1D Directional Filter Banks  2009  IEEE Trans. Image Process. Vol. 18(2), pp. 269280 
article  DOI 
Abstract: In this paper, effective multiresolution image representations using a combination of 2D filter bank (FB) and directional wavelet transform (WT) are presented. The proposed methods yield simple implementation and low computation costs compared to previous 1D and 2D FB combinations or adaptive directional WT methods. Furthermore, they are nonredundant transforms and realize quad tree like multiresolution representations. In applications on nonlinear approximation, image coding, and denoising, the proposed filter banks show visual quality improvements and have higher PSNR than the conventional separable WT or the contourlet.  
BibTeX:
@article{Tanaka_Y_2009_tip_mul_irc2d1ddfb, author = {Tanaka, Y. and Ikehara, M. and Nguyen, T. Q.}, title = {Multiresolution Image Representation Using Combined 2D and 1D Directional Filter Banks}, journal = {IEEE Trans. Image Process.}, year = {2009}, volume = {18}, number = {2}, pages = {269280}, doi = {http://dx.doi.org/10.1109/TIP.2008.2008078} } 

Taubman, D.  Adaptive, nonseparable lifting transforms for image compression  1999  Vol. 3Proc. Int. Conf. Image Process., pp. 772776 
inproceedings  DOI 
Abstract: In the context of high performance image compression algorithms, such as that emerging as the JPEG 2000 standard, the wavelet transform has demonstrated excellent compression performance with natural images. Like all waveform coding techniques, however, performance suffers in the neighbourhood of oriented edges clad with artificial imagery such as text and graphics. In this paper, we explore some of the opportunities offered by the framework of lifting for developing adaptive wavelet transforms to improve performance under these conditions  
BibTeX:
@inproceedings{Taubman_D_1999_picip_ada_nsltic, author = {Taubman, D.}, title = {Adaptive, nonseparable lifting transforms for image compression}, booktitle = {Proc. Int. Conf. Image Process.}, year = {1999}, volume = {3}, pages = {772776}, doi = {http://dx.doi.org/10.1109/ICIP.1999.817221} } 

Taubman, D. & Zakhor, A.  Orientation adaptive subband coding of images  1994  IEEE Trans. Image Process. Vol. 3(4), pp. 421437 
article  DOI 
Abstract: In the subband coding of images, directionality of image features has thus far been exploited very little. The proposed subband coding scheme utilizes orientation of local image features to avoid the highly objectionable Gibbslike phenomena observed at reconstructed image edges with conventional subband schemes at low bit rates, At comparable bit rates, the subjective image quality obtained by our orientation adaptive scheme is considerably enhanced over a conventional separable subband coding scheme, as well as other separable approaches such as the JPEG compression standard  
BibTeX:
@article{Taubman_D_1994_jieeetip_ori_asci, author = {Taubman, D. and Zakhor, A.}, title = {Orientation adaptive subband coding of images}, journal = {IEEE Trans. Image Process.}, year = {1994}, volume = {3}, number = {4}, pages = {421437}, doi = {http://dx.doi.org/10.1109/83.298396} } 

Taubman, D.S. & Marcellin, M.W.  JPEG2000: Image Compression Fundamentals, Standards and Practice [BibTeX] 
2002  book  
BibTeX:
@book{Taubman_D_2002_book_jpe_2000icfsp, author = {Taubman, D. S. and Marcellin, M. W.}, title = {JPEG2000: Image Compression Fundamentals, Standards and Practice}, publisher = {Kluwer Academic}, year = {2002} } 

Treitel, S. & Shanks, J.L.  The Design of Multistage Separable Planar Filters  1971  IEEE Trans. Geosci. Electron. Vol. 9(1), pp. 10627 
article  DOI 
Abstract: A twodimensional, or planar, digital filter can be described in terms of its planar response function, which is in the form of a matrix of weighting coefficients, or filter array. In many instances the dimensions of these matrices are so large that their implementation as ordinary planar convolutional filters becomes computationally inefficient. It is possible to expand the given coefficient matrix into a finite and convergent sum of matrixvalued stages. Each stage can be separated with no error into the product of an mlength column vector multiplied into an nlength row vector, where m is the number of rows and n is the number of columns of the original filter array. Substantial savings in computer storage and speed result if the given filter array can be represented with a tolerably small error by the first few stages of the expansion. Since each constituent stage consists of two vectorvalued factors, further computational economies accrue if the onedimensional sequences described by these vectors are in turn approximated by onedimensional recursive filters. Two geophysical examples have been selected to illustrate how the present design techniques may be reduced to practice.  
BibTeX:
@article{Treitel_S_1971_jieeetge_des_mspf, author = {Treitel, S. and Shanks, J. L.}, title = {The Design of Multistage Separable Planar Filters}, journal = {IEEE Trans. Geosci. Electron.}, year = {1971}, volume = {9}, number = {1}, pages = {10627}, doi = {http://dx.doi.org/10.1109/TGE.1971.271457} } 

Tropp, J.A.  Just relax: convex programming methods for identifying sparse signals in noise  2006  IEEE Trans. Inform. Theory Vol. 52(3), pp. 10301051 
article  DOI 
Abstract: This paper studies a difficult and fundamental problem that arises throughout electrical engineering, applied mathematics, and statistics. Suppose that one forms a short linear combination of elementary signals drawn from a large, fixed collection. Given an observation of the linear combination that has been contaminated with additive noise, the goal is to identify which elementary signals participated and to approximate their coefficients. Although many algorithms have been proposed, there is little theory which guarantees that these algorithms can accurately and efficiently solve the problem. This paper studies a method called convex relaxation, which attempts to recover the ideal sparse signal by solving a convex program. This approach is powerful because the optimization can be completed in polynomial time with standard scientific software. The paper provides general conditions which ensure that convex relaxation succeeds. As evidence of the broad impact of these results, the paper describes how convex relaxation can be used for several concrete signal recovery problems. It also describes applications to channel coding, linear regression, and numerical analysis  
BibTeX:
@article{Tropp_J_2006_tit_jus_rcpmissn, author = {Tropp, J. A.}, title = {Just relax: convex programming methods for identifying sparse signals in noise}, journal = {IEEE Trans. Inform. Theory}, year = {2006}, volume = {52}, number = {3}, pages = {10301051}, doi = {http://dx.doi.org/10.1109/TIT.2005.864420} } 

Tropp, J.A.  Greed is good: algorithmic results for sparse approximation  2004  IEEE Trans. Inform. Theory Vol. 50(10), pp. 22312242 
article  DOI 
Abstract: This article presents new results on using a greedy algorithm, orthogonal matching pursuit (OMP), to solve the sparse approximation problem over redundant dictionaries. It provides a sufficient condition under which both OMP and Donoho's basis pursuit (BP) paradigm can recover the optimal representation of an exactly sparse signal. It leverages this theory to show that both OMP and BP succeed for every sparse input signal from a wide class of dictionaries. These quasiincoherent dictionaries offer a natural generalization of incoherent dictionaries, and the cumulative coherence function is introduced to quantify the level of incoherence. This analysis unifies all the recent results on BP and extends them to OMP. Furthermore, the paper develops a sufficient condition under which OMP can identify atoms from an optimal approximation of a nonsparse signal. From there, it argues that OMP is an approximation algorithm for the sparse problem over a quasiincoherent dictionary. That is, for every input signal, OMP calculates a sparse approximant whose error is only a small factor worse than the minimal error that can be attained with the same number of terms.  
BibTeX:
@article{Tropp_J_2005_jieeetit_gre_garsa, author = {Tropp, J. A.}, title = {Greed is good: algorithmic results for sparse approximation}, journal = {IEEE Trans. Inform. Theory}, year = {2004}, volume = {50}, number = {10}, pages = {22312242}, doi = {http://dx.doi.org/10.1109/TIT.2004.834793} } 

Unser, M., Chenouard, N. & Ville, D.V.D.  Steerable Pyramids and Tight Wavelet Frames in $L_2(R^d)$  2011  IEEE Trans. Image Process.  article  
Abstract: We present a functional framework for the design of tight steerable wavelet frames in any number of dimensions. The 2D version of the method can be viewed as a generalization of Simoncelli's steerable pyramid that gives access to a larger palette of steerable wavelets via a suitable parametrization. The backbone of our construction is a primal isotropic wavelet frame that provides the multiresolution decomposition of the signal. The steerable wavelets are obtained by applying a onetomany mapping (Nthorder generalized Riesz transform) to the primal ones. The shaping of the steerable wavelets is controlled by an M × M unitary matrix (where M is the number of wavelet channels) that can be selected arbitrarily; this allows for a much wider range of solutions than the traditional equiangular configuration (steerable pyramid). We give a complete functional description of these generalized wavelet transforms and derive their steering equations. We describe some concrete examples of transforms, including some built around a Mallattype multiresolution analysis of $L_2(R^d)$, and provide a fast FFTbased decomposition algorithm. We also propose a principalcomponentbased method for signaladapted wavelet design. Finally, we present some illustrative examples together with a comparison of the denoising performance of various brands of steerable transforms. The results are in favor of an optimized wavelet design (equalized PCA) which consistently performs best.  
BibTeX:
@article{Unser_M_2011_jieeetip_ste_ptwfl2rd, author = {M. Unser and N. Chenouard and D. Van De Ville}, title = {Steerable Pyramids and Tight Wavelet Frames in $L_2(R^d)$}, journal = {IEEE Trans. Image Process.}, year = {2011}, note = {Preprint, in press} } 

Unser, M., Sage, D. & Van De Ville, D.  Multiresolution Monogenic Signal Analysis Using the RieszLaplace Wavelet Transform  2009  IEEE Trans. Image Process. Vol. 18(11), pp. 24022418 
article  DOI 
Abstract: The monogenic signal is the natural 2D counterpart of the 1D analytic signal. We propose to transpose the concept to the wavelet domain by considering a complexified version of the Riesz transform which has the remarkable property of mapping a realvalued (primary) wavelet basis of $L_2(BBR^2)$ into a complex one. The Riesz operator is also steerable in the sense that it give access to the Hilbert transform of the signal along any orientation. Having set those foundations, we specify a primary polyharmonic spline wavelet basis of $L_2(BBR^2)$ that involves a single Mexicanhatlike mother wavelet (Laplacian of a Bspline). The important point is that our primary wavelets are quasiisotropic: they behave like multiscale versions of the fractional Laplace operator from which they are derived, which ensures steerability. We propose to pair these realvalued basis functions with their complex Riesz counterparts to specify a multiresolution monogenic signal analysis. This yields a representation where each wavelet index is associated with a local orientation, an amplitude and a phase. We give a corresponding waveletdomain method for estimating the underlying instantaneous frequency. We also provide a mechanism for improving the shift and rotationinvariance of the wavelet decomposition and show how to implement the transform efficiently using perfect reconstruction filterbanks. We illustrate the specific feature extraction capabilities of the representation and present novel examples of waveletdomain processing; in particular, a robust, tensorbased analysis of directional image patterns, the demodulation of interferograms, and the reconstruction of digital holograms.  
BibTeX:
@article{Unser_M_2009_tip_mul_msarlwt, author = {Unser, M. and Sage, D. and Van De Ville, D.}, title = {Multiresolution Monogenic Signal Analysis Using the RieszLaplace Wavelet Transform}, journal = {IEEE Trans. Image Process.}, year = {2009}, volume = {18}, number = {11}, pages = {24022418}, doi = {http://dx.doi.org/10.1109/TIP.2009.2027628} } 

Unser, M. & Van De Ville, D.  HigherOrder Riesz Transforms and Steerable Wavelet Frames  2009  Proc. Int. Conf. Image Process., pp. 37573760  inproceedings  
Abstract: We introduce an Nthorder extension of the Riesz transform in d dimensions. We prove that this generalized transform has the following remarkable properties: shiftinvariance, scaleinvariance, innerproduct preservation, and steerability. The pleasing consequence is that the transform maps any primary wavelet frame (or basis) of L2(?d) into another "steerable" wavelet frame, while preserving the frame bounds. The concept provides a rigorous functional counterpart to Simoncelli's steerable pyramid whose construction was entirely based on digital filter design. The proposed mechanism allows for the specification of wavelets with any order of steerability in any number of dimensions; it also yields a perfect reconstruction filterbank algorithm. We illustrate the method using a Mexicanhatlike polyharmonic spline wavelet transform as our primary frame.  
BibTeX:
@inproceedings{Unser_M_2009_picip_hig_ortswf, author = {M. Unser and Van De Ville, D.}, title = {HigherOrder Riesz Transforms and Steerable Wavelet Frames}, booktitle = {Proc. Int. Conf. Image Process.}, year = {2009}, pages = {37573760} } 

Unser, M. & Van De Ville, D.  The Pairing of a Wavelet Basis with a Mildly Redundant Analysis via Subband Regression  2008  IEEE Trans. Image Process. Vol. 17(11), pp. 20402052 
article  
Abstract: A distinction is usually made between wavelet bases and wavelet frames. The former are associated with a onetoone representation of signals, which is somewhat constrained but most efficient computationally. The latter are overcomplete, but they offer advantages in terms of flexibility (shape of the basis functions) and shiftinvariance. In this paper, we propose a framework for improved wavelet analysis based on an appropriate pairing of a wavelet basis with a mildly redundant version of itself (frame). The processing is accomplished in four steps: 1) redundant wavelet analysis, 2) waveletdomain processing, 3) projection of the results onto the wavelet basis, and 4) reconstruction of the signal from its nonredundant wavelet expansion. The wavelet analysis is pyramidlike and is obtained by simple modification of Mallat's filterbank algorithm (e.g., suppression of the downsampling in the wavelet channels only). The key component of the method is the subband regression filter (Step 3) which computes a wavelet expansion that is maximally consistent in the least squares sense with the redundant wavelet analysis. We demonstrate that this approach significantly improves the performance of softthreshold wavelet denoising with a moderate increase in computational cost. We also show that the analysis filters in the proposed framework can be adjusted for improved feature detection; in particular, a new quincunx Mexicanhatlike wavelet transform that is fully reversible and essentially behaves the (??2)th Laplacian of a Gaussian.  
BibTeX:
@article{Unser_M_2008_tip_pai_wbmrasr, author = {Unser, M. and Van De Ville, D.}, title = {The Pairing of a Wavelet Basis with a Mildly Redundant Analysis via Subband Regression}, journal = {IEEE Trans. Image Process.}, year = {2008}, volume = {17}, number = {11}, pages = {20402052} } 

Vaidyanathan, P.P.  Multirate systems and filter banks [BibTeX] 
1993  book  
BibTeX:
@book{Vaidyanathan_P_1993_book_mul_sfb, author = {Vaidyanathan, P. P.}, title = {Multirate systems and filter banks}, publisher = {Prentice Hall}, year = {1993} } 

Valois, R.L.D., Albrecht, D.G. & Thorell, L.G.  Spatial frequency selectivity of cells in macaque visual cortex  1982  Vis. Res. Vol. 22(5), pp. 545559 
article  
Abstract: We measured the spatial frequency contrast sensitivity of cells in the primate striate cortex at two different eccentricities to provide quantitative statistics from a large population of cells. Distributions of the peak frequencies and bandwidths are presented and examined in relationship to (a) each other, (b) absolute contrast sensitivity, (c) orientation tuning, (d)retinal eccentricity, and (e) cell type. Simple and complex cells are examined in relationship to linear/nonlinear (that is, X/Y) properties; a procedure is described which provides a simple, reliable and quantitative method for classifying and describing striate cells. Among other things, it is shown that (a) many stirate cells have quite narrow spatial bandwidths and (b) at a given retinal eccentricity, the distribution of peak frequency covers a wide range of frequencies; these findings support the basic multiple channel notion. The orientation tuning and spatial frequency tuning which occurs at the level of striate cortex (in a positively correlated fashion) suggests that the cells might best be considered as twodimensional spatial filters.  
BibTeX:
@article{DeValois_R_1982_jvisres_spa_fscmvc, author = {R. L. De Valois and D. G. Albrecht and L. G. Thorell}, title = {Spatial frequency selectivity of cells in macaque visual cortex}, journal = {Vis. Res.}, year = {1982}, volume = {22}, number = {5}, pages = {545559} } 

Van De Ville, D. & Unser, M.  Complex Wavelet Bases, Steerability, and the MarrLike Pyramid  2008  IEEE Trans. Image Process. Vol. 17(11), pp. 20632080 
article  
Abstract: Our aim in this paper is to tighten the link between wavelets, some classical imageprocessing operators, and David Marr's theory of early vision. The cornerstone of our approach is a new complex wavelet basis that behaves like a smoothed version of the GradientLaplace operator. Starting from first principles, we show that a singlegenerator wavelet can be defined analytically and that it yields a semiorthogonal complex basis of L2(?2), irrespective of the dilation matrix used. We also provide an efficient FFTbased filterbank implementation. We then propose a slightly redundant version of the transform that is nearly translationinvariant and that is optimized for better steerability (Gaussianlike smoothing kernel). We call it the Marrlike wavelet pyramid because it essentially replicates the processing steps in Marr's theory of early vision.We use it to derive a primal wavelet sketch which is a compact description of the image by a multiscale, subsampled edge map. Finally, we provide an efficient iterative algorithm for the reconstruction of an image from its primal wavelet sketch.  
BibTeX:
@article{VanDeVille_D_2008_tip_com_wbsmlp, author = {Van De Ville, D. and Unser, M.}, title = {Complex Wavelet Bases, Steerability, and the MarrLike Pyramid}, journal = {IEEE Trans. Image Process.}, year = {2008}, volume = {17}, number = {11}, pages = {20632080} } 

Vandergheynst, P. & Frossard, P.  Image Coding Using Redundant Dictionaries [BibTeX] 
2006  Document and image compression  incollection  
BibTeX:
@incollection{Vandergheynst_P_2006_incoll_ima_crd, author = {Vandergheynst, P. and Frossard, P.}, title = {Image Coding Using Redundant Dictionaries}, booktitle = {Document and image compression}, publisher = {CRC Press}, year = {2006} } 

Vandergheynst, P. & Gobbers, J.F.  Directional dyadic wavelet transforms: design and algorithms  2002  IEEE Trans. Image Process. Vol. 11(4), pp. 363372 
article  DOI 
Abstract: We propose a simple and efficient technique for designing translation invariant dyadic wavelet transforms (DWTs) in two dimensions. Our technique relies on an extension of the work of DuvalDestin et al. (1993) where dyadic decompositions are constructed starting from the continuous wavelet transform. The main advantage of this framework is that it allows for a lot of freedom in designing twodimensional (2D) dyadic wavelets. We use this property to construct directional wavelets, whose orientation filtering capabilities are very important in image processing. We address the efficient implementation of these decompositions by constructing approximate QMFs through an L 2 optimization. We also propose and study an efficient implementation in the Fourier domain for dealing with large filters  
BibTeX:
@article{Vandergheynst_P_2002_jieeetip_dir_dwtda, author = {Vandergheynst, P. and Gobbers, J.F.}, title = {Directional dyadic wavelet transforms: design and algorithms}, journal = {IEEE Trans. Image Process.}, year = {2002}, volume = {11}, number = {4}, pages = {363372}, doi = {http://dx.doi.org/10.1109/TIP.2002.999670} } 

Vandergheynst, P. & Wiaux, Y.  Wavelets on the sphere [BibTeX] 
2010  Four short courses in harmonic analysis: wavelets, frames, timefrequency methods, and applications to signal and image analysis  incollection  
BibTeX:
@incollection{Vandergheynst_P_2010_incoll_wav_s, author = {P. Vandergheynst and Y. Wiaux}, title = {Wavelets on the sphere}, booktitle = {Four short courses in harmonic analysis: wavelets, frames, timefrequency methods, and applications to signal and image analysis}, publisher = {Birkhäuser}, year = {2010} } 

Velisavljević, V., BeferullLozano, B., Vetterli, M. & Dragotti, P.L.  Directionlets: Anisotropic multidirectional representation with separable filtering  2006  IEEE Trans. Image Process. Vol. 15(7), pp. 19161933 
article  DOI 
Abstract: In spite of the success of the standard wavelet transform (WT) in image processing in recent years, the efficiency of its representation is limited by the spatial isotropy of its basis functions built in the horizontal and vertical directions. Onedimensional (1D) discontinuities in images (edges and contours) that are very important elements in visual perception, intersect too many wavelet basis functions and lead to a nonsparse representation. To efficiently capture these anisotropic geometrical structures characterized by many more than the horizontal and vertical directions, a more complex multidirectional (MDIR) and anisotropic transform is required. We present a new latticebased perfect reconstruction and critically sampled anisotropic MDIR WT. The transform retains the separable filtering and subsampling and the simplicity of computations and filter design from the standard twodimensional WT, unlike in the case of some other directional transform constructions (e.g., curvelets, contourlets, or edgelets). The corresponding anisotropic basis functions (directionlets) have directional vanishing moments along any two directions with rational slopes. Furthermore, we show that this novel transform provides an efficient tool for nonlinear approximation of images, achieving the approximation power O(N1.55), which, while slower than the optimal rate O(N2), is much better than O(N1) achieved with wavelets, but at similar complexity.  
BibTeX:
@article{Velisavljevic_V_2006_tip_dir_amdrsf, author = {Velisavljević, V. and BeferullLozano, B. and Vetterli, M. and Dragotti, P. L.}, title = {Directionlets: Anisotropic multidirectional representation with separable filtering}, journal = {IEEE Trans. Image Process.}, year = {2006}, volume = {15}, number = {7}, pages = {19161933}, doi = {http://dx.doi.org/10.1109/TIP.2006.877076} } 

Vetterli, M. & Kovačević, J.  Wavelets and Subband Coding [BibTeX] 
1995  book  
BibTeX:
@book{Vetterli_M_1995_book_wav_sc, author = {M. Vetterli and J. Kovačević}, title = {Wavelets and Subband Coding}, publisher = {PrenticeHall}, year = {1995} } 

Wakin, M., Romberg, J., Choi, H. & Baraniuk, R.  Waveletdomain Approximation and Compression of Piecewise Smooth Images [BibTeX] 
2006  IEEE Trans. Image Process. Vol. 15(5), pp. 10711087 
article  
BibTeX:
@article{Wakin_M_2006_jieeetip_wav_dacpsi, author = {M. Wakin and J. Romberg and H. Choi and R. Baraniuk}, title = {Waveletdomain Approximation and Compression of Piecewise Smooth Images}, journal = {IEEE Trans. Image Process.}, year = {2006}, volume = {15}, number = {5}, pages = {10711087} } 

Wang, Z. & Simoncelli, E.P.  Translation Insensitive Image Similarity in Complex Wavelet Domain  2005  Vol. 2Proc. Int. Conf. Acoust. Speech Signal Process., pp. 573576 
inproceedings  DOI 
Abstract: We propose a complex wavelet domain image similarity measure, which is simultaneously insensitive to luminance change, contrast change and spatial translation. The key idea is to make use of the fact that these image distortions lead to consistent magnitude and/or phase changes of local wavelet coefficients. Since small scaling and rotation of images can be locally approximated by translation, the proposed measure also shows robustness to spatial scaling and rotation when these geometric distortions are small relative to the size of the wavelet filters. Compared with previous methods, the proposed measure is computationally efficient, and can evaluate the similarity of two images without a precise registration process at the front end. 

BibTeX:
@inproceedings{Wang_Z_2005_picassp_tra_iiscwd, author = {Wang, Z. and Simoncelli, E. P.}, title = {Translation Insensitive Image Similarity in Complex Wavelet Domain}, booktitle = {Proc. Int. Conf. Acoust. Speech Signal Process.}, year = {2005}, volume = {2}, pages = {573576}, doi = {http://dx.doi.org/10.1109/ICASSP.2005.1415469} } 

Watson, A.B.  The cortex transform: rapid computation of simulated neural images [BibTeX] 
1987  Comput. Vision Graph. Image Process. Vol. 39(3), pp. 311327 
article  DOI 
BibTeX:
@article{Watson_A_1987_jcomputvisiongraphimageprocess_cor_trcsni, author = {Watson, A. B.}, title = {The cortex transform: rapid computation of simulated neural images}, journal = {Comput. Vision Graph. Image Process.}, publisher = {Academic Press Professional, Inc.}, year = {1987}, volume = {39}, number = {3}, pages = {311327}, doi = {http://dx.doi.org/10.1016/S0734189X(87)801846} } 

Wedekind, J., Amavasai, B.P. & Dutton, K.  Steerable filters generated with the hypercomplex dualtree wavelet transform  2007  Proc. IEEE Int. Conf. Signal Process. Commun. (ICSPC), pp. 12911294  inproceedings  
Abstract: The use of wavelets in the image processing domain is still in its infancy, and largely associated with image compression. With the advent of the dualtree hypercomplex wavelet transform (DHWT) and its improved shift invariance and directional selectivity, applications in other areas of image processing are more conceivable. This paper discusses the problems and solutions in developing the DHWT and its inverse. It also offers a practical implementation of the algorithms involved. The aim of this work is to apply the DHWT in machine vision. Tentative work on a possible new way of feature extraction is presented. The paper shows that 2D hypercomplex basis wavelets can be used to generate steerable filters which allow rotation as well as translation.  
BibTeX:
@inproceedings{Wedekind_J_2007_picspc_ste_fghdtwt, author = {Wedekind, J. and Amavasai, B. P. and Dutton, K.}, title = {Steerable filters generated with the hypercomplex dualtree wavelet transform}, booktitle = {Proc. IEEE Int. Conf. Signal Process. Commun. (ICSPC)}, year = {2007}, pages = {12911294} } 

Weickert, J., Ishikawa, S. & Imiya, A.  Scalespace has been discovered in Japan [BibTeX] 
1997  (DIKUTR97/18)  techreport  
BibTeX:
@techreport{Weickert_J_1997_tr_sca_sdj, author = {Weickert, J. and Ishikawa, S. and Imiya, A.}, title = {Scalespace has been discovered in Japan}, year = {1997}, number = {DIKUTR97/18} } 

Weiss, J.  The Hilbert transform of wavelets are wavelets [BibTeX] 
1995  techreport  
BibTeX:
@techreport{Weiss_J_1995_tr_hil_tww, author = {Weiss, J.}, title = {The Hilbert transform of wavelets are wavelets}, year = {1995} } 

Wiaux, Y., Jacques, L. & Vandergheynst, P.  Correspondence principle between spherical and Euclidean wavelets [BibTeX] 
2005  Astrophys J. Vol. 632(1), pp. 1528 
article  
BibTeX:
@article{Wiaux_Y_2005_jastrophysj_cor_pbsew, author = {Wiaux, Y. and Jacques, L. and Vandergheynst, P.}, title = {Correspondence principle between spherical and Euclidean wavelets}, journal = {Astrophys J.}, publisher = {UChicago Press}, year = {2005}, volume = {632}, number = {1}, pages = {1528} } 

Wiaux, Y., Jacques, L., Vielva, P. & Vandergheynst, P.  Fast directional correlation on the sphere with steerable filters [BibTeX] 
2006  Astrophys J. Vol. 652(1), pp. 820832 
article  
BibTeX:
@article{Wiaux_Y_2006_jastrophysj_fas_dcssf, author = {Wiaux, Y. and Jacques, L. and Vielva, P. and Vandergheynst, P.}, title = {Fast directional correlation on the sphere with steerable filters}, journal = {Astrophys J.}, publisher = {UChicago Press}, year = {2006}, volume = {652}, number = {1}, pages = {820832} } 

Wiaux, Y., McEwen, J.D., Vandergheynst, P. & Blanc, O.  Exact reconstruction with directional wavelets on the sphere [BibTeX] 
2008  Mon. Not. Roy. Astron. Soc. Vol. 388(2), pp. 770788 
article  
BibTeX:
@article{Wiaux_Y_2008_jmonnotroyastronsoc_exa_rdws, author = {Wiaux, Y. and McEwen, J. D. and Vandergheynst, P. and Blanc, O.}, title = {Exact reconstruction with directional wavelets on the sphere}, journal = {Mon. Not. Roy. Astron. Soc.}, publisher = {London: Priestley and Weale, 1833}, year = {2008}, volume = {388}, number = {2}, pages = {770788} } 

Wiaux, Y., Vielva, P., Barreiro, R.B., MartinezGonzález, E. & Vandergheynst, P.  NonGaussianity analysis on local morphological measures of WMAP data  2008  Mon. Not. Roy. Astron. Soc. Vol. 385(2), pp. 939947 
article  
Abstract: The decomposition of a signal on the sphere with the steerable wavelet constructed from the second Gaussian derivative gives access to the orientation, signedintensity, and elongation of the signal's local features. In the present work, the nonGaussianity of the WMAP temperature data of the cosmic microwave background (CMB) is analyzed in terms of the first four moments of the statistically isotropic random fields associated with these local morphological measures, at wavelet scales corresponding to angular sizes between 27.5 arcminutes and 30 degrees on the celestial sphere. While no detection is made neither in the orientation analysis nor in the elongation analysis, a strong detection is made in the excess kurtosis of the signedintensity of the WMAP data. The nonGaussianity is observed with a significance level below 0.5% at a wavelet scale corresponding to an angular size around 10 degrees, and confirmed at neighbour scales. This supports a previous detection of an excess of kurtosis in the wavelet coefficient of the WMAP data with the axisymmetric Mexican hat wavelet (Vielva et al. 2004). Instrumental noise and foreground emissions are not likely to be at the origin of the excess of kurtosis. Largescale modulations of the CMB related to some unknown systematics are rejected as possible origins of the detection. The observed nonGaussianity may therefore probably be imputed to the CMB itself, thereby questioning the basic inflationary scenario upon which the present concordance cosmological model relies. Taking the CMB temperature angular power spectrum of the concordance cosmological model at face value, further analysis also suggests that this nonGaussianity is not confined to the directions on the celestial sphere with an anomalous signedintensity.  
BibTeX:
@article{Wiaux_Y_2008_jmonnotroyastronsoc_non_galmmwmapd, author = {Wiaux, Y. and Vielva, P. and Barreiro, R. B. and MartinezGonzález, E. and Vandergheynst, P.}, title = {NonGaussianity analysis on local morphological measures of WMAP data}, journal = {Mon. Not. Roy. Astron. Soc.}, publisher = {London: Priestley and Weale, 1833}, year = {2008}, volume = {385}, number = {2}, pages = {939947} } 

Wickerhauser, M.V.  INRIA Lectures on Wavelet Packet Algorithms  1991  , pp. 3199Lecture notes, INRIA  misc  
Abstract: We begin by defining continuous wavelet packets on R. These are squareintegrable functions with prescribed smoothness and other properties, which we shall develop to establish the main notions. Our construction will be directed toward numerical applications, so we will restrict ourselves to the quadrature mirror filter algorithm. Next we will define several discrete algorithms and explore their advantages and disadvantages. We will show the correspondence between wavelet packets and coefficients computed from sampled signals, and relate the convergence of this approximation to the smoothness of the signal. We will define information cost functions and the &quot;bestbasis&quot; method. We will count operations and consider practical matters like the memory requirements of the algorithms, periodizing, the spreading of the support of aperiodic wavelet packets, and the combinatorics of constructing wavelet packet bases of increasing generality. In parallel, we will develop smooth orthogonal local trigonometric transforms. These are properly considered transposes of wavelet packet methods, or alternatively conjugates of wavelet packet methods by the Fourier transform. We will describe both continuous and  
BibTeX:
@misc{Wickerhauser_M_1991_misc_lec_nwp, author = {Wickerhauser, M. V.}, title = {INRIA Lectures on Wavelet Packet Algorithms}, year = {1991}, pages = {3199} } 

Willett, R.M. & Nowak, R.D.  Platelets: a multiscale approach for recovering edges and surfaces in photonlimited medical imaging  2003  IEEE Trans. Med. Imag. Vol. 22(3), pp. 332350 
article  DOI 
Abstract: The nonparametric multiscale platelet algorithms presented in this paper, unlike traditional waveletbased methods, are both well suited to photonlimited medical imaging applications involving Poisson data and capable of better approximating edge contours. This paper introduces platelets, localized functions at various scales, locations, and orientations that produce piecewise linear image approximations, and a new multiscale image decomposition based on these functions. Platelets are well suited for approximating images consisting of smooth regions separated by smooth boundaries. For smoothness measured in certain Holder classes, it is shown that the error of mterm platelet approximations can decay significantly faster than that of mterm approximations in terms of sinusoids, wavelets, or wedgelets. This suggests that platelets may outperform existing techniques for image denoising and reconstruction. Fast, plateletbased, maximum penalized likelihood methods for photonlimited image denoising, deblurring and tomographic reconstruction problems are developed. Because platelet decompositions of Poisson distributed images are tractable and computationally efficient, existing image reconstruction methods based on expectationmaximization type algorithms can be easily enhanced with platelet techniques. Experimental results suggest that plateletbased methods can outperform standard reconstruction methods currently in use in confocal microscopy, image restoration, and emission tomography.  
BibTeX:
@article{Willett_R_2003_tmi_pla_maresplmi, author = {Willett, R. M. and Nowak, R. D.}, title = {Platelets: a multiscale approach for recovering edges and surfaces in photonlimited medical imaging}, journal = {IEEE Trans. Med. Imag.}, year = {2003}, volume = {22}, number = {3}, pages = {332350}, doi = {http://dx.doi.org/10.1109/TMI.2003.809622} } 

Wilson, R., Calway, A.D. & Pearson, E.R.S.  A generalized wavelet transform for Fourier analysis: the multiresolution Fourier transform and its application to image and audio signal analysis  1992  IEEE Trans. Inform. Theory Vol. 38(2), pp. 674690 
article  DOI 
Abstract: A wavelet transform specifically designed for Fourier analysis at multiple scales is described and shown to be capable of providing a local representation which is particularly well suited to segmentation problems. It is shown that, by an appropriate choice of analysis window and sampling intervals, it is possible to obtain a Fourier representation which can be computed efficiently and overcomes the limitations of using a fixed scale of window, yet by virtue of its symmetry properties allows simple estimation of such fundamental signal parameters as instantaneous frequency and onset time/position. The transform is applied to the segmentation of both image and audio signals, demonstrating its power to deal with signal events which are localized in either time/space or frequency. Feature extraction and segmentation are performed through the introduction of a class of multiresolution Markov models, whose parameters represent the signal events underlying the segmentation  
BibTeX:
@article{Wilson_R_1992_jieeetit_gen_wtfamftaiasa, author = {Wilson, R. and Calway, A. D. and Pearson, E. R. S.}, title = {A generalized wavelet transform for Fourier analysis: the multiresolution Fourier transform and its application to image and audio signal analysis}, journal = {IEEE Trans. Inform. Theory}, year = {1992}, volume = {38}, number = {2}, pages = {674690}, doi = {http://dx.doi.org/10.1109/18.119730} } 

Witkin, A.P.  Scalespace filtering: A new approach to multiscale description  1984  Vol. 9Proc. Int. Conf. Acoust. Speech Signal Process., pp. 150153 
inproceedings  
Abstract: The extrema in a signal and its first few derivatives provide a useful general purpose qualitative description for many kinds of signals. A fundamental problem in computing such descriptions is scale: a derivative must be taken over some neighborhood, but there is seldom a principled basis for choosing its size. Scalespace filtering is a method that describes signals qualitatively, managing the ambiguity of scale in an organized and natural way. The signal is first expanded by convolution with gaussian masks over a continuum of sizes. This "scale space" image is then collapsed, using its qualitative structure, into a tree providing a concise but complete qualitative description covering all scales of observation. The description is further refined by applying a stability criterion, to identify events that persist of large changes in scale.  
BibTeX:
@inproceedings{Witkin_A_1984_picassp_sca_sfnamsd, author = {Witkin, A. P.}, title = {Scalespace filtering: A new approach to multiscale description}, booktitle = {Proc. Int. Conf. Acoust. Speech Signal Process.}, year = {1984}, volume = {9}, pages = {150153} } 

Wornell, G.  Signal Processing with Fractals: A Wavelet Based Approach [BibTeX] 
1995  book  
BibTeX:
@book{Wornell_G_1995_book_sig_pfwba, author = {Wornell, G.}, title = {Signal Processing with Fractals: A Wavelet Based Approach}, publisher = {Prentice Hall}, year = {1995} } 

Xia, X.G. & Suter, B.W.  A familly of twodimensional nonseparable Malvar wavelets [BibTeX] 
1995  Appl. Comp. Harm. Analysis Vol. 2, pp. 243256 
article  
BibTeX:
@article{Xia_X_1995_acha_f_tdnmw, author = {Xia, X. G. and Suter, B. W.}, title = {A familly of twodimensional nonseparable Malvar wavelets}, journal = {Appl. Comp. Harm. Analysis}, year = {1995}, volume = {2}, pages = {243256} } 

Xiong, Z., Guleryuz, O.G. & Orchard, M.T.  A DCTbased embedded image coder  1996  Signal Process. Lett. Vol. 3(11), pp. 289290 
article  DOI 
Abstract: Since Shapiro (see ibid., vol.41, no.12, p. 445, 1993) published his work on embedded zerotree wavelet (EZW) image coding, there have been increased research activities in image coding centered around wavelets. We first point out that the wavelet transform is just one member in a family of linear transformations, and the discrete cosine transform (DCT) can also be coupled with an embedded zerotree quantizer. We then present such an image coder that outperforms any other DCTbased coder published in the literature, including that of the Joint Photographers Expert Group (JPEG). Moreover, our DCTbased embedded image coder gives higher peak signaltonoise ratios (PSNR) than the quoted results of Shapiro's EZW coder  
BibTeX:
@article{Xiong_Z_1996_jieeespl_dct_beic, author = {Z. Xiong and Guleryuz, O. G. and Orchard, M. T.}, title = {A DCTbased embedded image coder}, journal = {Signal Process. Lett.}, year = {1996}, volume = {3}, number = {11}, pages = {289290}, doi = {http://dx.doi.org/10.1109/97.542157} } 

Xu, D. & Do, M.N.  Anisotropic 2D wavelet packets and rectangular tiling: theory and algorithms  2003  Proc. SPIE, Wavelets: Appl. Signal Image Process., pp. 619630  inproceedings  DOI 
Abstract: We propose a new subspace decomposition scheme called anisotropic wavelet packets which broadens the existing definition of 2D wavelet packets. By allowing arbitrary order of row and column decompositions, this scheme fully considers the adaptivity, which helps find the best bases to represent an image. We also show that the number of candidate tree structures in the anisotropic case is much larger than isotropic case. The greedy algorithm and doubletree algorithm are then presented and experimental results are shown.  
BibTeX:
@inproceedings{Xu_D_2003_pspiewasip_ani_2dwprtta, author = {Xu, D. and Do, M. N.}, title = {Anisotropic 2D wavelet packets and rectangular tiling: theory and algorithms}, booktitle = {Proc. SPIE, Wavelets: Appl. Signal Image Process.}, year = {2003}, pages = {619630}, doi = {http://dx.doi.org/10.1117/12.506601} } 

Xu, J., Yang, L. & Wu, D.  Ripplet: A new transform for image processing  2010  J. Vis. Comm. Image Repr. Vol. 21(7), pp. 627639 
article  DOI URL 
Abstract: Efficient representation of images usually leads to improvements in storage efficiency, computational complexity and performance of image processing algorithms. Efficient representation of images can be achieved by transforms. However, conventional transforms such as Fourier transform and wavelet transform suffer from discontinuities such as edges in images. To address this problem, we propose a new transform called ripplet transform. The ripplet transform is a higher dimensional generalization of the curvelet transform, designed to represent images or twodimensional signals at different scales and different directions. Specifically, the ripplet transform allows arbitrary support c and degree d while the curvelet transform is just a special case of the ripplet transform (Type I) with c = 1 and d = 2. Our experimental results demonstrate that the ripplet transform can provide efficient representation of edges in images. The ripplet transform holds great potential for image processing such as image restoration, image denoising and image compression.  
BibTeX:
@article{Xu_J_2010_jvcir_rip_ntip, author = {J. Xu and L. Yang and D. Wu}, title = {Ripplet: A new transform for image processing}, journal = {J. Vis. Comm. Image Repr.}, year = {2010}, volume = {21}, number = {7}, pages = {627639}, url = {http://www.sciencedirect.com/science/article/B6WMK4YWXSG01/2/1aadc76f25ac3b6920def08e0a78b0e8}, doi = {DOI: 10.1016/j.jvcir.2010.04.002} } 

Yeo, B.T.T., Ou, W. & Golland, P.  On the Construction of Invertible Filter Banks on the 2Sphere  2008  IEEE Trans. Image Process. Vol. 17(3), pp. 283300 
article  DOI 
Abstract: The theories of signal sampling, filter banks, wavelets, and ldquoovercomplete waveletsrdquo are well established for the Euclidean spaces and are widely used in the processing and analysis of images. While recent advances have extended some filtering methods to spherical images, many key challenges remain. In this paper, we develop theoretical conditions for the invertibility of filter banks under continuous spherical convolution. Furthermore, we present an analogue of the Papoulis generalized sampling theorem on the 2Sphere. We use the theoretical results to establish a general framework for the design of invertible filter banks on the sphere and demonstrate the approach with examples of selfinvertible spherical wavelets and steerable pyramids. We conclude by examining the use of a selfinvertible spherical steerable pyramid in a denoising experiment and discussing the computational complexity of the filtering framework  
BibTeX:
@article{Yeo_B_2008_jieeetip_con_ifb2s, author = {Yeo, B. T. T. and Wanmei Ou and Golland, P.}, title = {On the Construction of Invertible Filter Banks on the 2Sphere}, journal = {IEEE Trans. Image Process.}, year = {2008}, volume = {17}, number = {3}, pages = {283300}, doi = {http://dx.doi.org/10.1109/TIP.2007.915550} } 

Yin, B.C., Li, X., Shi, Y.H., Zhang, F. .Z. & Zhang, N.  Directional liftingbased wavelet transform for multiple description image coding [BibTeX] 
2008  Signal Process. Image Comm. Vol. 23(1), pp. 4257 
article  
BibTeX:
@article{Yin_B_2008_jspic_dir_lbwtmdic, author = {Yin, B. C. and Li, X. and Shi, Y. H. and Zhang, F .Z. and Zhang, N.}, title = {Directional liftingbased wavelet transform for multiple description image coding}, journal = {Signal Process. Image Comm.}, year = {2008}, volume = {23}, number = {1}, pages = { 4257} } 

Zhang, X.P., Desai, M.D. & Peng, Y.N.  Orthogonal complex filter banks and wavelets: some properties and design  1999  IEEE Trans. Signal Process. Vol. 47(4), pp. 10391048 
article  DOI 
Abstract: Previous wavelet research has primarily focused on realvalued wavelet bases. However, complex wavelet bases offer a number of potential advantageous properties. For example, it has been suggested that the complex Daubechies wavelet can be made symmetric. However, these papers always imply that if the complex basis has a symmetry property, then it must exhibit linear phase as well. In this paper, we prove that a linearphase complex orthogonal wavelet does not exist. We study the implications of symmetry and linear phase for both complex and realvalued orthogonal wavelet bases. As a byproduct, we propose a method to obtain a complex orthogonal wavelet basis having the symmetry property and approximately linear phase. The numerical analysis of the phase response of various complex and real Daubechies wavelets is given. Both real and complexsymmetric orthogonal wavelet can only have symmetric amplitude spectra. It is often desired to have asymmetric amplitude spectra for processing general complex signals. Therefore, we propose a method to design general complex orthogonal perfect reconstruct filter banks (PRFBs) by a parameterization scheme. Design examples are given. It is shown that the amplitude spectra of the general complex conjugate quadrature filters (CQFs) can be asymmetric with respect the zero frequency. This method can be used to choose optimal complex orthogonal wavelet basis for processing complex signals such as in radar and sonar  
BibTeX:
@article{Zhang_X_1999_tsp_ort_cfbwpd, author = {Zhang, X.P. and Desai, M. D. and Peng, Y.N.}, title = {Orthogonal complex filter banks and wavelets: some properties and design}, journal = {IEEE Trans. Signal Process.}, year = {1999}, volume = {47}, number = {4}, pages = {10391048}, doi = {http://dx.doi.org/10.1109/78.752601} } 

Zhang, Z., Ma, S., Liu, H. & Gong, Y.  An edge detection approach based on directional wavelet transform  2009  Comput. Math. Appl. Vol. 57(8), pp. 12651271 
article  DOI 
Abstract: The standard 2D wavelet transform (WT) has been an effective tool in image processing. In recent years, many new transforms have been proposed successively, such as curvelets, bandlets, directional wavelet transform etc, which inherit the merits of the standard WT, and are more adequate at the 2D image processing tasks. Intuitively, it seemed that applying these novel tools to edge detection should acquire finer performance. In this paper, we propose an edge detection approach based on directional wavelet transform which retains the separable filtering and the simplicity of computations and filter design from the standard 2D WT. In addition, the corresponding gradient magnitude is redefined and a new algorithm for nonmaximum suppression is described. The experimental results of edge detection for several test images are provided to demonstrate our approach.  
BibTeX:
@article{Zhang_Z_2009_jcomputmathappl_edg_dabdwt, author = {Zhang, Z. and Ma, S. and Liu, H. and Gong, Y.}, title = {An edge detection approach based on directional wavelet transform}, journal = {Comput. Math. Appl.}, publisher = {Pergamon Press, Inc.}, year = {2009}, volume = {57}, number = {8}, pages = {12651271}, doi = {http://dx.doi.org/10.1016/j.camwa.2008.11.013} } 

Zhou, J. & Do, M.N.  Multidimensional Oversampled Filter Banks  2005  Vol. 5914Proc. SPIE, Wavelets: Appl. Signal Image Process., pp. 591424.1591424.12 
inproceedings  
Abstract: We present the characterization and design of multidimensional oversampled FIR filter banks. In the polyphase domain, the perfect reconstruction condition for an oversampled filter bank amounts to the invertibility of the analysis polyphase matrix, which is a rectangular FIR matrix. For a nonsubsampled FIR filter bank, its analysis polyphase matrix is the FIR vector of analysis filters. A major challenge is how to extend algebraic geometry techniques, which only deal with polynomials (that is, causal filters), to handle general FIR filters. We propose a novel method to map the FIR representation of the nonsubsampled filter bank into a polynomial one by simply introducing a new variable. Using algebraic geometry and Groebner bases, we propose the existence, computation, and characterization of FIR synthesis filters given FIR analysis filters. We explore the design problem of MD nonsubsampled FIR filter banks by a mapping approach. Finally, we extend these results to general oversampled FIR filter banks.  
BibTeX:
@inproceedings{Zhou_J_2005_spiewav_mul_ofb, author = {Zhou, J. and Do, M. N.}, title = {Multidimensional Oversampled Filter Banks}, booktitle = {Proc. SPIE, Wavelets: Appl. Signal Image Process.}, year = {2005}, volume = {5914}, pages = {591424.1591424.12} } 

Zuidwijk, R.A.  Directional and TimeScale Wavelet Analysis  2000  SIAM J. Math. Anal. Vol. 31(2), pp. 416430 
article  DOI URL 
Abstract: Combined use of the Xray (Radon) transform and the wavelet transform has proved to be useful in application areas such as diagnostic medicine and seismology. The wavelet Xray transform performs onedimensional wavelet transforms along lines in $ which are parameterized in the same fashion as for the Xray transform. The reconstruction formula for this transform gives rise to a continuous family of elementary projections. These projections provide the building blocks of a directional wavelet analysis of functions in several variables. Discrete wavelet Xray transforms are described which make use of wavelet orthonormal bases and, more generally, of biorthogonal systems of wavelet Riesz bases. Some attention is given to approximation results which involve wavelet Xray analysis in several directions.  
BibTeX:
@article{Zuidwijk_R_2000_jsiammathanal_dir_tswa, author = {Rob A. Zuidwijk}, title = {Directional and TimeScale Wavelet Analysis}, journal = {SIAM J. Math. Anal.}, publisher = {SIAM}, year = {2000}, volume = {31}, number = {2}, pages = {416430}, url = {http://link.aip.org/link/?SJM/31/416/1}, doi = {http://dx.doi.org/10.1137/S0036141098333359} } 

Bayesian Inference in Wavelet Based Models [BibTeX] 
1999  Vol. 141 
book  
BibTeX:
@book{Muller_P_1999_book_bay_iwbm,, title = {Bayesian Inference in Wavelet Based Models}, publisher = {Springer Verlag}, year = {1999}, volume = {141}, edition = {1st} } 

Wavelet image and video compression [BibTeX] 
1998  book  
BibTeX:
@book{Topiwala_P_1998_book_wav_ivc,, title = {Wavelet image and video compression}, publisher = {Kluwer Academic}, year = {1998} } 

Beyond wavelets  2003  (10)  book  
Abstract: Description "Beyond Wavelets" presents stateoftheart theories, methods, algorithms, and applications of mathematical extensions for classical wavelet analysis. Wavelets, introduced 20 years ago by Morlet and Grossmann and developed very rapidly during the 1980's and 1990's, has created a common link between computational mathematics and other disciplines of science and engineering. Classical wavelets have provided effective and efficient mathematical tools for timefrequency analysis which enhances and replaces the Fourier approach. However, with the current advances in science and technology, there is an immediate need to extend wavelet mathematical tools as well. "Beyond Wavelets" presents a list of ideas and mathematical foundations for such extensions, including: continuous and digital ridgelets, brushlets, steerable wavelet packets, contourlets, enowavelets, splinewavelet frames, and quasiaffine wavelets. Wavelet subband algorithms are extended to pyramidal directional and nonuniform filter banks. In addition, this volume includes a method for tomographic reconstruction using a mechanical image model and a statistical study for independent adaptive signal representation. Investigators already familiar with wavelet methods from areas such as engineering, statistics, and mathematics will benefit by owning this volume. Audience anyone interested in wavelet technology, including mathematicians, physical scientists, engineers, etc. Contents Preface Digital Ridgelet Transform based Trude Ridge Functions, D.L. Donoho and A.G. Flesia. Digital Implementation of Ridgelet Packets, A.G. Flesia, H. HelOr, A. Averbuch, E.J. Cand s, R.R. Coifman andD.L. Donoho. Brushlets: Steerable Wavelet Packets, F.G. Meyer and R.R. Coifman. Countourlets, M.N. Do and M. Vetterli. ENOwavelet Tranforms and Some Applications, T.F. Chan and HaoMin Zhou. A Mechanical Image Model for Bayesian Tomographic Reconstruction, S. Zhao and H.Cai. Sparsity vs. Statistical Independence in Adaptive Signal Representations: A Case Study of the Spike Process, B. B nichou and N. Saito. Nonuniform Filter Banks: New Results and Open Problems, S. Akkarakaran and P.P. Vaidyanathan. Recent Development of Spline Wavelet Frames with Compact Support, C.K> Chui and J. St ckler. Affine, QuasiAffine and CoAffine Wavelets, P. Gressman, D. Labate, G. Weiss and E.N. Wilson. Index.  
BibTeX:
@book{Welland_G_2003_book_bey_w,, title = {Beyond wavelets}, publisher = {Academic Press}, year = {2003}, number = {10} } 
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