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If you cannot find anything more, look for something else (Bridget Fountain) 


Mistakelets may occur in wavelet names below. Send YOUR correctionlets, additionlets and commentlets at: lcd (ad) ieee (dod) org. Next paper in mind: Wavelet without casualties, due to this strange weaponry related connection between Morlet (as a genealogist) and Wavelet (as a father name), bothers in arms
Otherlets: wavelet names not in *let  Artlets: wavelet uses (and misuses) in art (music, painting,...)  Forgottenlets: waiting for adoption  Linklets: other starlet/wavelet pages 

[Multiselective wavelets] [SOHO wavelets]  [AguaSonic Acoustics] [BIG Art Gallery] [Le Spy art]  [Arclet] [Beanlet] [Besselet] [Bricklet] [Cordlet] [Disklet] [Droplet] [Gauntlet] [Islet] [Multiplet] [Squarelet] [Stringlet] [Toylet] [Winglet] [& other future starlet]  [Xiaobo Qu] [Agnieszka Lisowska] 
Years of wavelet developments have generated an inflation of "waveletlike" names. They are generally built in a diminutive form based on the suffix "let" or "lette". Hence the term "starlet", from the "★let" wildcard combination, and the ★(star)like status of wavelets in signal or image processing, as well as in many other fields. More generally, suffixes et, ette, let, ling, and ule reffer to "little". A very tiny wavelet could then be baptised "lingulet". And a generic one a starling, the globish form for the more common étourneau in French. Étournelette, what a beautiful, beautiful name...
"WITS: Where is the starlet?" stands here for an approximate translation of the basic French sentence "Où est l'étoilette ?" In French again, many synonyms exist, such as "le petit coin" (somewhat equivalent to "de la menue monnaie", for the simple "change" in English). Now we have an approximation, what are the details? What kinds of "★let" names exist? What do they mean? A first (obvious, yet) answer is provided by Wim Sweldens (twitter) in the introduction for his PhD thesis, Construction and Application of Wavelets in Numerical Analysis, in 1994:
Uit de wiskundige analyse volgde dat de integraal van deze functie nul moet zijn en dat deze functie naar nul moet convergeren als het argument naar oneindig gaat. M.a.w. deze functie moet een beetje "schommelen" en dan geleidelijk uitsterven; het is een soort "lokaal golfje".
CQFD/QED/USW
More seriously, one of my favorite, yet not very specific, definition is due to Wim Sweldens too: "Wavelet are building blocks that can quickly decorrelate data." (The Lifting Scheme: A New Philosophy in Biorthogonal Wavelet Constructions, 1995, Proc. SPIE Wavelet Applications in Signal and Image Processing). The following provides a quick reference to numerous wavelet names and some of their contributors. Of course, it cannot be exhaustive, and should be considered only as a starting point. Some names are not exactly wavelets (but what is a wavelet exactly?), but belong to this domain. Given properties are stated in a very coarse sense, and should not be taken as 100% accurate. However, corrections and especially additions are very welcome (send a message to lcd (ad) ieee (dod) org).
In short:  The mother (wavelet) of them all (see below) 
Etymology:  The "lette" (or "let") suffix association generally means "petite" ("small"). "Ondelette" is built upon "onde" (French for "wave"). It thus means "small wave", hence "wavelet". The "let" suffix is somewhat about its decay. Wavelets by other names (in other languages): ondicula, golfje, ondeta/tes, pndosimilajo 
Origin:  It is often attributed to Jean Morlet, engineer at the (late)
French oil company ElfAquitaine, now merged within Total (personal note: ELF used to be associated (apocryphally) with Essences et Lubrifiants de France). The
most famous references arise from the collaboration of Alex
Grossman and Jean Morlet, Decomposition of Hardy functions
into square integrable wavelets of constant shape (pdf), SIAM
Journal of Mathematical Analysis, vol. 15, no. 4, pp. 723736,
July 1984.
Abstract: An arbitrary square integrable realvalued function (or, equivalently, the associated Hardy function) can be conveniently analyzed into a suitable family of square integrable wavelets of constant shape, (i.e. obtained by shifts and dilations from anyone of them.) The resulting integral transform is isometric and selfreciprocal if the wavelets satisfy an "admissibility condition" given here. Explicit expressions are obtained in the case of a particular analyzing family that plays a role analogous to that of coherent states (Gabor wavelets) in the usual Lztheory. They are written in terms of a modified ffunction that is introduced and studied. From the point of view of group theory, this paper is concerned with square integrable coefficients of an irreducible representation of the nonunimodular ax +bgroup Some earlier works need be mentioned:
Abstract: ### 
Contributors:  Probably too many to mention, with the great risk of forgetting some of them. See the lists by Andreas Klappenecker: Some Wavelet People, or Palle Jorgensen: Some Wavelet Researchers, with Their EMail Addresses 
Some properties:  Basically, wavelets are basis functions that are localized both in time (or spaces of higher dimension) and frequency. Wavelet atoms are generally related by scale properties. 
Anecdote:  The term wavelet is ubiquituous in the field on geosphysics, more
specifically in reflection seismology. It refers to the seismic pulse (once
called impulsion sismique in French) sent through the ground
subsurface in order to detect (after its reflections on interfaces) earth "
structures". Its accurate determination is thus crucial for the wavefield
deconvolution. The word wavelet is attested in early works
such as the one by N. Ricker, A note on the determination of the viscocity
of shale from the measurement of the wavelet breadth, Geophysics, Society
of Exploration Geophysicists, vol. 06, pp. 254258, 1941.
Abstract: From the breadth of a wavelet for a given travel time, it is possible to calculate the viscosity of the formation through which the seismic disturbance has passed. This calculation has been carried out for the Cretaceous Shale of Eastern Colorado, and the value thus found ranges from 2.7 X 10 7 to 4.9 X 10 7 , with a mean value of 3.8 X 10 7 grams per cm. per second.The Ricker wavelet (aka the Mexican hat) is often used in geophysics modelling. The first known wavelet basis (under a different name) is the Haar basis, for instance in Alfred Haar, Zur Theorie der orthogonalen FunktionenSysteme, Math. Ann., vol. 69, pp. 331371, 1910 (english translation: On the Theory of Orthogonal Function Systems by Georg Zimmermann, with local copy) Abstract: Die vorliegende Arbeit ist, bis auf unwesentliche Änderungen, ein Abdruck meiner im Juli 1909 erschienenen Göttinger Inauguraldissertation.Early nearly wavelets include Philip Franklin's construction of piecewise polynomial orthonormal splines on a bounded interval (1928), taken to its asymptotics on the whole line by J.O. Strömberg (1981). For other earlier wavelet bases (indeed including Haar, Franklin and Strömberg systems), read a nice paper by Hans G. Feichtinger, Precursors in mathematics: early wavelet bases Abstract: The plain fact that wavelet families are very interesting orthonormal systems for L 2 (R) makes it natural to view them as an important contribution to the field of orthogonal expansions of functions. This classical field of mathematical analysis was particularly flourishing in the first 30 years of the 20th century, when detailed discussions of the convergence of orthogonal series, in particular of trigonometric series, were undertaken. Alfred Haar describes the situation in his 1910 paper in Math. Annalen appropriately as follows: for any given (family of) orthonormal system(s) of functions on the unit interval [0, 1] one has to ask the following questions: convergence theory (sufficient conditions that a series is convergent); divergence theory (in contrast to convergence theory it exhibits examples of relatively "decent" functions for which nevertheless no good convergence, e.g., at that time mostly in the pointwise or uniform sense, takes place); summability theory (to which extent can summation methods help to overcome the problems of divergence);The concept of "wavelet" in the sense of a small light pulse also appears in Christian Huygens's (Dutch physicist) light propagation theory. The term was apparently introduced by Huygens in 1678, but this matter needs further investigations. It has been widely recognized that wavelets have aggregated numerous works from the fields of harmonic analysis, coherent states in quantum mechanics, electrical engineering or computer vision. 2005/05/25: i have just discovered that many french speaking people use "ondulette" instead of "ondelette". It probably comes form the verb "onduler". But some googling tells you quite fast that the term is also used for certain types of "stores" ("Venetian Blind"). This deserves further investigation. 
Usage:  Probably too many to mention, considering the great risk of forgetting some of them. 
See also:  There are many information sources, either books, articles,
web sites or even bedtime stories. We shall mention here the DMOZ
Open Directory  Science: Math: Numerical Analysis: Wavelets,
the Wavelet Digest, which contributes a lot
to the diffusion of wavelet related information.
The Wikipedia: wavelet transform
provides useful links on wavelets.
A recent article, La surprenante ascension des ondelettes, in the
La Recherche monthly (number 383, Feb. 2005, p. 5559) by Mathieu
Nowak and Yves Meyer recalls the early days of the wavelet and
its recent applications. Here are a few short reviews or tutorials on wavelets:

Comments: 
Sources for wavelet and wavelet packets code: Wavelab 850 (Matlab 6.x or 7), C++ Source Code for the Wavelet Packet Transform, WAILI  Wavelets with Integer Lifting, with WAILI.xl, an extension for very large images,
YAWTB: "Yet Another Wavelet Toolbox" (Matlab),
Computational Toolsmiths, WavBox (Matlab).
Matlab source code for the Ricker wavelet. Spherical Wavelets Code Release, version 1.2.2 available by B. T. Thomas Yeo 
Author  Title  Year  Journal/Proceedings  Reftype  DOI/URL 

Abry, P., Clausel, M., Jaffard, S., Roux, S. and Vedel, B.  Hyperbolic wavelet transform: an efficient tool for multifractal analysis of anisotropic textures  2012  PREPRINT  article  
Abstract: Global and local regularities of functions are analyzed in anisotropic function spaces, under a common framework, that of hyperbolic wavelet bases. Local and directional regularity features are characterized by means of global quantities constructed upon the coefficients of hyperbolic wavelet decompositions. A multifractal analysis is introduced, that jointly accounts for scale invariance and anisotropy. Its properties are studied in depth.  
BibTeX:
@article{Abry_P_2012_PREPRINT_hyperbolic_wtetmaat, author = {Abry, P. and Clausel, M. and Jaffard, S. and Roux, S. and Vedel, B.}, title = {Hyperbolic wavelet transform: an efficient tool for multifractal analysis of anisotropic textures}, journal = {PREPRINT}, year = {2012} } 

Averbuch, A., Beylkin, G., Coifman, R., Fischer, P. and Israeli, M.  Adaptive Solution of Multidimensional PDEs via Tensor Product Wavelet Decomposition [BibTeX] 
2008  Intern. J. of Pure and Applied Mathematics Vol. 44(1), pp. 75115 
article  
BibTeX:
@article{Averbuch_A_2008_jinterjpureapplmath_adaptive_smpdetpwd, author = {A. Averbuch and G. Beylkin and R. Coifman and P. Fischer and M. Israeli}, title = {Adaptive Solution of Multidimensional PDEs via Tensor Product Wavelet Decomposition}, journal = {Intern. J. of Pure and Applied Mathematics}, year = {2008}, volume = {44}, number = {1}, pages = {75115} } 

Ben Slimane, M. and Ben Braiek, H.  Directional and Anisotropic Regularity and Irregularity Criteria in Triebel Wavelet Bases  2012  J. Fourier Anal. Appl. Vol. 18, pp. 893914 
article  URL 
Abstract: Many natural mathematical objects, as well as many multidimensional signals and images from real physical problems, need to distinguish local directional behaviors (for tracking contours in image processing for example). Using some results of Jaffard and Triebel, we obtain criteria of directional and anisotropic regularities by decay conditions on Triebel anisotropic wavelet coefficients (resp. wavelet leaders).  
BibTeX:
@article{BenSlimane_B_2012_jfouranalappl_directional_arictwb, author = {Ben Slimane, Mourad and Ben Braiek, Hnia}, title = {Directional and Anisotropic Regularity and Irregularity Criteria in Triebel Wavelet Bases}, journal = {J. Fourier Anal. Appl.}, publisher = {Birkhäuser Boston}, year = {2012}, volume = {18}, pages = {893914}, note = {10.1007/s0004101292265}, url = {http://dx.doi.org/10.1007/s0004101292265} } 

Beylkin, G.  Wavelets and fast numerical algorithms  1993  Vol. 47#psympapplmath# 
inproceedings  
Abstract: Wavelet based algorithms in numerical analysis are similar to other transform methods in that vectors and operators are expanded into a basis and the computations take place in this new system of coordinates. However, due to the recursive definition of wavelets, their controllable localization in both space and wave number (time and frequency) domains, and the vanishing moments property, wavelet based algorithms exhibit new and important properties.For example, the multiresolution structure of the wavelet expansions brings about an efficient organization of transformations on a given scale and of interactions between different neighbouring scales. Moreover, wide classes of operators which naively would require a full (dense) matrix for their numerical description, have sparse representations in wavelet bases. For these operators sparse representations lead to fast numerical algorithms, and thus address a critical numerical issue.We note that wavelet based algorithms provide a systematic generalization of the Fast Multipole Method (FMM) and its descendents.These topics will be the subject of the lecture. Starting from the notion of multiresolution analysis, we will consider the socalled nonstandard form (which achieves decoupling among the scales) and the associated fast numerical algorithms. Examples of nonstandard forms of several basic operators (e.g. derivatives) will be computed explicitly.  
BibTeX:
@inproceedings{Beylkin_G_1993_psympapplmath_wavelets_fna, author = {Beylkin, G.}, title = {Wavelets and fast numerical algorithms}, booktitle = {#psympapplmath#}, year = {1993}, volume = {47} } 

Beylkin, G., Coifman, R. and Rokhlin, V.  Fast wavelet transforms and numerical algorithms I  1991  Comm. Pure Appl. Math. Vol. 44(2), pp. 141183 
article  DOI URL 
Abstract: A class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors. Previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices. In contrast, the methods presented here are based on the recently developed theory of wavelets and are applicable to all CalderonZygmund and pseudodifferential operators. The algorithms of this paper require order O(N) or O(N log N) operations to apply an N $ N matrix to a vector (depending on the particular operator and the version of the algorithm being used), and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here.  
BibTeX:
@article{Beylkin_G_1991_jcommpureapplmath_fast_wtna1, author = {Beylkin, G. and Coifman, R. and Rokhlin, V.}, title = {Fast wavelet transforms and numerical algorithms I}, journal = {Comm. Pure Appl. Math.}, publisher = {Wiley Subscription Services, Inc., A Wiley Company}, year = {1991}, volume = {44}, number = {2}, pages = {141183}, url = {http://dx.doi.org/10.1002/cpa.3160440202}, doi = {http://dx.doi.org/10.1002/cpa.3160440202} } 

Dahlke, S., Friedrich, U., Maaß, P., Raasch, T. and Ressel, R.A.  An adaptive wavelet solver for a nonlinear parameter identification problem for a parabolic differential equation with sparsity constraints  2012  J. Inv. IllPosed Problems  article  
Abstract: In this paper, we combine concepts from two different mathematical research topics: adaptive wavelet techniques for wellposed problems and regularization theory for nonlinear inverse problems with sparsity constraints. We are concerned with identifying certain parameters in a parabolic reactiondiffusion equation from measured data. Analytical properties of the related parametertostate operator are summarized, which justify the application of an iterated soft shrinkage algorithm for minimizing a Tikhonov functional with sparsity constraints. The forward problem is treated by means of a new adaptive wavelet algorithm which is based on tensor wavelets. In its general form, the underlying PDE describes gene concentrations in embryos at an early state of development. We implemented an algorithm for the related nonlinear parameter identification problem and numerical results are presented for a simplified test equation.  
BibTeX:
@article{Dahlke_S_2012_jinvillposedproblems_adaptive_wsnpippdesc, author = {Dahlke, Stephan and Friedrich, Ulrich and Maaß, Peter and Raasch, Thorsten and Ressel, Rudolf A.}, title = {An adaptive wavelet solver for a nonlinear parameter identification problem for a parabolic differential equation with sparsity constraints}, journal = {J. Inv. IllPosed Problems}, year = {2012} } 

Dahlke, S., Friedrich, U., Maaß, P., Raasch, T. and Ressel, R.A.  An adaptive wavelet method for parameteridentification problems in parabolic partialdifferential equations [BibTeX] 
2011  PREPRINT  article  
BibTeX:
@article{Dahlke_S_2011_PREPRINT_adaptive_wmpipppde, author = {S. Dahlke and U. Friedrich and P. Maaß and T. Raasch and R. A. Ressel}, title = {An adaptive wavelet method for parameteridentification problems in parabolic partialdifferential equations}, journal = {PREPRINT}, year = {2011} } 

Davis, A.B., Marshak, A. and Clothiaux, E.E.  Anisotropic multiresolution analysis in 2D: application to longrange correlations in cloud millimeterradar fields  1999  Vol. 3723Proc. SPIE Wavelet Applications VI, pp. 194207 
inproceedings  DOI 
Abstract: Taking a wavelet standpoint, we survey on the one hand various approaches to multifractal analysis, as a means of characterizing longrange correlations in data, and on the other hand various ways of statistically measuring anisotropy in 2Dfields. In both instances, we present new and related techniques: (i) a simple multifractal analysis methodology based onDiscrete Wavelet Transforms (DWTs), and (ii) a specific DWT adapted to strongly anisotropic fields sampled on rectangular grids with large aspect ratios. This DWT uses a tensor product of the standard dyadic Haar basis (dividing ratio 2) and a nonstandardiriadic counterpart (dividing ratio 3) which includes the famous "French tophat" wavelet. The new DWT is amenableto an anisotropic version of MultiResolution Analysis (MRA) in image processing where the natural support of the field is2z pixels (vertically) by Y' pixels (horizontally), n being the number of levels in the MRA. The complete 2D basis has onescaling function and five wavelets. The new MRA is used in synthesis mode to generate random multifractal fields thatmimic quite realistically the structure and distribution of boundarylayer clouds even though only a few parameters are used tocontrol statistically the wavelet coefficients of the liquid water density field.  
BibTeX:
@inproceedings{Davis_A_1999_pspiewa_anisotropic_ma2dalrccmrf, author = {Davis, A. B. and A. Marshak and Clothiaux, E. E.}, title = {Anisotropic multiresolution analysis in 2D: application to longrange correlations in cloud millimeterradar fields}, booktitle = {Proc. SPIE Wavelet Applications VI}, year = {1999}, volume = {3723}, pages = {194207}, doi = {http://dx.doi.org/10.1117/12.342928} } 

DeVore, R., Konyagin, S.V. and Temlyakov, V.N.  Hyperbolic wavelet approximation  1998  Constructive Approximation Vol. 14, pp. 126 
article  
Abstract: We study the multivariate approximation by certain partial sums (hyperbolicwavelet sums) of wavelet bases formed by tensor products of univariate wavelets.We characterize spaces of functions which have a prescribed approximation error byhyperbolicwavelet sums in terms of a Kfunctional and interpolation spaces. The resultsparallel those for hyperbolic trigonometric cross approximation of periodic functions[DPT].  
BibTeX:
@article{DeVore_R_1998_jconstapprox_hyperbolic_wa, author = {DeVore, R. and Konyagin, S. V. and Temlyakov, V. N.}, title = {Hyperbolic wavelet approximation}, journal = {Constructive Approximation}, year = {1998}, volume = {14}, pages = {126} } 

Duarte, M.F. and Baraniuk, R.G.  Kronecker Compressive Sensing  2012  IEEE Trans. Instrum. Meas. Vol. 21(2), pp. 494504 
article  
Abstract: Compressive sensing (CS) is an emerging approachfor the acquisition of signals having a sparse or compressible representationin some basis.While the CS literature has mostly focusedon problems involving 1D signals and 2D images, many importantapplications involve multidimensional signals; the constructionof sparsifying bases and measurement systems for such signalsis complicated by their higher dimensionality. In this paper,wepropose the use of Kronecker product matrices in CS for two purposes.First, such matrices can act as sparsifying bases that jointlymodel the structure present in all of the signal dimensions. Second,such matrices can represent themeasurement protocols used in distributedsettings. Our formulation enables the derivation of analyticalbounds for the sparse approximation of multidimensionalsignals and CS recovery performance, as well as a means of evaluatingnovel distributed measurement schemes.  
BibTeX:
@article{Duarte_M_2012_jieeetim_kronecker_cs, author = {Duarte, M. F. and Baraniuk, R. G.}, title = {Kronecker Compressive Sensing}, journal = {IEEE Trans. Instrum. Meas.}, year = {2012}, volume = {21}, number = {2}, pages = {494504} } 

Fournier, A.  Wavelets and their Applications in Computer Graphics [BibTeX] 
1995  SIGGRAPH’95 Course Notes  misc  
BibTeX:
@misc{Fournier_A_1995_lect_wavelets_acg, author = {Alain Fournier}, title = {Wavelets and their Applications in Computer Graphics}, year = {1995} } 

Fridman, J. and Manolakos, E.S.  On the Scalability of 2D Discrete Wavelet Transform Algorithms  1997  Multidimension. Syst. Signal Process. Vol. 8, pp. 185217 
article  DOI URL 
Abstract: TBC: The ability of a parallel algorithm to make efficient use of increasing computational resources is known as its scalability. In this paper, we develop four parallel algorithms for the 2dimensional Discrete Wavelet Transform algorithm (2D DWT), and derive their scalability properties on Mesh and Hypercube interconnection networks. We consider two versions of the 2D DWT algorithm, known as the Standard (S) and Nonstandard (NS) forms, mapped onto P processors under two data partitioning schemes, namely checkerboard (CP) and stripped (SP) partitioning. The two checkerboard partitioned algorithms M2=?(PlogP) (Nonstandard form, NSCP), and as M2=?(Plog2P) (Standard form, SCP); while on the storeandforwardrouted (SFrouted) Mesh and Hypercube they are scalable as 3?? (NSCP), and as 2?? (SCP), respectively, where M 2 is the number of elements in the input matrix, and ? ? (0,1) is a parameter relating M to the number of desired octaves J as J=??logM? . On the CTrouted Hypercube, scalability of the NSform algorithms shows similar behavior as on the CTrouted Mesh. The Standard form algorithm with stripped partitioning (SSP) is scalable on the CTrouted Hypercube as M 2 = ?(P 2), and it is unscalable on the CTrouted Mesh. Although asymptotically the stripped partitioned algorithm SSP on the CTrouted Hypercube would appear to be inferior to its checkerboard counterpart SCP, detailed analysis based on the proportionality constants of the isoefficiency function shows that SSP is actually more efficient than SCP over a realistic range of machine and problem sizes. A milder form of this result holds on the CT and SFrouted Mesh, where SSP would, asymptotically, appear to be altogether unscalable.  
Review: Could be applied to TRAN HuyQuan for processing speedup, or in scalable data compression, for seismic data  
BibTeX:
@article{Fridman_J_1997_jmultsystsp_scalability_2ddwta, author = {Fridman, J. and Manolakos, E. S.}, title = {On the Scalability of 2D Discrete Wavelet Transform Algorithms}, journal = {Multidimension. Syst. Signal Process.}, publisher = {Kluwer Academic Publishers}, year = {1997}, volume = {8}, pages = {185217}, url = {http://dx.doi.org/10.1023/A%3A1008229209464}, doi = {http://dx.doi.org/10.1023/A:1008229209464} } 

Grohs, P.  Tree approximation with anisotropic decompositions  2012  Appl. Comp. Harm. Analysis Vol. 33, pp. 4457 
article  
Abstract: In recent years anisotropic transforms like the shearlet or curvelet transform have received a considerable amount of interest due to their ability to efficiently capture anisotropic features in terms of nonlinear Nterm approximation. In this paper we study treeapproximation properties of such transforms where the Nterm approximant has to satisfy the additional constraint that the set of kept indices possesses a tree structure. The main result of this paper is that for shearlet and related systems, this additional constraint does not deteriorate the approximation rate. As an application of our results we construct (almost) optimal encoding schemes for cartoon images.  
BibTeX:
@article{Grohs_P_2012_jacha_tree_aad, author = {Grohs, P.}, title = {Tree approximation with anisotropic decompositions}, journal = {Appl. Comp. Harm. Analysis}, year = {2012}, volume = {33}, pages = {4457} } 

Hochmuth, R.  Anisotropic wavelet bases and thresholding  2007  Math. Nachr. Vol. 280(56), pp. 523533 
article  DOI URL 
Abstract: We consider thresholding with respect to anisotropic wavelet bases measuring the approximation error in anisotropic Hardy spaces $ H^a_p $ for $p > 0$, which are known to be equal to $L_p$ for $p >$ 1. In particular, we characterize those functions that provide a specific convergence rate by intrinsic smoothness properties. To this end we apply restricted nonlinear approximation, see [3], which is a generalization of $n$term approximation in which a weight function is used to control the terms of the approximations.  
BibTeX:
@article{Hochmuth_R_2007_jmathnachr_anisotropic_wbt, author = {Hochmuth, Reinhard}, title = {Anisotropic wavelet bases and thresholding}, journal = {Math. Nachr.}, publisher = {WILEYVCH Verlag}, year = {2007}, volume = {280}, number = {56}, pages = {523533}, url = {http://dx.doi.org/10.1002/mana.200410500}, doi = {http://dx.doi.org/10.1002/mana.200410500} } 

Hochmuth, R.  $N$term Approximation in Anisotropic Function Spaces  2002  Math. Nachr. Vol. 244(1), pp. 131149 
article  DOI URL 
Abstract: In $L_2((0, 1)^2)$ infinitely many different biorthogonal wavelet bases may be introduced by taking tensor products of onedimensional biorthogonal wavelet bases on the interval $(0, 1)$. Most wellknown are the standard tensor product bases and the hyperbolic bases. In [23, 24] further biorthogonal wavelet bases are introduced, which provide wavelet characterizations for functions in anisotropic Besov spaces. Here we address the following question: Which of those biorthogonal tensor product wavelet bases is the most appropriate one for approximating nonlinearly functions from anisotropic Besov spaces? It turns out, that the hyperbolic bases lead to nonlinear algorithms which converge as fast as the corresponding schemes with respect to specific anisotropy adapted bases.  
BibTeX:
@article{Hochmuth_R_2002_jmathnachr_nterm_aafs, author = {Hochmuth, Reinhard}, title = {$N$term Approximation in Anisotropic Function Spaces}, journal = {Math. Nachr.}, publisher = {WILEYVCH Verlag}, year = {2002}, volume = {244}, number = {1}, pages = {131149}, url = {http://dx.doi.org/10.1002/15222616(200210)244:1<131::AIDMANA131>3.0.CO;2G}, doi = {http://dx.doi.org/10.1002/15222616(200210)244:1} } 

Hochmuth, R.  Wavelet Characterizations for AnisotropicBesov Spaces  2002  Appl. Comp. Harm. Analysis Vol. 12, pp. 179208 
article  DOI 
Abstract: The goal of this paper is to provide wavelet characterizations for anisotropicBesov spaces. Depending on the anisotropy, appropriate biorthogonal tensorproduct bases are introduced and Jackson and Bernstein estimates are provedfor twoparameter families of finitedimensional spaces. These estimates leadto characterizations for anisotropic Besov spaces by anisotropydependent linearapproximation spaces and lead further on to interpolation and embedding results.Finally, wavelet characterizations for anisotropic Besov spaces with respect to $L_p$spaceswith $0
 
BibTeX:
@article{Hochmuth_R_2012_jacha_wavelet_cabs, author = {Hochmuth, R.}, title = {Wavelet Characterizations for AnisotropicBesov Spaces}, journal = {Appl. Comp. Harm. Analysis}, year = {2002}, volume = {12}, pages = {179208}, doi = {http://dx.doi.org/10.1006/acha.2001.0377} } 

Neumann, M.H.  MULTIVARIATE WAVELET THRESHOLDING IN ANISOTROPIC FUNCTION SPACES  2000  Statist. Sinica Vol. 10, pp. 399431 
article  
Abstract: It is well known that multivariate curve estimation under standard (isotropic) smoothness conditions suffers from the ``curse of dimensionality''. This is reflected by rates of convergence that deteriorate seriously in standard asymptotic settings. Better rates of convergence than those corresponding to isotropic smoothness priors are possible if the curve to be estimated has different smoothness properties in different directions and the estimation scheme is capable of making use of a lower complexity in some of the directions. We consider typical cases of anisotropic smoothness classes and explore how appropriate wavelet estimators can exploit such restrictions on the curve that require an adaptation to different smoothness properties in different directions. It turns out that nonlinear thresholding with an anisotropic multivariate wavelet basis leads to optimal rates of convergence under smoothness priors of anisotropic type. We derive asymptotic results in the model ``signal plus Gaussian white noise'', where a decreasing noise level mimics the standard asymptotics with increasing sample size.  
BibTeX:
@article{Neumann_M_2000_jstatistsinica_multivariate_wtafs, author = {Neumann, M. H.}, title = {MULTIVARIATE WAVELET THRESHOLDING IN ANISOTROPIC FUNCTION SPACES}, journal = {Statist. Sinica}, year = {2000}, volume = {10}, pages = {399431} } 

Neumann, M.H. and von Sachs, R.  Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra  1997  Ann. Stat. Vol. 25(1), pp. 3876 
article  
Abstract: We derive minimax rates for estimation in anisotropic smoothness classes. These rates are attained by a coordinatewise thresholded wavelet estimator based on a tensor product basis with separate scale parameter for every dimension. It is shown that this basis is superior to its onescale multiresolution analog, if different degrees of smoothness in different directions are present. As an important application we introduce a new adaptive waveletestimator of the timedependent spectrum of a locally stationary time series. Using this model which was recently developed by Dahlhaus, we show that the resulting estimator attains nearly the rate, which is optimal in Gaussian white noise, simultaneously over a wide range of smoothness classes. Moreover, by our new approach we overcome the difficulty of how to choose the right amount of smoothing, that is, how to adapt to the appropriate resolution, for reconstructing the local structure of the evolutionary spectrum in the timefrequency plane.  
BibTeX:
@article{Neumann_M_1997_jannstat_wavelet_tafcaaees, author = {M. H. Neumann and von Sachs, R.}, title = {Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra}, journal = {Ann. Stat.}, year = {1997}, volume = {25}, number = {1}, pages = {3876} } 

Nowak, R.D. and Baraniuk, R.G.  Waveletbased transformations for nonlinear signal processing  1999  IEEE Trans. Signal Process. Vol. 47(7), pp. 18521865 
article  DOI 
Abstract: Nonlinearities are often encountered in the analysis and processing of realworld signals. We introduce two new structures for nonlinear signal processing. The new structures simplify the analysis, design, and implementation of nonlinear filters and can be applied to obtain more reliable estimates of higher order statistics. Both structures are based on a twostep decomposition consisting of a linear orthogonal signal expansion followed by scalar polynomial transformations of the resulting signal coefficients. Most existing approaches to nonlinear signal processing characterize the nonlinearity in the time domain or frequency domain; in our framework any orthogonal signal expansion can be employed. In fact, there are good reasons for characterizing nonlinearity using more general signal representations like the wavelet expansion. Wavelet expansions often provide very concise signal representations and thereby can simplify subsequent nonlinear analysis and processing. Wavelets also enable local nonlinear analysis and processing in both time and frequency, which can be advantageous in nonstationary problems. Moreover, we show that the wavelet domain offers significant theoretical advantages over classical time or frequency domain approaches to nonlinear signal analysis and processing  
BibTeX:
@article{Nowak_R_1999_jieeetsp_waveletbased_tnsp, author = {Nowak, R. D. and Baraniuk, R. G.}, title = {Waveletbased transformations for nonlinear signal processing}, journal = {IEEE Trans. Signal Process.}, year = {1999}, volume = {47}, number = {7}, pages = {18521865}, doi = {http://dx.doi.org/10.1109/78.771035} } 

Rosiene, C.P. and 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} } 

Roux, S., Clausel, M., Vedel, B., Jaffard, S. and Abry, P.  Wavelet analysis for anisotropic images [BibTeX] 
2012  INPREPARATION  article  
BibTeX:
@article{Roux_S_2012_INPREPARATION_wavelet_aai, author = {Roux, S. and Clausel, M. and Vedel, B. and Jaffard, S. and Abry, P.}, title = {Wavelet analysis for anisotropic images}, journal = {INPREPARATION}, year = {2012} } 

Roux, S.G., Clausel, M., Vedel, B., Jaffard, S. and Abry, P.  SelfSimilar Anisotropic Texture Analysis: the Hyperbolic Wavelet Transform Contribution  2013  PREPRINT  article  
Abstract: Textures in images can often be well modeled using selfsimilar processes while they may at the same time display anisotropy. The present contribution thus aims at studying jointly selfsimilarity and anisotropy by focusing on a specific classical class of Gaussian anisotropic selfsimilar processes. It will first be shown that accurate joint estimates of the anisotropy and selfsimilarity parameters are performed by replacing the standard 2Ddiscrete wavelet transform by the hyperbolic wavelet transform, which permits the use of different dilation factors along the horizontal and vertical axis. Defining anisotropy requires a reference direction that needs not a priori match the horizontal and vertical axes according to which the images are digitized, this discrepancy defines a rotation angle. Second, we show that this rotation angle can be jointly estimated. Third, a non parametric bootstrap based procedure is described, that provides confidence interval in addition to the estimates themselves and enables to construct an isotropy test procedure, that can be applied to a single texture image. Fourth, the robustness and versatility of the proposed analysis is illustrated by being applied to a large variety of different isotropic and anisotropic selfsimilar fields. As an illustration, we show that a true anisotropy builtin selfsimilarity can be disentangled from an isotropic selfsimilarity to which an anisotropic trend has been superimposed.  
BibTeX:
@article{Roux_S_2013_PREPRINT_selfsimilar_atahwtc, author = {Roux, S. G. and Clausel, M. and Vedel, B. and Jaffard, S. and Abry, P.}, title = {SelfSimilar Anisotropic Texture Analysis: the Hyperbolic Wavelet Transform Contribution}, journal = {PREPRINT}, year = {2013} } 

Schremmer, C.  Decomposition strategies for waveletbased image coding  2001  Vol. 2Signal Processing and its Applications, Sixth International, Symposium on. 2001, pp. 529 532 
inproceedings  DOI 
Abstract: The wavelet transform has become the most interesting new algorithm for still image compression. Yet there are many parameters within a wavelet analysis and synthesis which govern the quality of a decoded image. In this paper, we discuss different decomposition strategies of a twodimensional signal and their implications for the decoded image: a pool of grayscale images has been wave lettransformed with different settings of the wavelet filter bank, quantization threshold and decomposition method. Contrary to the new standard JPEG2000, where nonstandard decomposition is implemented, our investigation proposes standard decomposition for lowbitrate coding  
BibTeX:
@inproceedings{Schremmer_C_2001_pisspa_decomposition_swbic, author = {Schremmer, C.}, title = {Decomposition strategies for waveletbased image coding}, booktitle = {Signal Processing and its Applications, Sixth International, Symposium on. 2001}, year = {2001}, volume = {2}, pages = {529 532}, doi = {http://dx.doi.org/10.1109/ISSPA.2001.950197} } 

Triebel, H.  Wavelet bases in anisotropic function spaces  2005  Proc. Conf. "Function spaces, differential operators and nonlinear analysis", Milovy, 2004, pp. 370387  inproceedings  
Abstract: The paper deals with wavelet characterisations for anisotropicspaces $B_pq^s,alpha$ and $F_pq^s,alpha$ in $R^n$ for all parameters $s$, $p$, $q$ and all anisotropies $.Some applications are given.  
BibTeX:
@inproceedings{Triebel_H_2005_PREPRINT_wavelet_bafs, author = {Triebel, H.}, title = {Wavelet bases in anisotropic function spaces}, booktitle = {Proc. Conf. "Function spaces, differential operators and nonlinear analysis", Milovy, 2004}, year = {2005}, pages = {370387} } 

Velisavljević, V.  Directionlets: Anisotropic Multidirectional Representation with Separable Filtering  2005  School: EPFL  phdthesis  
Abstract: Efficient representation of geometrical information in images is very important in many image processing areas, including compression, denoising and feature extraction. However, the design of transforms that can capture these geometrical features and represent them with a sparse description is very challenging. Recently, the separable wavelet transform achieved a great success providing a computationally simple tool and allowing for a sparse representation of images. However, in spite of the success, the efficiency of the representation is limited by the spatial isotropy of the wavelet basis functions built in the horizontal and vertical directions as well as the lack of directionality. Onedimensional discontinuities in images (edges and contours), which are very important elements in visual perception, intersect with too many wavelet basis functions leading to a nonsparse representation. To capture efficiently these anisotropic geometrical structures characterized by many more than the horizontal and vertical directions, more flexible multidirectional and anisotropic transforms are required. We present a new latticebased perfect reconstruction and critically sampled anisotropic multidirectional wavelet transform. The transform retains the separable filtering, subsampling and simplicity of computations and filter design from the standard twodimensional wavelet transform, unlike in the case of some other existing directional transform constructions (e.g. curvelets, contourlets or edgelets). The corresponding anisotropic basis functions, which we call 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 decay of meansquare errorO(N 1.55), which, while slower than the optimal rate O(N2), is much better than O(N1) achieved with wavelets, but at similar complexity. Owing to critical sampling, directionlets can easily be applied to image compression since it is possible to use Lagrange optimization as opposed to the case of overcomplete expansions. The compression algorithms based on directionlets outperform the methods based on the standard wavelet transform achieving better numerical results and visual quality of the reconstructed images. Moreover, we have adapted image denoising algorithms to be used in conjunction with an undecimated version of directionlets obtaining results that are competitive with the current stateoftheart image denoising methods while having lower computational complexity.  
BibTeX:
@phdthesis{Velisavljevic_V_2005_phd_dir_amdrsf, author = {Velisavljević, V.}, title = {Directionlets: Anisotropic Multidirectional Representation with Separable Filtering}, school = {EPFL}, year = {2005} } 

Velisavljević, V., BeferullLozano, B., Vetterli, M. and 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} } 

Velisavljević, V., BeferullLozano, B., Vetterli, M. and Dragotti, P.L.  Directionlets: anisotropic multidirectional representation with separable filtering [BibTeX] 
2006  misc  
BibTeX:
@misc{Velisavljevic_V_2006_pp_dir_amdrsf, author = {Velisavljević, V. and BeferullLozano, B. and Vetterli, M. and Dragotti, P. L.}, title = {Directionlets: anisotropic multidirectional representation with separable filtering}, year = {2006}, note = {Preprint, better reference teVelisavljevic_V_2006_tip_dir_amdrsf} } 

Wegmann, B. and Zetzsche, C.  Efficient image sequence coding by vector quantization of spatiotemporal bandpass outputs  1992  Proc. SPIE 1818, Visual Communications and Image Processing, pp. 11461154  inproceedings  DOI URL 
Abstract: A coding scheme for image sequences is designed in analogy to human visual information processing. We propose a featurespecific vector quantization method applied to multichannel representation of image sequences. The vector quantization combines the corresponding local/momentary amplitude coefficients of a set of threedimensional analytic bandpass filters being selective for spatiotemporal frequency, orientation, direction and velocity. Motion compensation and decorrelation between successive frames is achieved implicitly by application of a nonrectangular subsampling to the 3Dbandpass outputs. The nonlinear combination of the outputs of filters which are selective for constantly moving one dimensional (i.e. spatial elongated) image structures allows a classification of the local/momentary signal features with respect to their intrinsic dimensionality. Based on statistical investigations a natural hierarchy of signal features is provided. This is then used to construct an efficient encoding procedure. Thereby, the different sensitivity of the human vision to the various signal features can be easily incorporated. For a first example, all multi dimensional vectors are mapped to constantly moving 1Dstructures.  
BibTeX:
@inproceedings{Wegmann_B_1992_pspievcip_efficient_iscvqsbo, author = {Wegmann, Bernhard and Zetzsche, Christoph}, title = {Efficient image sequence coding by vector quantization of spatiotemporal bandpass outputs}, booktitle = {Proc. SPIE 1818, Visual Communications and Image Processing}, year = {1992}, pages = {11461154}, url = {http://dx.doi.org/10.1117/12.131386}, doi = {http://dx.doi.org/10.1117/12.131386} } 

Welk, M., Weickert, J. and Steidl, G.  From TensorDriven Diffusion to Anisotropic Wavelet Shrinkage [BibTeX] 
2006  Proc. Eur. Conf. Comput. Vis.  inproceedings  
BibTeX:
@inproceedings{Welk_M_2006_peccv_ten_ddaws, author = {Welk, M. and Weickert, J. and Steidl, G.}, title = {From TensorDriven Diffusion to Anisotropic Wavelet Shrinkage}, booktitle = {Proc. Eur. Conf. Comput. Vis.}, year = {2006} } 

Westerink, P.H.  Subband coding of images [BibTeX] 
1989  School: Delft University of Technology  phdthesis  
BibTeX:
@phdthesis{Westerink_P_1989_phd_subband_ci, author = {P. H. Westerink}, title = {Subband coding of images}, school = {Delft University of Technology}, year = {1989} } 

Xu, D. and 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} } 

Zavadsky, V.  Image Approximation by Rectangular Wavelet Transform  2007  J. Math. Imaging Vis. Vol. 27, pp. 129138 
article  URL 
Abstract: We study image approximation by a separable wavelet basis $$ 2^k_1xi)2^k_2yj), xi)2^k_2yj), 2^k_1(xi)yj), xi)yi),$ where $k_1, k_2 in Z_+; i,jinZ; $$ and ?,? are elements of a standard biorthogonal wavelet basis in L 2 (?). Because k 1 ? k 2 , the supports of the basis elements are rectangles, and the corresponding transform is known as the rectangular wavelet transform . We provide a selfcontained proof that if onedimensional wavelet basis has M dual vanishing moments then the rate of approximation by N coefficients of rectangular wavelet transform is $$ O(N^M) $$ for functions with mixed derivative of order M in each direction. These results are consistent with optimal approximation rates for such functions. The square wavelet transform yields the approximation rate is $$ O(N^M/2) $$ for functions with all derivatives of the total order M . Thus, the rectangular wavelet transform can outperform the square one if an image has a mixed derivative. We provide experimental comparison of image approximation which shows that rectangular wavelet transform outperform the square one.  
BibTeX:
@article{Zavadsky_V_2007_jmathimagingvis_image_arwt, author = {Zavadsky, Vyacheslav}, title = {Image Approximation by Rectangular Wavelet Transform}, journal = {J. Math. Imaging Vis.}, publisher = {Springer Netherlands}, year = {2007}, volume = {27}, pages = {129138}, note = {10.1007/s108510070777z}, url = {http://dx.doi.org/10.1007/s108510070777z} } 
In short:  Wavelets inspired by the shape of canonical hemodynamic response functions 
Etymology:  Active wavelet 
Origin:  Khalidov, Ildar and Van De Ville, Dimitri and Fadili, Jalal M. and Unser, Michael A.
Activelets and sparsity: a new way to detect brain activation from fMRI data, SPIE Optics and Photonics, Wavelets XII
Conference 6701  Proceedings of SPIE Volume 6701, 2629 August 2007 [(pdf)]
Abstract: FMRI time course processing is traditionally performed using linear regression followed by statistical hypothesis testing. While this analysis method is robust against noise, it relies strongly on the signal model. In this paper, we propose a nonparametric framework that is based on two main ideas. First, we introduce a problemspecific type of wavelet basis, for which we coin the term "activelets". The design of these wavelets is inspired by the form of the canonical hemodynamic response function. Second, we take advantage of sparsitypursuing search techniques to find the most compact representation for the BOLD signal under investigation. The nonlinear optimization allows to overcome the sensitivityspecificity tradeoff that limits most standard techniques. Remarkably, the activelet framework does not require the knowledge of stimulus onset times; this property can be exploited to answer to new questions in neuroscience. Activelets: Wavelets for Sparse Representation of Hemodynamic Responses (DOI:10.1016/j.sigpro.2011.03.008), Ildar Khalidov and Jalal Fadili and François Lazeyras and Dimitri Van De Ville and Michael Unser (Related work) Abstract: We propose a new framework to extract the activityrelated component in the BOLD functional Magnetic Resonance Imaging (fMRI) signal. As opposed to traditional fMRI signal analysis techniques, we do not impose any prior knowledge of the event timing. Instead, our basic assumption is that the activation pattern is a sequence of short and sparselydistributed stimuli, as is the case in slow eventrelated fMRI. We introduce new wavelet bases, termed ``activelets'', which sparsify the activityrelated BOLD signal. These wavelets mimic the behavior of the differential operator underlying the hemodynamic system. To recover the sparse representation, we deploy a sparsesolution search algorithm. The feasibility of the method is evaluated using both synthetic and experimental fMRI data. The importance of the activelet basis and the nonlinear sparse recovery algorithm is demonstrated by comparison against classical Bspline wavelets and linear regularization, respectively. 
Contributors:  Ildar Khalidov, Dimitri Van De Ville Jalal Fadili Michael Unser 
Some properties:  
Anecdote:  
Usage:  Detect brain activation from fMRI data 
See also:  
Comments: 
In short:  Nonlinear and nonparametric estimator of additive models with wavelets 
Etymology:  Additive Model wavelet estimator (also with a Robust extension) 
Origin:  Sardy, Sylvain and Tseng, Paul,
AMlet and GAMlet: Automatic Nonlinear Fitting of Additive Models and Generalized Additive Models with Wavelets, Journal of Computational and Graphical Statistics,
2004, [local AMlet GAMlet copy in pdf and ps]
Abstract: A simple and yet powerful method is presented to estimate nonlinearly and nonparametrically the components of additive models using wavelets. The estimator enjoys the good statistical and computational properties of the Waveshrink scatterplot smoother and it can be efficiently computed using the block coordinate relaxation optimization technique. A rule for the automatic selection of the smoothing parameters, suitable for data mining of large datasets, is derived. The waveletbased method is then extended to estimate generalized additive models. A primaldual logbarrier interior point algorithm is proposed to solve the corresponding convex programming problem. Based on an asymptotic analysis, a rule for selecting the smoothing parameters is derived, enabling the estimator to be fully automated in practice. We illustrate the finite sample property with a Gaussian and a Poisson simulation. 
Contributors:  Sylvain Sardy, Paul Tseng 
Some properties:  Provides universal thresholding rules for Gaussian and Poisson distributions 
Anecdote:  
Usage:  Statistics, fitting of additive models 
See also:  Its generalization, called GAMlet 
Comments:  Not truly a wavelet by itself 
In short:  Orthogonal multiwavelet for which polynomial perturbation of the input does not affect the wavelet decomposition with highpass output 
Etymology:  Analysis Ready Multiwavelet 
Origin:  Lian, J. A. and Chui, C. K. AnalysisReady Multiwavelets
(Armlets) for processing scalarvalued signals , Signal
Processing Letters, vol. 11, no. 2, pp. 205208, Feb. 2004
Abstract: The notion of armlets is introduced in this letter as a precise formulation of orthonormal multiwavelets that guarantee wavelet decomposition with highpass output not being effected by polynomial perturbation of the input. A mathematical scheme for constructing armlets is given, and it is shown that the notions of armlets and balanced multiwavelets are different. In particular, while balanced wavelets are armlets, the converse is false in general. One advantage of armlets is that the weaker assumption provides flexibility to facilitate wavelet and filter construction. 
Contributors:  Jianao Lian, and Charles K. Chui 
Some properties:  Defined to satisfy the n th order wavelet consistency requirement (n WAC). More general than n balanced multiwavelets. Correspond to the Daubechies orthogonal wavelets (daublets) in the scalar setting 
Anecdote:  
Usage:  
See also:  
Comments: 
In short:  2D multiscale basis vectors adaptively elongated in the direction of (image) geometric flows 
Etymology:  From bandelet, little stripes, generally made of soft matter (in French bandelette), or the ringshaped molding one can find at the top of columns 
Origin:  Le Pennec, Erwan and Mallat, Stéphane, Image
compression with geometrical wavelets, International
Conference on Image Processing (ICIP), September 2000,
VancouverAbstract: We introduce a sparse image representation that takes advantage of the geometrical regularity of edges in images. A new class of onedimensional wavelet orthonormal bases, called foveal wavelets, are introduced to detect and reconstruct singularities. Foveal wavelets are extended in two dimensions, to follow the geometry of arbitrary curves. The resulting two dimensional “bandelets” define orthonormal families that can restore close approximations of regular edges with few nonzero coefficients. A double layer image coding algorithm is described. Edges are coded with quantized bandelet coefficients, and a smooth residual image is coded in a standard twodimensional wavelet basisBandlet Image Estimation with Model Selection (DOI:10.1016/j.sigpro.2011.01.013) [back to the starlet list] Charles Dossal and Stéphane Mallat and Erwan Le Pennec Abstract: To estimate geometrically regular images in the white noise model and obtain an adaptive near asymptotic minimaxity result, we consider a model selection based bandlet estimator. This bandlet estimator combines the best basis selection behaviour of the model selection and the approximation properties of the bandlet dictionary. We derive its near asymptotic minimaxity for geometrically regular images as an example of model selection with general dictionary of orthogonal bases. This paper is thus also a self contained tutorial on model selection with orthogonal bases dictionary. 
Contributors:  Erwan Le Pennec, Stéphane Mallat, Charles Dossal, Gabriel Peyré 
Some properties:  Bandelets have a support parallel to flow lines in images. Approximation rate: M ^{a} for images having discontinuities along C^{a} contours, and being C^{a} away from the contours 
Anecdote:  According to one of the authors, most of the obvious names in "let" were already taken at the time of its invention, making it difficult to find this one 
Usage:  Image coding, denoising, deconvolution, 3D surface compression 
See also:  Charles Dossal, for further bandelet developments, Gabriel Peyré, for the development of second generation bandelets, and Let it wave (Zoran), a startup devoted to bandelet applications, including low bitrate identity pictures 
Comments:  A secondgeneration Matlab bandelet toolbox is available from Gabriel Peyré at MatlabCentral 
In short:  A fat edgelet/beamlet 
Etymology:  From bar (a solid, more or less rigid object with a uniform crosssection smaller than its length) and the ewig let 
Origin:  Multiscale Geometric Image Compression using Wavelets and Wedgelets, Richard Baraniuk, Hyeokho Choi, Justin Romberg, Mike Wakin. [pdf] 
Contributors:  
Some properties:  
Anecdote:  
Usage:  
See also:  
Comments: 
In short:  An orthogonal or biorthogonal wavelet designed, through a balanced weighted uncertainty (time and frequency spread) approach, to improve its coding capabilities  
Etymology:  From the University of Bath, School of Electronic and Electrical Engineering, where the design has been proposed  
Origin: 
Orthonormal wavelets with balanced uncertainty, D. M. Monro, B. E. Bassil and G. J.
Dickson, IEEE International Conference on Image Processing, 1996, Vol.2, pp.581
584 (local copy).
Abstract: This paper addresses the question: "What makes a good wavelet for image compression?", by considering objective and subjective measurements of quality. A new metric is proposed for the design of the Finite Impulse Response (FIR) filters used in the Discrete Wavelet Transform (DWT). The metric is the diagonal of the Heisenberg uncertainty rectangle, with time weighted by a factor k relative to frequency. Minimization of the metric balances the time and frequency spreads of the filter response. The metric can be computed directly from the filter coefficients, so it can be used to optimize wavelets for image compression without the cost of repeatedly compressing and decompressing images. A psychovisual evaluation carried out with 24 subjects demonstrates that orthonormal FIR filters designed this way give good subjective results with zerotree image compression.With suitably chosen k, both better subjective quality and lower RMS error are achived than with wavelets chosen for maximum regularity.Spacefrequency balance in biorthogonal wavelets, D. M. Monro and B. G. Sherlock, IEEE International Conference on Image Processing, 1997, Vol.1, pp.624627 (local copy). Abstract: This paper shows how to design good biorthogonal FIR filters for wavelet image compression by balancing the space and frequency dispersions of analysis and synthesis lowpass filters. A quality metric is proposed which can be computed directly from the filter coefficients. By optimizing over the space of FIR filter coefficients, a filter bank can be found which minimizes the metric in about 60 seconds on a high performance workstation. The metric contains three parameters which weight the space and frequency dispersions of the low pass analysis and synthesis filters. A series of biorthogonal, symmetric wavelet filters of length 10 was found, each optimized for different weightings. Each of these filter banks was then evaluated by compressing and decompressing five test images at three compression ratios. Selecting each optimum provides fifteen sets of parameters corresponding to filter banks which maximize the PSNR in each case. The average of these parameters was used to define a ‘mean’ filter bank, which was then evaluated on the test images. Individual images can produce substantially different weightings of the time dispersion at the optimum, but the PSNR of the mean filter is normally close to the optimum. The mean filter also compares favourably with a maximum regularity biorthogonal filter of the same length. 

Contributors:  D. M. Monro, B. E. Bassil, G. J. Dickson  
Some properties:  Based on an Heisenberg uncertainty metric, efficient FIR filters are designed to improve image coding, as compared to maximum regularity filters, via the balancing of both the time and frequency spread of the function. Provides apparently better subjective quality than maximum regularity wavelets.  
Anecdote:  The word "bathlet" (the correct spelling is
bat'leth, but the mistake is quite common, perhaps due to the analogy with a small "battle") belongs to the Klingon vocabulary (from
the Star Trek space soap opera). It is a personal weapon that
every Klingon carries on with him. You never know! Notice (on the
right) the smoothness of the contours and the sharpness of the
edges. For others bathlet pictures... (Klingonwaffen in german,
what a beautiful, beautiful name)
Trivia: Colorado 7eleven (7 11 math problem here) stores fear a Klingonweaponed robber threatening clerks with the spiky, crescent shaped Star Trek inspired sword called bat'leth or Klingon's personal sword of honor. Details at The Denver Channel. 

Usage:  Image coding  
See also:  The Bath Wavelet Warehouse, for Bath wavelets coefficient tables, orthogonal and biorthogonal wavelet coefficients. A WhereIsTheStarlet entry: WITS: Bathlet wavelets from La vertu d'un LA.  
Comments: 
In short:  Collection of dyadicallyorganized line segments, occupying a range of dyadic locations and scales, and occuring at a range of orientations 
Etymology:  From beam a piece of timber used for construction, or directly beamlet, a small beam of light 
Origin:  Donoho, David and Huo, Xiaoming, Beamlets and Multiscale
Image Analysis, 2001, Stanford, Research reportAbstract: We describe a framework for multiscale image analysis in which line segments play a role analogous to the role played by points in wavelet analysis. The framework has 5 key components. The beamlet dictionary is a dyadically organized collection of line segments, occupying a range of dyadic locations and scales, and occurring at a range of orientations. The beamlet transform of an image f(x, y) is the collection of integrals of f over each segment in the beamlet dictionary; the resulting information is stored in a beamlet pyramid. The beamlet graph is the graph structure with pixel corners as vertices and beamlets as edges; a path through this graph corresponds to a polygon in the original image. By exploiting the ?rst four components of the beamlet framework, we can formulate beamletbased algorithms which are able to identify and extract beamlets and chains of beamlets with special properties. In this paper we describe a fourlevel hierarchy of beamlet algorithms. The ?rst level consists of simple procedures which ignore the structure of the beamlet pyra mid and beamlet graph; the second level exploits only the parentchild dependence between scales; the third level incorporates collinearity and cocurvity relationships; and the fourth level allows global optimization over the full space of polygons in an image. These algorithms can be shown in practice to have suprisingly powerful and apparently unprecedented capabilities, for example in detection of very faint curves in very noisy data. We compare this framework with important antecedents in image processing (Brandt and Dym; Horn and collaborators; G¨otze and Druckenmiller) and in geo metric measure theory (Jones; David and Semmes; and Lerman). 
Contributors:  David Donoho, Xiaoming Huo 
Some properties:  
Anecdote:  Beamlet is also the name of a singlebeam laser 
Usage:  Filament or object boundary extraction in noise. Analysis of largescale structures of the Universe, esp. in 3D 
See also:  Wedgelets, which share a similar dyadic recursive decomposition. Also recent chordlets. 
Comments:  Beamlab: a Matlab (TM) toolbox code for the implementation of various feature oriented transforms 
In short:  A wavelet with "binary filter" coefficients or generated by "binary" wavelet coefficients filter bank 
Etymology:  From the contraction binary filter (symmetric) wavelet 
Origin: 
Le Gall, D. and Tabatabai, A. Subband coding of digital images using symmetric short kernel filters and arithmetic coding techniques, Proc. ICASSP 1988
Abstract: A simple and efficient method of subband coding of digital images is reported. First, a technique for designing symmetric short tap filters is presented, and it is shown that such filters can be easily implemented by using simple arithmetic operations (e.g. addition and multiplication). By applying the above filters, the input image is decomposed into four bands, which are then coded by using arithmetic coding in combination with discrete PCM coding of the lowest band and PCM coding of higher bands. Simulation results demonstrate that by using the method mentioned above good quality pictures can be obtained in the range of 0.7 to 0.8 bits/pelStrang, G. and Nguyen, T., Wavelets and filter banks, p. 217 or p. 249, WellesleyCambridge Presss, 1996 A. R. Calderbank and Ingrid Daubechies and Wim Sweldens and BoonLock Yeo Wavelet Transforms That Map Integers to Integers, ACHA, 1998 Abstract: Invertible wavelet transforms that map integers to integers have important applications in lossless coding. In this paper we present two approaches to build integer to integer wavelet transforms. The first approach is to adapt the precoder of Laroiaet al.,which is used in information transmission; we combine it with expansion factors for the high and low pass band in subband filtering. The second approach builds upon the idea of factoring wavelet transforms into socalled lifting steps. This allows the construction of an integer version of every wavelet transform. Finally, we use these approaches in a lossless image coder and compare the results to those given in the literature. 
Contributors:  Gilbert Strang, Truong Nguyen, and many others, sometimes under the name of reversible wavelets. 
Some properties:  DSPfriendly wavelet filter banks with integer coefficients (like the Haar wavelet) or integers divided by powers of 2, with the form c = n/2^{k} (with n and k integers), up to a normalization scaling coefficient (sometimes irrational). Such transforms are easily computed by adds or binary shifts. Related works mention reversible ITIwavelets (integertointer wavelets, or filterbanks in general), multiplierless transforms, SOPOT (sumofpowersoftwo) coefficients. 
Anecdote:  Apparently, a 9/7 wavelet filter pair was found by Gilbert Strang by solving the halfband equation, and discovered later that Wim Sweldens created earlier a whole family of binary symmetric filters in 1995. One of them, an integer biorthogonal reversible 5/3 filter bank (known as the 5/3 Le GallTabatabai filter bank) is used for lossless compression in the JPEG 2000 standard, with coefficients [1 2 1}/2 and [1 2 6 2 1]/8. The binary 9/7 filters are [1 0 8 16 46 16 8 0 1]/64 and [1 0 9 16 9 0 1]/32. The Le Gall 5/3 analysis filters [1 2 6 2 1]/8 and [1 2 1]/3 
Usage:  Binlets are especially useful for finite arithmetic reversible transforms, especially for lossless compression 
See also:  Some other integertointeger transforms (Generalized S Transform) have been developed by Michael Adams, who develops the JPEG 2000 JasPer project 
Comments:  Often used in "the 9/7 binlet" expression. Also used for the Haar wavelet, some biorthogonal spline wavelets; also used for the S+P transform from A. Said and W. Pearlman SPIHT image compression and other (NB: the S+P transform is nonlinear). Thus, binlet is a relatively illdefined term. "Binary" structures may be generated by the lifting scheme, developed by Wim Sweldens in 1995. 
In short:  Biorthogonal basis with good spatial localization and precise localization, providing a decomposition with different orientations, frequencies, sizes and positions 
Etymology:  From brush, from the brush stroke aspect of the 2D tensor products 
Origin:  Meyer, François G. and Coifman, Ronald R.,
Brushlets: a tool for directional image analysis and image
compression, Applied and Computational Harmonic Analysis,
vol. 4, pp. 147187, 1997
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. 
Contributors:  François G. Meyer, Ronald R. Coifman Lasse Borup 
Some properties:  Works directly in the Fourier domain 
Anecdote:  
Usage:  Image coding (esp. for highly textured images) 
See also:  
Comments:  Applied for denoising and segmentation of cardiac ultrasound 
In short:  A blend of standard MRA (multiresolution analysis), framelets and hierarchical bases, based of a set of three filters, a lowpass decomposition, a lowpass prediction and an alignment filter 
Etymology:  From the contraction CAP, from Coarsification, Alignment, Prediction (in the first papers). More recent works use CAP for Compression, Alignment, Prediction, and CAMP for Compression, Alignment, Modified Prediction 
Origin:  Ron, A. Caplets: wavelets without wavelets, 29th
Annual Spring Lecture Series, Recent Developments in Applied
Harmonic Analyis, Multiscale Geometric Analysis, April 1517,
2004 (CAPlet local copy)
Abstract: Wavelet decompositions are implemented and inverted by fast algorithms, the socalled fast wavelet transform (FWT). The FWT is the primary reason for the popularity of waveletbased methods in so many different scientific and engineering disciplines. The second most important reason for the popularity of wavelets is their mathematical theory: that theory shows that the wavelet coefficients record faithfully the precise smoothness class of the underlying dataset/function. These characterizations are instrumental for the mathematical analysis of waveletbased algorithms in the areas of image and signal analysis. The third most important reason for the popularity of wavelets (which is closely related to the first one) is the vehicle of MultiResolution Analysis (MRA) which allows for the construction of a wide variety of wavelet systems. This approach is epitomized in the univariate Mallat's algorithm. The effective construction of wavelet systems is more cumbersome in higher dimensions. For example, in 4D (and dyadic downsampling) one employs (at least) 15 different highpass filters in any MRAbased wavelet system. And the struggle in higher dimensions to balance optimally between time localization (short filters) and frequency localization is hampered by the need to adhere to the MRAbased construction principles. 
Contributors:  Amos Ron (University of Wisconsin), Youngmi Hur (Johns Hopkins University) 
Some properties:  Caplet coefficients provide characterization of function spaces analogous to wavelet's. Redundant description, with redundancy decreasing with the spatial dimension. 
Anecdote:  Caplet information is hard to find on the Internet, since it
is often mixed with advertising on medicines (tablets), especially
on Amazon web pages. See for instance the answer for a Google
search on wavelet and caplet, performed on
2005/02/02.
Amazon.com: Editorial Reviews: Multirate and Wavelet Signal ... ... Customers interested in Multirate and Wavelet Signal Processing ... in ... Aleve All Day Strong Pain Reliever, Fever Reducer, Caplet, 100pack ... www.amazon.com/exec/obidos/tg/ detail//0126775605?v=glance&vi=reviews ... 
Usage:  
See also: 
Hur, Yougmi and Ron, Amos, CAP representations (The mathematical theory of pyramidal algorithms), Wavelet Theory and Applications: Singapore, August 2004
(CAP representations (The mathematical theory of pyramidal algorithms))
Hur, Yougmi and Ron, Amos, CAPlets: wavelet representations without wavelets (CAPlet local copy) Abstract: MultiResolution (MR) is among the most effective and the most popular approaches for data representation. In that approach, the given data are organized into a sequence of resolution layers, and then the "difference" between each two consecutive layers is recorded in terms of detail coefficients. Wavelet decomposition is the best known representation methodology in the MR category. The major reason for the popularity of wavelet decompositions is their implementation and inversion by a fast algorithm, the socalled fast wavelet transform (FWT). Another central reason for the success of wavelets is that the wavelet coefficients capture very accurately the smoothness class of the function hidden behind the data. This is essential for the understanding of the performance of key waveletbased algorithms in compression, in denoising, and in other applications. On the downside, constructing wavelets with good spacefrequency localization properties becomes involved as the spatial dimension grows. An alternative to the sometimehardtoconstruct wavelet representations is the alwayseasytoconstruct (and slightly older) nonorthogonal pyramidal algorithms. Similar to wavelets, the (linear, regular, isotropic) pyramidal representations are based on some method for linear coarsening (by a decomposition filter) of their data, and a complementary method for linear prediction (by a prediction filter) of the original data from the coarsened one. The first step creates the resolution layers and the second allows for trivial extractions of suitable detail coefficients. The decomposition and reconstruction algorithms in the pyramidal approach are as fast as those of wavelets. In contrast with orthonormal wavelets, the representation is redundant, viz. the total number of detail coefficients exceeds the original size of the data: denoting by s the ratio between the size of the data at two consecutive resolution layers, the “redundancy ratio” in the pyramidal representation is s/(s  1). In this paper, we introduce and study a general class of pyramidal representations that we refer to as Compression AlignmentPrediction (CAP) representations. The CAP representation is based on the selection of three filters: the lowpass decomposition filter, the lowpass prediction filter, and the fullpass alignment filter. Like previous pyrami dal algorithms, CAP are implemented by a simple, fast, waveletlike decomposition and a trivial reconstruction. The primary goal of this paper is to establish the precise way in which the CAP representations encode the smoothness class of the underlying function. Remarkably, the CAP coefficients provide the same characterizations of Triebel Lizorkin spaces and Besov spaces as the wavelet coefficients do, provided that the three CAP filters satisfy certain requirements. This means, at least in principle, that the performance of CAPbased algorithms should be similar to their wavelet counterparts, despite of the fact that, when compared with wavelets, it is much easier to develop CAP representations with “customized” or “optimal” properties. Moreover, upon assuming the prediction filter to be interpolatory, we extract from the CAP representation a sister CAMP representation (“M” for “modified”). Those CAMP representations strike a phenomenal balance between performance (viz., smoothness characterization) and space localization. Our analysis of the CAP representations is based on the existing theory of framelet (redundant wavelet) representations. 
Comments: 
In short:  A windowed portion of a chirp 
Etymology:  From chirp, an oscillating function whose "period" varies with the variable (e.g. time) position 
Origin:  Mann, Steve and Haykin, Simon, The chirplet transform: a
generalization of Gabor's Logon transform (local copy), Proc. Vision
Interface'91, June 37, pp. 205212, 1991.
Abstract: We propose a novel transform, an expansion of an arbitrary function onto a basis of multiscale chirps (swept frequency wave packets). We apply this new transform to a practical problem in marine radar: the detection of floating objects by their "acceleration signature" (the "chirpyness" of their radar backscatter), and obtain results far better than those previously obtained by other current Doppler radar methods. Each of the chirplets essentially models the underlying physics of motion of a floating object. Because it so closely captures the essence of the physical phenomena, the transform is near optimal for the problem of detecting floating objects. Besides applying it to our radar image processing interests, we also found the transform provided a very good analysis of actual sampled sounds, such as bird chirps and police sirens, which have a chirplike nonstationarity, as well as Doppler sounds from people entering a room, and from swimmers amid sea clutter.Mihovilovic, D. and Bracewell, R., Adaptive chirplet representation of signals on timefrequency plane (local copy), Electronic Letters, 27(13), pp. 11591161, June 1991. Abstract: Dynamic spectra, which exhibit the spectral content of a signal as time elapses, are based on subdivision of the timefrequency plane into minimumarea rectangular cells. The cell dimensions in time and frequency are usually held constant throughout. A more general spectral analysis would allow the cells to change aspect ratio with time. Elementary cells assuming oblique forms (chirplets) are proposed, together with an adaptive method for selecting their aspect ratio and obliquity to suit the data.The chirplet transform: physical considerations (local copy), Mann, S. and Haykin, S., IEEE Trans. Signal Processing, 1995 Abstract: We consider a multidimensional parameter space formed by inner products of a parameterizable family of chirp functions with a signal under analysis. We propose the use of quadratic chirp functions (which we will call qchirps for short), giving rise to a parameter space that includes both the timefrequency plane and the timescale plane as 2D subspaces. The parameter space contains a “timefrequencyscale volume” and thus encompasses both the shorttime Fourier transform (as a slice along the time and frequency axes) and the wavelet transform (as a slice along the time and scale axes). In addition to time, frequency, and scale, there are two other coordinate axes within this transform space: shear in time (obtained through convolution with a qchirp) and shear in frequency (obtained through multiplication by a qchirp). Signals in this multidimensional space can be obtained by a new transform, which we call the “qchirplet transform” or simply the “chirplet transform”. The proposed chirplets are generalizations of wavelets related to each other by 2D affine coordinate transformations (translations, dilations, rotations, and shears) in the timefrequency plane, as opposed to wavelets, which are related to each other by 1D affine coordinate transformations (translations and dilations) in the time domain only 
Contributors:  Steve
Mann and Simon Haykin Domingo Mihovilovic and Ronald Bracewell (wiki) 
Some properties:  Offers a mapping from a continuous function of one real
variable to a continuous function of 56 real variables.
Quadratic (as opposed to linear) chirplets are also of interest for radar
applications. Adaptive or even 
Anecdote:  The chirplet formulation was motivated by the discovery that the Doppler radar backscatter from a small piece of ice floating in an ocean environment is chirplike. Examples of chirps are the sounds made by birds where the resonant cavity changes size while oscillating 
Usage:  Radar applications, projective geometry acting on a periodic
structure (e.g. arcades in a perspective
picture) 
See also:  Several publications on chirplets on Steve Mann's page, and a wikipedia page chirplets with a reference to wchirplets as warblets 
Comments:  The "independent" discovery and naming controversy of chirplets by two groups at about the same time is not even discussed here 
In short:  Multiscale arcbased dictionary with constrainted curvature and endpoints 
Etymology:  From chord. It ought to be straight line connecting two points on a curve. Here a chord (reminiscent of a beamlet) subtends a set of arcs 
Origin:  He, Z. and Bystrom, M. The chordlet transform with an application to shape compression, Signal Processing: Image Communication, 2012. (chordlet local copy)Due to their abilities to succinctly capture features at different scales and directions, waveletbased decomposition or representation methods have found wide use in image analysis, restoration, and compression. While there has been a drive to increase the representation ability of these methods via directional filters or elongated basis functions, they still have been focused on essentially piecewise linear representation of curves in images. We propose to extend the linebased dictionary of the beamlet framework to one that includes sets of arcs that are quantized in height. The proposed chordlet dictionary has elements that are constrained at their endpoints and limited in curvature by system rate or distortion constraints. This provides a more visually natural representation of curves in images and, furthermore, it is shown that for a class of images the chordlet representation is more efficient than the beamlet representation under tight distortion constraints. A data structure, the fat quadtree and an algorithm for determining an optimal chordlet representation of an image are proposed. Codecs have been implemented to illustrate applications to both lossy and lossless low bitrate compressions of binary edge images, and better rate or rate–distortion performance over the JBIG2 standard and a beamletbased compression method are demonstrated.He, Z. Texture and structurebased image representation with applications to image retrieval and compression, PhD Thesis, Boston university, 2007. (chordlet local copy) The design of an efficient image representation methods using small numbers of features can facilitate image processing tasks such as compression of images and contentbased retrieval of images from databases. In this dissertation, three methods for capturing and concisely representing two distinguishing characteristics of images, namely texture and structure, are developed. Applications of these compact representations of image characteristics to image compression as well as retrieval of images and handsketches of images from databases are given and performance is compared with other compression and retrieval methods. The first method to be introduced is a directional, hiddenMarkovmodelbased method for succinctly describing image texture using a small number of features. This method employs the well known, multiscale contourlet and steerablepyramid transforms to isolate in different subbands the edges that comprise the image texture. Statistical inter and intrasubband dependencies are captured via hidden Markov models, and model parameters are used to represent texture in small feature sets. Application of this method to contentbased retrieval of images with homogeneous textures from database is shown. At the similar computation cost, about 10% higher retrieval rates over comparable methods are demonstrated; when approximately one third fewer features are used, similar retrieval rates can be obtained using the proposed method. A method for concisely describing large image structures, that is, significant image edges, is then proposed. This method decomposes an image using the contourlet transform into directional subbands which contain edges of different orientations. Each subband is then projected onto its associated primary and orthogonal directions and the resulting projections are filtered and then modeled using piecewise linear approximations or Gaussian mixture models. The model parameters then form the concise feature sets used to represent the image's structure. An application of this imagerepresentation method to retrieval of images from databases based on users' sketches of the images is shown. An retrieval rate increase of 13% using the proposed method is demonstrated over a current spatialhistogrambased method. Finally, a new multiscale curve representation framework, the chordlet, is constructed for succinct curvebased image structure representation. This framework can be viewed as an extension to curves of the well known beamlet transform, a multiscale line representation system. In this dissertation, the representation efficiency, in terms of bits versus distortion, of the chordlet transform is compared with that of the beamlet transform. An algorithm for performing a fast chordlet transform has been developed. A chordletbased coding system is constructed for application of the chordlet transform to compression of image shapes. By using the proposed method increased compression is obtained at lower distortion when compared with two well known methods. 
Contributors:  Zhihua He and M. Bystrom 
Some properties: 
Uses a fat quadtree

Anecdote:  
Usage:  Image compression, especially contour/shape compression (JBIG2, JBEAM) 
See also:  Chordlets extends beamlet dictionary. Directionlets and bandlets do not stand afar. 
Comments: 
In short:  The result of a convolution between a limited width circular shape and a wavelet 
Etymology:  Wavelet in circles 
Origin: 
Chauris, H., Karoui, I., Garreau, P., Wackernagel, H., Craneguy, P. and Bertino, L.,
The circlet transform: A robust tool for detecting features with circular shapes (local copy), Computers & Geosciences, 201103, Vol. 37, N. 3, P. 331342 Hervé Chauris et al., Ocean eddy tracking with circlets, GeoInformatics for Environmental Surveillance (StatDIS 2009) 
Contributors:  Hervé Chauris 
Some properties:  
Anecdote:  The Circlet (wikipedia), a.k.a. stephanos is a ancient type of crown without arches or cap, often used as a bridal or fairy attributes (aren't they the same?) According to Medieval Bridal Fashions, "It will work with any hairstyle." With any Haar Styl too? 
Usage:  Coastal oceanography and ocean eddy tracking 
See also:  
Comments: 
In short:  Orthogonal compactly supported wavelet with vanishing moments equally distributed for the scaling function and the wavelet 
Etymology:  Contraction from the name of R. R. Coifman 
Origin:  Daubechies, Ingrid, Orthonormal bases of compactly supported wavelets II. Variations on a theme (local copy), SIAM, J. Math. Anal., vol. 24, no. 2, pp. 499519, March 1993 
Contributors:  Ingrid Daubechies 
Some properties:  For p vanishing moments, the minimum support size of the wavelet is 3p1 (instead of 2p1 for Daubechies wavelets). Scaling functions with vanishing moments help establish precise quadrature formulas 
Anecdote:  In 1989, R. Coifman proposed the idea of constructing orthogonal wavelets with vanishing moments equally distributed for the scaling function and wavelet 
Usage:  Numerical analysis 
See also:  Other classical compactly supported orthogonal Daubechies wavelets (aka daublet), with minimum phase property or the nearly symmetric symmlets. The cooklet stands for a biorthogonal nearly coiflet 
Comments: 
In short:  A discrete domain waveletlike expansion allowing contour description, based on a Laplacian pyramid and a directional filter bank 
Etymology:  
Origin:  Do, M. N. and Vetterli, M. Contourlets: A Directional Multiresolution Image Representation, Proc. of IEEE International Conference on Image Processing ( ICIP), Rochester, September 2002 
Contributors:  Minh N. Do, Martin Vetterli, with Arthur L. Cunha and Jianping Zhou for the contourlet nonsubsampled version, and Yue Lu for the critically sampled CRISPcontourlet 
Some properties:  Approximation rate: M ^{2}(log M)^{3} for images having discontinuities along C^{2} curves. Slightly redundant due to the Laplacian pyramid. 
Anecdote:  
Usage:  Image coding, denoising 
See also:  The CRISPcontourlet, a critically sampled avatar (by Y. Lu and M. N. Do, SPIE 2003) 
Comments:  Contourlet toolbox Matlab code available at www.ifp.uiuc.edu/~minhdo/software/, with a Nonsubsampled Contourlet Transform Matlab toolbox at MatlabCentral 
In short:  Biorthogonal nearly coiflet 
Etymology:  Named after Dr. T. Cooklev for his construction of the oddlength biorthogonal coiflets, and the let< /td> 
Origin:  Winger, L. L. and Venetsanopoulos, A. N. Biorthogonal nearly coiflet wavelets for image compression, Signal Processing: Image Communication, Volume 16, Issue 9, June 2001, Pages 859869, see also an early version: Winger, L. L. and Venetsanopoulos, A. N. Biorthogonal modified coiflet filters for image compression 
Contributors:  Lowell L. Winger, Anastasios (Tas) Venetsanopoulos 
Some properties:  
Anecdote:  
Usage:  Image compression 
See also:  
Comments: 
In short:  Crap stuff in the wavelet domain, esp. broken wavelet code 
Etymology:  Simply from crap 
Origin:  Meerwald, Peter, The craplet page (assorted broken Wavelet code) 
Contributors:  Peter Meerwald 
Some properties:  Searches for crappy wavelet code 
Anecdote:  
Usage:  For clean wavelet code. See Craplets by Peter Meerwald for examples 
See also:  
Comments:  Akin to Sturgeon's Law: Ninety percent of everything is crap (or crude) 
In short:  Multiscale elongated and rotated functions that defines (bases or) frames in L^{2}(R^{2}) 
Etymology:  Simply from curved wavelets 
Origin:  Candès, E. J. and Donoho, D. L., Curvelets  a surprinsingly effective nonadative representation for objects with edges, in Curve and Surface fitting, A. Cohen, C. Rabut and L. L. Schumaker (Eds.), 1999 
Contributors:  Emmanuel Candès, David
Donoho, JeanLuc Starck Laurent Demanet 
Some properties:  Approximation rate: M ^{2}(log M)^{3} for images having discontinuities along C^{2} curves 
Anecdote:  
Usage:  
See also:  
Comments:  Curvelets have evolved both in concept and implemetation since the earlier works, dealing with what's now called "curvelets 99", which relied to some extend on ridgelets. Second generation curvelet code is available at http://www.curvelet.org, with version 2.0 
In short:  Orthogonal compactly supported wavelet with a maximal number of vanishing moments for some given (finite) support. A Daublet is each member of Daubechies's extremal phase family. 
Etymology:  Nickname for orthogonal Daubechies wavelets 
Origin:  Contraction from the name of Ingrid Daubechies 
Contributors:  () 
Some properties:  
Anecdote:  
Usage:  
See also:  Other classical compactly supported orthogonal Daubechies wavelets with approximate symmetry, the symmlets, or with vanishing moments equally distributed on the scaling function and of the wavelet, the coiflets. Armlets are multiwavelets that restrict to Daubechies wavelets in the scalar case 
Comments: 
In short:  
Etymology:  
Origin:  Velisavljevic, Vladan and BeferullLozano, Baltasar and Vetterli, Martin and Dragotti, Pier Luigi, Directionlets: Anisotropic multidirectional representation with separable filtering, submitted to IEEE Transactions on Image Processing (Dec. 2004) 
Contributors:  Vladan Velisavljevic, Baltasar BeferullLozano, Martin Vetterli, Pier Luigi Dragotti 
Some properties:  
Anecdote:  
Usage:  
See also:  No public toolbox available, but additional details on Vladan Velisavljevic webpage 
Comments: 
In short:  Element for a collection of edgels (small line segments forming an edge) connecting vertices on the boundary of a dyadic square 
Etymology:  From edge or edgel, an edge element in the computer vision literature 
Origin:  David L. Donoho, Manuscript, Stanford University, Fast edgelet transform and applications, Manuscript, September 1998 
Contributors:  David Donoho 
Some properties:  
Anecdote:  
Usage:  
See also:  
Comments:  Edgelets might be combined with wavelet for an overcomplete image representation, as in Donoho, D. and Huo, X., Combined Image representation using edgelets and wavelets ??? 
In short:  A linear and invertible timefrequency transformation adapted to human auditory perception, for masking and perceptual sparsity 
Etymology:  From the ERB scale or Equivalent Rectangular Bandwidth filter banks, devised for auditory basedrepresentation, following the philosophy of thirdoctave filter banks. See also Frequency Analysis and Masking  MIT, Brian C. J. Moore, 1995 and Bark and ERB Bilinear Transforms  Stanford University, by J. O. Smith III 
Origin:  Thibaud Necciari, Design and implementation of the ERBlet transform, FLAME 12 (Frames and Linear Operators for Acoustical Modeling and Parameter Estimation), 2012
Timefrequency representations are widely used in audio applications involving sound analysissynthesis. For such applications, obtaining a timefrequency transform that accounts for some aspects of human auditory perception is of high interest. To that end, we exploit the theory of nonstationary Gabor frames to obtain a perceptionbased, linear, and perfectly invertible timefrequency transform. Our goal is to design a nonstationary Gabor transform (NSGT) whose timefrequency resolution best matches the timefrequency analysis properties by the ear. The peripheral auditory system can be modeled in a first approximation as a bank of bandpass filters whose bandwidth increases with increasing center frequency. These socalled “auditory filters” are characterized by their equivalent rectangular bandwidths (ERB) that follow the ERB scale. Here, we use a NSGT with resolution evolving across frequency to mimic the ERB scale, thereby naming the resulting paradigm "ERBlet transform". Preliminary results will be presented. Following discussion shall focus on finding the "best" transform settings allowing to achieve perfect reconstruction while minimizing redundancy.Thibaud Necciari with P. Balazs, B. Laback, P. Soendergaard, R. KronlandMartinet, S. Meunier, S. Savel, and S. Ystad, The ERBlet transform, auditory timefrequency masking and perceptual sparsity, 2nd SPLab Workshop, October 24–26, 2012, Brno The ERBlet transform, timefrequency masking and perceptual sparsity Timefrequency (TF) representations are widely used in audio applications involving sound analysissynthesis. For such applications, obtaining an invertible TF transform that accounts for some aspects of human auditory perception is of high interest. To that end, we combine results of nonstationary signal processing and psychoacoustics. First, we exploit the theory of nonstationary Gabor frames to obtain a linear and perfectly invertible nonstationary Gabor transform (NSGT) whose TF resolution best matches the TF analysis properties by the ear. The peripheral auditory system can be modeled in a first approximation as a bank of bandpass filters whose bandwidth increases with increasing center frequency. These socalled “auditory filters” are characterized by their equivalent rectangular bandwidths (ERB) that follow the ERB scale. Here, we use a NSGT with resolution evolving across frequency to mimic the ERB scale, thereby naming the resulting paradigm “ERBlet transform”. Second, we exploit recent psychoacoustical data on auditory TF masking to find an approximation of the ERBlet that keeps only the audible components (perceptual sparsity criterion). Our longterm goal is to obtain a perceptually relevant signal representation, i.e., as close as possible to “what we see is what we hear”. Auditory masking occurs when the detection of a sound (referred to as the “target” in psychoacoustics) is degraded by the presence of another sound (the “masker”). To accurately predict auditory masking in the TF plane, TF masking data for masker and target signals with a good localization in the TF plane are required. To our knowledge, these data are not available in the literature. Therefore, we conducted psychoacoustical experiments to obtain a measure of the TF spread of masking produced by a Gaussian TF atom. The ERBlet transform and the psychoacoustical data on TF masking will be presented. The implementation of the perceptual sparsity criterion in the ERBlet will be discussed. 
Contributors:  Thibaud Necciari with P. Balazs, B. Laback, P. Soendergaard, R. KronlandMartinet, S. Meunier, S. Savel, and S. Ystad 
Some properties:  Develops a nonstationary Gabor transform (NSGT) [Theory, Implementation and Application of Nonstationary Gabor Frames, P. Balazs et al., J. Comput. Appl. Math., 2011] with resolution evolving over frequency to mimic the ERB scale (Equivalent Rectangular Bandwidth, after B. C. J. Moore and B. R. Glasberg, "Suggested formulae for calculating auditoryfilter bandwidths and excitation patterns", J. Acoustical Society of America 74:750753, 1983). Linear and invertible timefrequency transform adapted to human auditory perception. 
Anecdote:  
Usage:  A few ERBlet Matlab scripts for ICASSP 2013 are downloadable at the ERBlet transform project listing. An implementation of the ERBlet transform is available in the excellent The Large TimeFrequency Analysis Toolbox, also known as the LTFAT toolbox ("All your frame are belong to us") 
See also:  
Comments: 
In short:  A basis made of M adjacent box function scalets (scaling functions) and $M$ piecewise constant functions with $M$ vanishing moments 
Etymology:  From flat, meaning... flat, and again, let 
Origin:  Steven J. Gortler, Peter Schröder, Michael F. Cohen, Pat Hanrahan Wavelet radiosity, Computer Graphics, SIGGRAPH 1993 
Contributors:  Steven J. Gortler, Peter Schröder, Michael F. Cohen, Pat Hanrahan, 
Some properties:  For the given example, 2 rows of the twoscale relationship are orthogonal to constant and linear variations 
Anecdote:  
Usage:  Sparse basis for hierarchical radiosity formulation, to solve the global illumination problem 
See also:  
Comments: 
In short:  Element of a wavelet frame or the wavelet frame by itself 
Etymology:  From frame, an extension from the (vector) base concept 
Origin:  Ingrid Daubechies, Bin Han, Amos Ron, Zuowei Shen, Framelets: MRABased Constructions of Wavelet Frames (local copy), 2000 
Contributors:  Ramesh A. Gopinath (phaselets of framelets) 
Some properties:  
Anecdote:  The framelet term was also introduced in the field of software framework to designate nonoverlapping groups of logically related design patterns and interfaces. Those interested could take a look at Alessandro Pasetti homepage. 
Usage:  
See also:  Many developments on framelets (inpainting, deconvolution, restoration, missing samples recovery) by Zuowei Shen and coauthors, for instance in Jianfeng Cai, Raymond Chan, Lixin Shen, Zuowei Shen, Convergence analysis of tight framelet approach for missing data recovery, Advances in Computational Mathematics,xx (200x) or in Anwei Chai, Zuowei Shen, Deconvolution: A wavelet frame approach, Numerische Mathematik, 106 (2007), 529587 
Comments: 
In short:  Waveletlike basis made of a wavelet basis combined with a unitary Fresnel transform. 
Etymology:  From the Fresnel transform, after the name of physicist Augustin Jean Fresnel (MacTutor History) 
Origin:  Liebling, M., Blu, T., Unser, M., Fresnelets — A New Wavelet Basis for Digital Holography, Proceedings of the SPIE Conference on Mathematical Imaging: Wavelet Applications in Signal and Image Processing IX, San Diego CA, USA, July 29 August 1, 2001, vol. 4478, pp. 347352 
Contributors:  Michael Liebling, Thierry Blu, Michael Unser 
Some properties:  
Anecdote:  
Usage:  Reconstruction and processing of optically generated Fresnel holograms recorded on CCDarrays 
See also:  Liebling, M., Blu, T., Unser, M., Fresnelets: New Multiresolution Wavelet bases for digital holography, Proceedings of the IEEE Transactions on Image processing, vol. 12, no. 1, January 2003 [pdf] 
Comments: 
In short:  Complex exponentials modulated by a "smooth" function, originally a Gaussian 
Etymology:  From the name of the godfather Denis Gabor, and especially his Theory of Communication paper, Journal of the IEE, vol. 93, pp. 429457, 1946 
Origin:  Not clear, but named in some papers, esp. by Bruno Torrésani, Timefrequency and timescale analysis, Signal Processing for multimedia, J. S. Byrnes (Ed.), IOS Press, 1999 
Contributors:  Bruno Torrésani 
Some properties:  
Anecdote:  
Usage:  
See also:  
Comments: 
In short:  Non linear and nonparametric estimator of generalized additive models with wavelets 
Etymology:  Generalized Additive Model wavelet estimator 
Origin:  Sardy, Sylvain and Tseng, Paul, Automatic Nonlinear Fitting of Additive Models and Generalized Additive Models with Wavelets, Journal of Computational and Graphical Statistics, 2004 (submitted) 
Contributors:  Sylvain Sardy, Paul Tseng 
Some properties:  Universal thresholding rule for Gaussian and Poisson distributions 
Anecdote:  
Usage:  Fitting of generalized additive models 
See also:  Its simpler version, called AMlet 
Comments:  Not truly a wavelet by itself 
In short:  
Etymology:  From the famous mathematician Johann Carl Friedrich Gauss (MacTutor History), and the ubiquituous bell curve named after him. Gauss is also believed to have discovered the Fast Fourier Transform (FFT algorithm) 
Origin:  Triebel H. Towards a Gausslet analysis : Gaussian representations of functions. In M. Cwikel, M. Englis, A. Kufner, L.E. Persson, and G. Sparr, editors, Function Spaces, Interpolation Theory and Related Topics. Proc. Conf. Lund, August 2000, 425450, de Gruyter Proceedings, 2002. 
Contributors:  Hans Triebel 
Some properties:  
Anecdote:  
Usage:  
See also:  
Comments: 
In short:  A nickname for wavelets on graphs 
Etymology:  From the Graph structure (as introduced by Sylvester in Nature, 1878) and let 
Origin: 
Wavelets on Graphs via Spectral Graph Theory, Applied and Computational Harmonic Analysis, 2011 (local copy, DOI)
David K. Hammond,
Pierre Vandergheynst
Rémi Gribonval
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. 
Contributors:  David K. Hammond, Pierre Vandergheynst Reacute;mi Gribonval 
Some properties:  
Anecdote:  http://en.wikipedia.org/wiki/Graphlets: The name "graphlet" for "wavelets on graphs" steals from another network technology.
Graphlets are small connected nonisomorphic induced subgraphs of a large network.[1][2] Graphlets differ from network motifs, since they must be induced subgraphs, whereas motifs are partial subgraphs. An induced subgraph must contain all edges between its nodes that are present in the large network, while a partial subgraph may contain only some of these edges. Moreover, graphlets do not need to be overrepresented in the data when compared with randomized networks, while motifs do. 
Usage:  
See also: 
The Spectral Graph Wavelets Matlab Toolbox page is now available, with a direct link to sgwt_toolbox1.01.zip (local copy). PySGWT, a python code port for graphlet (aka Spectral Graph Wavelet Transform). PySGWT
Narang, S. K. and Ortega, A.: Perfect Reconstruction TwoChannel Wavelet FilterBanks for Graph Structured Data, 2012, 32 pages double spaced 12 Figures, to appear in IEEE Transactions of Signal Processing Abstract: In this work we propose the construction of twochannel wavelet filterbanks for analyzing functions defined on the vertices of any arbitrary finite weighted undirected graph. These graph based functions are referred to as graphsignals as we build a framework in which many concepts from the classical signal processing domain, such as Fourier decomposition, signal filtering and downsampling can be extended to graph domain. Especially, we observe a spectral folding phenomenon in bipartite graphs which occurs during downsampling of these graphs and produces aliasing in graph signals. This property of bipartite graphs, allows us to design critically sampled twochannel filterbanks, and we propose quadrature mirror filters (referred to as graphQMF) for bipartite graph which cancel aliasing and lead to perfect reconstruction. For arbitrary graphs we present a bipartite subgraph decomposition which produces an edgedisjoint collection of bipartite subgraphs. GraphQMFs are then constructed on each bipartite subgraph leading to "multidimensional" separable wavelet filterbanks on graphs. Our proposed filterbanks are critically sampled and we state necessary and sufficient conditions for orthogonality, aliasing cancellation and perfect reconstruction. The filterbanks are realized by Chebychev polynomial approximations. Yue M. Lu: Spectral graph wavelet frames with compact supports, Wavelets and Sparsity, Proc. SPIE 2011 
Comments:  The Spectral Graph Wavelets Toolbox page (SGWT) is not to be mistaken with the SGWT = Second Generation Wavelet Transform. Also different from other Graphlets which are small connected nonisomorphic induced subgraphs of a large network 
In short:  Multiscale grouped coefficients through association fields 
Etymology:  From a grouping of (wavelet) coefficients) 
Origin:  Mallat, Stéphane, Geometrical Grouplets, submitted to ACHA  Applied and Computational Harmonic Analysis (Oct. 2006) 
Contributors:  Stéphane Mallat 
Some properties:  
Anecdote:  
Usage:  
See also:  
Comments: 
In short:  A notsocommon nickname for the Haar wavelet 
Etymology:  From hungarian mathematician Alfréd Haar (MacTutor History) 
Origin:  Haar,
Alfréd, Zur Theorie der orthogonalen FunktionenSysteme, Math.
Ann., vol. 69, pp. 331371, 1910 (On the Theory of Orthogonal
Function Systems, translated for the magnificent collection of papers in Fundamental Papers in Wavelet Theory edited by
Christopher Heil and David F. Walnut)
In RealTime Body Pose Recognition Using 2D or 3D Haarlets (Internation Journal on Computer Vision, 2009), Van den Bergh et al. abbreviate a combination of Average Neighborhood Margin Maximization (ANMM) and (Viola and Jones 2001) Haar waveletlike features as "Haarlets". 
Contributors:  Alfred Haar 
Some properties:  A Schauder basis, unconditional for Lp spaces, p > 1. Discontinuous 
Anecdote: 
Celebrate Haar wavelet centenary with the following Memorial plaque in honor of A. Haar and F. Riesz
found at Szeged University: the inscription says:
"A szegedi matematikai iskola világhírű
megalapítói (The worldwide
famous founders of the mathematical school in Szeged)" [picture courtesy of Professor
Károly Szatmáry]. The picture is a natural testbench for directional/textural analysis.

Usage:  Often considered of poor performance in "real life" applications, the Haar wavelet may prove very efficient if used cleverly (for instance Fast Haarwavelet denoising of multidimensional fluorescence microscopy data, F. Luisier et al., ISBI 2009). Much sooner, an avatar of the 2D Haar transform, under the name of "HTransform" (at MathWorld), as been used for astronomical image compression (Hcompress Image Compression Software ), originated in Fritze, K.; Lange, M.; Möstle, G.; Oleak, H.; and Richter, G. M. "A Scanning Microphotometer with an OnLine Data Reduction for Large Field Schmidt Plates." Astron. Nachr. 298, 189196, 1977. 
See also:  Wikipedia: Haar wavelet or the Multilevel Haar Transform at Connexions (Rice University) 
Comments: 
In short:  
Etymology:  From a pun on mathematicians Alfréd Haar and Jacques Hadamard: Ha(dam)ard. Reminicent to the Waleymard transform, build upon J. L. Walsh, Raymond E.A.C. Paley and Jacques Hadamard, depending on the basis ordering (resp. sequency, dyadic or natural), see Wolfram Walsh page for instance 
Origin:  
Contributors:  GrandAdmiral Petry 
Some properties:  
Anecdote:  
Usage:  
See also:  
Comments: 
In short:  The heat evolution of an initial wavelet state  
Etymology:  From the heat equation and the diminutive let  
Origin:  On Wavelet Fundamental Solutions to the Heat EquationHeatlets, Shen, Jianhong and Strang, Gilbert, Journal of Differential Equations, 2000
We present an application of wavelet theory in partial differential equations. We study the wavelet fundamental solutions to the heat equation. The heat evolution of an initial wavelet state is called a heatlet. Like wavelets for the L2 space, heatlets are "atomic'' heat evolutions in the sense that any general heat evolution can be "assembled'' from a heatlet according to some simple rules. We study the basic properties and algorithms of heatlets and related functions 

Contributors:  Jackie (Jianhong) Shen  Gilbert Strang 
Some properties:  
Anecdote:  
Usage:  
See also:  A few lines from Image Compression and Wavelet Applications at UCLA  
Comments: 
In short:  Biorthogonal wavelet with the Hut function as the father wavelet 
Etymology:  From Hut, German for hat 
Origin:  MeyerBäse, Uwe Die Hutlets  eine biorthogonale WaveletFamilie: Effiziente Realisierung durch multipliziererfreie, perfekt rekonstruierende Quadratur Mirror Filter , Frequenz., vol; 51, p. 3949, 1997, also in MeyerBäse, Uwe and Taylor, F., The Hutlets  a biorthogonal wavelet family and their high speed implementation with RNS, multiplierfree, perfect reconstruction QMF 
Contributors:  Uwe MeyerBäse 
Some properties:  The Hut function has an asymptotically fast decrease in amplitude. Multiplierfree implementation with the residue number system (RNS). Synthesis filters are IIR 
Anecdote:  Notice the first author name; is MeyerBäse related to
the Meyer wavelet basis? Other wavelets reveal a similar kind of hat trick: the Mexican hat wavelet (also known as the Ricker wavelet) and the 
Usage:  Envelope discontinuity detection in amplitude modulation 
See also:  A scaling function in the hutlet may be view as an instance of a binlet 
Comments:  The Hut function was defined by W. Hilberg, Impulse und Impulsfolgen, die durch Integration oder Differentiation in einem veränderten Zeitmasstab reproduziert werden, Arch. für Eltr. Übertr. (AEÜ), vol. 25, pp. 3948, 1971. It results from the infinite convolution of rectangles with area one (2^{k}/T)r(T/2^{ k}), k varying from 1 to infinity 
Comments: 
In short:  An example of multicomposite wavelets with hyperbolic scaling law 
Etymology:  From the hyperbola (wiki entry), with a potential reference (article no available on 2011/05/26) to the parabolic scaling law of the shearlets 
Origin:  Glenn R. Easley, Demetrio Labate, Vishal M. Patel: Multicomposite wavelet estimation, Proceedings of SPIE Volume 8138, Wavelets and Sparsity XIV, Aug. 2011 (local copy)
Abstract: In this work, we present a new approach to image denoising by using a general representation known as wavelets with composite dilations. These representations allow for waveforms to be defined not only at various scales and locations but also at various orientations. For this talk, we present many new representations such as hyperbolets and propose combining multiple estimates from various representations to form a unique denoised image. In particular, we can take advantage of different representations to sparsely represent important features such as edges and texture independently and then use these estimates to derive an improved estimate.The hyperbolet construction is further refined in: G. R. Easley, D. Labate and V. M. Patel, Hyperbolic shearlets, IEEE International Conference on Image Processing (ICIP), Orlando, FL, 2012, submitted (local copy) G. R. Easley, D. Labate, and V. M. Patel, Directional multiscale processing of images using wavelets with composite dilations, submitted 2011 (local copy) 
Contributors:  Glenn R. Easley (no personal page), Demetrio Labate, Vishal M. Patel 
Some properties:  Tiling of the frequency domain associated with an hyperbolic system of wavelets with composite dilations. Closely related to shearlets 
Anecdote:  
Usage:  
See also:  The above work might be related to Glenn R. Easley, Demetrio Labate: Critically Sampled Wavelets with Composite Dilations (local copy), preprint, 2011, which develops interesting critically sampled directional wavelet schemes (DWTShear, CShear, QDWTShear) 
Comments:  See also: Hyperbolets (on WITS: Where is the Starlet) 
In short:  Independent Component Analysis by Wavelets 
Etymology:  Concatenation of ICA, a standard method for blind source separation, and let 
Origin: 
Independent Component Analysis by Wavelets, Pascal Barbedor, Preprint, 2005, published in Test, 2009
This paper introduces a new approach in solving the ICA problem using a method that fits in the contrast and minimize paradigm, mostly found in the ICA literature. In our case, the contrast is a L_2 norm dependence measure, which constitutes an alternative to the usual criteria, based on mutual information. We propose a non parametric evaluation of the L_2 contrast, using a wavelet projection estimator. The mean square error of the procedure is bounded under Besov assumptions. Finally, we provide a set of simulations to show how the method performs in practice.Independent component analysis and estimation of a quadratic functional, Pascal Barbedor, Preprint, 2006 Independent component analysis (ICA) is linked up with the problem of estimating a non linear functional of a density, for which optimal estimators are well known. The precision of ICA is analyzed from the viewpoint of functional spaces in the wavelet framework. In particular, it is shown that, under Besov smoothness conditions, parametric rate of convergence is achieved by a Ustatistic estimator of the wavelet ICA contrast, while the previously introduced plugin estimator C^{2}_{j}, with moderate computational cost, has a rate in n^{4s/(4s+d)}.Independent component analysis by wavelets, Pascal Barbedor, PhD thesis, 2006 Independent component analysis (ICA) is a form of multivariate analysis that emerged as a concept in the eighties/nineties. It is a type of inverse problem where one observes a variable X whose components are linear mixtures of an unobservable variable S. The components of S are mutually independent. The relation between both variables is expressed by X=AS, where A is an unknown mixing matrix. The main problem in ICA is to estimate the matrix A, seeing an i.i.d. sample of X, to reach S which constitutes a better explicative system than X, in the study of some phenomena. The problem is generally resolved through the minimization of a criteria coming from some dependence measure. ICA looks like principal component analysis (PCA) in the formulation. In PCA, one seeks after uncorrelated components, that is to say pairwise independent at order 2 ; as for ICA, one seeks after mutually independent components, which is much more constraining, and there is not any more a simple algebraic solution in the general case. The main problems in the identification of A are removed by restrictions imposed in the classical ICA model. The approach which is proposed in this thesis adopts a non parametric point of view. Under Besov assumptions, we study several estimators of an exact dependence criteria given by the L2 norm between a density and the product of its marginals. This criteria constitutes an alternative to mutual information which represented so far the exact criteria of reference for the majority of ICA methods. We give an upper bound of the mean squared error of different estimators of the L2 contrast. This bound takes into account the approximation bias between the Besov space and the projection space which, here, stems from a multiresolution analysis (MRA) generated by the tensorial product of Daubechies wavelets. This type of bound, taking into account the approximation bias, is generally absent from recent non parametric methods in ICA (kernel methods, mutual information). The L2 norm criteria makes it possible to get closer to wellknown problems in the statistical literature, estimation of integral of squared f, L2 norm homogeneity tests, convergence rates for estimators adopting block thresholding. We propose estimators of the L2 contrast that reach the optimal minimax rate of the problem integral of squared f. These estimators, of Ustatistic type, have numerical complexities quadratic in n, which can be a problem for the contrast minimization to follow, to obtain a concrete estimation of matrix A. However these estimators also admit a blockthresholded version, where knowledge of the regularity s of the underlying multivariate density is useless to obtain an optimal rate. We propose a plugin type estimator whose convergence rate is suboptimal but with a numerical complexity linear in n. The plugin estimator also admits a term by term thresholded version, which dampens the convergence rate but yields an adaptive criteria. In its linear version, the plugin estimator already seems autoadaptive in facts, that is to say under the constraint 2^{jd} &<; n, where d is the dimension of the problem and n the number of observations, the majority of resolutions j allow to estimate A after minimization. To obtain these results, we had to develop specific combinatorial tools, that allow to bound the rth moment of a Ustatistic or a Vstatistic. Standard results on Ustatistics are indeed not directly usable and not easily adaptable in the context of study of the thesis. The tools that were developed are usable in other contexts. The wavelet method builds upon the usual paradigm, estimation of an independence criteria, then minimization. So we study in the thesis the elements useful for minimization. In particular we give filter aware formulations of the gradient and the hessian of the contrast estimator, that can be computed with a complexity equivalent to that of the estimator itself. Simulations proposed in the thesis confirm the applicability of the method and give excellent results. All necessary information for the implementation of the method, and the commented code of key parts of the program (notably ddimensional algorithms) also appear in the document. 
Contributors:  Pascal Barbedor (old page: http://www.proba.jussieu.fr/pageperso/barbedor/) 
Some properties:  
Anecdote:  
Usage:  
See also:  
Comments:  Fortran source code and Mac OS X Pascal Barbedor icalet binaries 
In short:  Interpolating wavelet transform 
Etymology:  Combination of Interpolation and let 
Origin:  Apparently, Donoho, D. L. (once again), Interpolating wavelet transforms 1992, technical report, Stanford university, although the name "interpolet" itself has been coined later (local pdf copy). 
Contributors:  David Donoho 
Some properties:  Loosely speaking, based on the autocorrelation of some scaling function or interpolating filter 
Anecdote:  Early mention of interpolets is found in "Savior of the Nations, Come"
by St. Ambrose, (340397). Seventh verse:
Praesepe iam fulget tuum, lumenque nox spirat suum, quod nulla nox interpolet fideque iugi luceat. 
Usage:  
See also:  
Comments: 
In short:  Properties of filter sets used in local structure estimation that are the most important 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 certainty of the estimate can be obtained byfinding the deviation from the signal model manifold. 
Etymology:  From Logarithmic wavelets 
Origin:  Knutsson, Hans and Andersson, Mats, Loglets  Generalized Quadrature and Phase for Local Spatiotemporal Structure Estimation, 2003, Scandinavian Conference on Image Analysis Knutsson, Hans and Andersson, Mats, Implications of invariance and uncertaintyfor local structure analysis filter sets, 2005, Signal Processing: Image Communication 
Contributors:  Hans Knutsson Mats Andersson 
Some properties:  Polar separable filter banks in the Fourier domain 
Anecdote:  
Usage:  
See also:  
Comments: 
In short:  A sort of Mband wavelet 
Etymology:  From MIMO (Multipleinput/Multipleouput) systems generating wavelets 
Origin:  The netherlands, the other cheese country 
Contributors:  Will remain anonymous (none of the famous dutch wavelet school) 
Some properties:  Wavelets with frequencies in the orange tones. 
Anecdote:  
Usage:  Tasteful for RAClet and TARTIFlet recomposition (pun borrowed from "TB from CH", aka "TB from HK"). Mband wavelets (such as the dualtree wavelets, see Mband dualtree and discrete complex wavelets, a blog entry: PhD thesis award on Mband dualtree wavelets or Wikipedia, Complex Wavelet) in filter bank form, since they are related to the LOT (Lapped Orthogonal Transform), may be called "bancs de LOT(tes)" ("lote/lotte" the fish, not the transform) in french 
See also:  A recent MIMOlet preprint 
Comments:  Still waiting for SISOlets, MISOlets and SIMOlets 
In short:  Short name for the Morlet wavelet 
Etymology:  A clever combinaison of the father Jean Morlet and the mother wavelet 
Origin:  Misprint found in a preprint 
Contributors:  Will remain anonymous 
Some properties:  No parentchild dependency known to date. Its dual basis (the lesslet?) remains to be described (or even defined). 
Anecdote:  Morelet is also the name of a crocodile, or Crocodylus moreletii, from the French naturalist P. M. A. Morelet (18091892), who discovered this species in 1850 in Mexico. I expect P. M. A. Morelet came back from Mexico with a Mexican hat (or Sombrero), which is one of the most famous wavelet shape, known as the sombrero, mexican hat, Ricker wavelet or Marrlet (from the work of David Marr) 
Usage:  The Morelet wavelet is becoming increasingly popular due to
three typical wavelet phenomena:

See also:  The future invention of the lesslet 
Comments:  (Relative) fun exists in DSP, as in the invention of Softy space (see Hardy spaces), or in company names like Let it wave 
In short:  A multiscale representation for diffeomorphisms 
Etymology:  A contraction of both Morphing or Morphism or and the mother wavelet 
Origin:  Jonathan R. Kaplan and David L. Donoho, The Morphlet Transform: A Multiscale Representation for Diffeomorphisms (local copy), Workshop on Image Registration in Deformable Environments, 2006 
Contributors:  Jonathan R. Kaplan and David L. Donoho 
Some properties:  
Anecdote:  
Usage:  
See also:  
Comments: 
In short:  Sort of vector extension to the standard "scalar wavelet" based on multiple scaling functions and wavelet functions rather than a single pair 
Etymology:  Multi+let 
Origin:  
Contributors:  
Some properties:  
Anecdote:  
Usage:  
See also:  Multiwavelet Alpert Transform Matlab toolbox by Gabriel Peyré, Multiwavelet matlab code by Vasily Strela (refering page not available, but a local copy of the matlab multiwavelet toolbox is made available), another Multiwavelet MATLAB Package at MatlabCentral, a last wavelet and multiwavelet Matlab package by Fritz Keinert at CRC Press. 
Comments: 
In short:  A breed of spherical wavelets, with needle shape 
Etymology:  From their needle shape + let 
Origin:  P. Baldi, G. Kerkyacharian, D. Marinucci, D. Picard, Asymptotics for Spherical Needlets Also in D. Marinucci, D. Pietrobon, A. Balbi, P. Baldi, P. Cabella, G. Kerkyacharian, P. Natoli, D. Picard, N. Vittorio, Spherical Needlets for CMB Data Analysis (arXiv page) 
Contributors:  
Some properties:  Do not rely on any tangent plane approximation. Computationally attractive. Same needlets functions are present in the direct and the inverse transform. Quasiexponentially concentrated (hence, the needle shape). Random needlets coefficients can be shown to be asymptotically uncorrelated 
Anecdote:  
Usage:  Cosmic Microwave Background (CMB) analysis, cosmological data processing 
See also:  
Comments: 
In short:  Sort of twisted wavelet packets, maximally incoherent system with respect to the Haar wavelet 
Etymology:  From the signalprocessingubiquitous noise + let 
Origin:  R. Coifman, F. Geshwind, and Y. Meyer, Noiselets, Appl. Comp. Harmonic Analysis, 10:2744, 2001 
Contributors:  Ronald Coifman, F. Geshwind, Yves Meyer 
Some properties:  Perfectly incoherent with the Haar basis (similar to the perfect incoherence of the canonical basis with respect to the Fourier basis), cf. T. Tuma and P. Hurley, On the incoherence of noiselet and Haar bases, Proc. SAMPTA 2009 (local copy) . Can be decomposed as a multirate filter bank. Binary valued real and imaginery parts (see the recent discussion Some comments on noiselets by Laurent Jacques, mentioned Yves Meyer: Compressed Sensing, Quasicrystals, Wavelets and Noiselets.) 
Anecdote:  Mark Noiselet is a makeup artist.
Have a look at this interesting page by artist Michael Thieke:
Very sparse. Very minimal. These musicians make sounds with their instruments that may not have been intended by the original inventors. They do this in a way that at first seems to be a very random. After a longer listen, the inspirations soak through. These “noiselets and sounduals” (my words entirely) may be improvised, but they are very potent in their expressive capability. In Art is Arp  When art (noiselets) meets wavelets and compressive sensing, paintings by François Morellet vaguely ressemble noiselets noiselets. 
Usage:  Compressed sensing 
See also:  The noiselets have been recently mentioned in a paper by J.P. Allouche and G. Skordev, Von Koch and ThueMorse revisited (arXiv page), which links fractal objects and automatic sequences, focused on the ThueMorse sequence and the Von Koch curve. See also: Sparsity and Incoherence in Compressive Sampling by Emmanuel Candès and Justin Romberg. 
Comments:  Basic Noiselet Matlab code for building orthogonal noiselet bases (or Zipped Matlab code (or eventually there Zipped Matlab code)). Other more interesting (faster, higher, stronger) codes are provided at Compressive Imaging Code by Justin Romberg, and especially at A Fast (1D and 2D) Noiselet Transform by Laurent Jacques. 
In short:  An approximately shiftinvariant redundant dyadic wavelet transform 
Etymology:  
Origin:  Gopinath, Ramesh A. The phaselet transform  an integral redundancy nearly shiftinvariant wavelet transform 
Contributors:  Ramesh A. Gopinath 
Some properties:  
Anecdote:  
Usage:  
See also:  
Comments: 
In short:  Compactly supported basis functions ressembling planar structures, for the representations of locally planar structures found in video sequences 
Etymology:  From plane, the common name for a flat surface 
Origin:  Rajpoot, N. and Wilson, R. and Yao, Zhen Planelets: A new analysis tool for planar feature extraction, International Workshop on Image Analysis for Multimedia Interactive Services (WIAMIS), 2004 
Contributors:  Nasir Rajpoot, Roland Wilson, Zhen Yao 
Some properties:  Non orthogonal basis and redundant by less than 14% (see the paper: can a basis really be redundant?) 
Anecdote:  
Usage:  Video sequence denoising 
See also:  
Comments: 
In short:  Partition based on a recursive, dyadic squares, allowing wedgeshaped final nodes (instead of squares), with piecewise planar value 
Etymology:  From Plate, accounting for the piecewise planar value 
Origin:  Willett, R. M. and Nowak, R. D. Platelets: A Multiscale Approach for Recovering Edges and Surfaces in PhotonLimited Medical Imaging, preprint ??? 
Contributors:  Rebecca M. Willett, Robert D. Nowak 
Some properties:  Well suited for the approximation of images consisting in smooth regions separated by smooth contours, especially in the case of Poisson distributions 
Anecdote:  Platelets used to be a major component of blood. They are not anymore 
Usage:  Analysis, denoising, reconstruction of images, esp. Poisson distributed (medical imaging) 
See also:  The wedgelet, which it generalizes upon 
Comments:  A platelet Matlab toolbox (for Mac, Unix, Windows) by Rebecca Willett and Robert Nowak. See also platelets for photonlimited image reconstruction 
In short:  Local line whose family is a basis for discrete signals 
Etymology:  From the Radon transform (which is performed along lines), after the Austrian mathematician Johann Radon 
Origin:  Do, M. N. and Vetterli, M. The contourlet transform: an efficient directional multiresolution image representation, IEEE Transactions Image Processing, vol. 14, no. 12, pp. 20912106, Dec. 2005 
Contributors:  Minh N. Do, Martin Vetterli 
Some properties:  Element of a family having almost linear support and different orientations, defined by translating some filters over some sampling lattices 
Anecdote:  The radonlet concept represents only an item on the above paper 
Usage:  
See also:  
Comments: 
In short:  Randlets are randomlychosen basis functions 
Etymology:  From random 
Origin:  Malkin, Michael and Venkatesan, Ramarathnam, The randlet transform, Allerton 2004, 
Contributors:  Michael Malkin, Ramarathnam Venkatesan 
Some properties:  
Anecdote:  
Usage:  Universal Perceptual Hashing, image verification, watermarking 
See also:  
Comments: 
In short:  Ranklets are a complete family of multiscale rank features characterized by Haarwavelet style orientation selectivity 
Etymology:  From rank, since they are related to Wilcoxon rank sum test 
Origin:  Smeraldi, F. Ranklets: orientation selective nonparametric features applied to face detection, Proceedings of the 16th International Conference on Pattern Recognition, Quebec QC, vol. 3, pages 379382, August 2002 
Contributors:  Fabri Smeraldi 
Some properties:  
Anecdote:  
Usage:  Face detection 
See also:  
Comments:  The ranklet page, software available upon request 
In short:  
Etymology:  
Origin:  () 
Contributors:  David Donoho, E.
Candès Minh N. Do, Martin Vetterli, Image denoising using orthonormal finite ridgelet transform, Proc. of SPIE Conference on Wavelet Applications in Signal and Image Processing VIII, San Diego, USA, August 2000 
Some properties:  
Anecdote:  In the La Recherche montly (Number 383, Feb. 2005, p. 5559), Mathieu Nowak and Yves Meyer propose the translation arêtelette 
Usage:  
See also:  
Comments: 
In short:  A kind prewavelet or, more recently, some generalizations to curvelets and ridgelets 
Etymology:  Contraction from ripple and let 
Origin:  Tentative name origin: Goodman, T. N. T. and Micchelli, C. A.,
On refinement equations determined by Pólya frequency sequence, SIAM J. Math. Anal., vol. 23, pp. 766784, 1992
And now for completely different rippling ones: rippletI and rippletII (or type1 and type2 ripplets), generalizations of curvelets and ridgelets, respectively. Jun Xu, Lei Yang, Dapeng Wu Ripplet: A new transform for image processing (local copy), Journal of Visual Communication and Image Representation, Oct. 2010, and Ripplet transform II Transform for Feature Extraction (local copy), IET Image processing, June 2012. matlab codes are available below. 
Contributors:  Tim N. T. Goodman, Charles A. Micchelli, Dapeng Oliver Wu 
Some properties:  
Anecdote:  The idea behind the original or first generation ripplet (Goodman and Micchelli) is an intermediate between concepts
on refinable functions (satisfying a refinement or scaling
equation) and the positivity of the coolocation matrices of their
shifts. The stronger concept is known as a Pólya frequency
function. In earth sciences, a ripplet denotes a tiny ripple. It is also, in computer science, "A powerful, high available, flexible, enterpriselevel load/stress test framework" (Java code). 
Usage:  First generation ripplets are used to build prewavelets by F. Pitolli,
Refinement masks of Hurwitz type in the cardinal
interpolation problem, Rendicondi di Matematica, Serie VII,
vol. 18, pp. 473287, Roma 1998. Ripplet properties are also
valuable in computeraided geometric design, for instance in
Goodman, T. N. T. Total positivity and the shape of
curves, in Total positivity and its applications,
M. Gasca and C. A. Micchelli (Eds.), p. 157186, 1996.
Applications of second generation ripplets to image fusion: Home > Vol. 30 > pp. 355370 Medical image fusion based on ripplet transform typeI By S. Das, M. Chowdhury, and M. K. Kundu, Progress In Electromagnetics Research B, 2011. 
See also:  Goodman, T. N. T. and Sun, Q., Total positivity and refinable functions with general dilation, 2004, preprint 
Comments:  Example of an everydaydaylife ripplet: a Bspline with integer knots
determined by a constant coefficient differential operator, whose characteristic polynomial only has real zeros (of first type).
For second generation ripplets, Matlab codes are available at Dapeng Oliver Wu publications page: Matlab code for type I Ripplet (local copy) and Matlab code for type II Ripplet (local copy) 
In short:  Efficient multiresolution representation for retargeting applications 
Etymology: seam and wavelet for a generalization of the DWT and seam carving function  
Origin:  Seamlets: ContentAware Nonlinear Wavelet Transform, David D. Conger, Hayder Radha, Mrityunjay Kumar, ICASSP 2010 
Contributors:  
Some properties:  
Anecdote:  
Usage:  
See also:  
Comments: 
In short:  A form of wavelet decomposition based on seismic wavefield and data properties 
Etymology:  From seismic, one of the origin of the wavelet transform 
Origin:  Fomel, Sergey, Towards the seislet transform SEG (Society of Exploration Geophysicists) Annual Conference (2006) or Seislet transform and seislet frame Geophysics 75, V25 (2010). 
Contributors:  Sergey Fomel, with a baptism by Huub Douma 
Some properties:  The seislet provides a multiscale transform aligned along seismic event slopes in seismic data. Definition based on the wavelet lifting scheme combined with local planewave destruction. 
Anecdote:  The name "seislet" was, according to Sergey Fomel, suggested by Huub Douma 
Usage:  
See also:  Page on the seislet transform at www.reproducibility.org or the Madagascar development blog 
Comments: 
In short:  2D set of functions based on the product of a gaussian with a Hermite (or Laguerre) polynomial (tensor product of 1D function) 
Etymology:  From shape 
Origin:  Refregier, Alexandre and Chang and Bacon, David, Shapelets: A New Method to Measure Galaxy Shapes. Proceedings of the Workshop "The Shapes of Galaxies and their Halos", Yale, May 2001 
Contributors:  Alexandre Refregier, David Bacon 
Some properties:  Possess 4 degrees of freedom. Standard image operations are possible in the shapelet space: translations, scaling, small angle rotations, convolutions, shear estimation, flux/radius/centroid measurements 
Anecdote:  Same functions arise in the solution of the quantum harmonic oscillator 
Usage:  Useful for the representation (and compression) of astronomical objects, object classification or galaxy morphology 
See also:  Shapelets webpage by Richard Massey and Alexandre Refregier, much pointers to papers, IDL shapelets software, animations Links on shapelets by Christopher Spitzer 
Comments:  Not yet public Matlab and C++ code available from Christopher Spitzer. Shapelets are also cited in programs by P. Kovesi for Computer Vision, IDL shapelet software by Massey and Refregier 
In short:  Nonseparable wavelets built out of parabolic scaling, shear, and translation operations 
Etymology:  From shear, a sheer distorsion 
Origin:  Labate, Demetrio and Lim, WQ. and Kutyniok, Gitta and Weiss Guido, Sparse multidimensional representation using shearlets (local copy) A handful lot of papers is available here: shealet papers. A first overview is given in Shearlets. The First Five Year (Oberwolfach Report, 2010, local copy). 
Contributors: 
Guido Weiss Gitta Kutyniok Demetrio Labate Kanghui Guo (Scholar) Gabriele Steidl Sören Häuser 
Some properties:  Unlike curvelets, shearlets form an affine system with a single generating mother shearlet function parameterized by a scaling, a shear, and a translation parameter. Provides the same approximation properties as curvelets, albeit with a different directional sensitivity. Exist in bandlimited or compact support flavors. Possess natural, canonical smoothness spaces, called shearlet coorbit spaces, similar to Besov spaces for wavelets. Apparently extend to arbitrary any dimensions. 
Anecdote:  
Usage:  Image denoising, restoration, Morphological component analysis 
See also:  The shearlet website, recently updated with ShearLab (... a rationally designed digital shearlet transform)
For discrete implementation, there exists for instance a Digital Shearlet Transforms or Development of a Digital Shearlet Transform Based on PseudoPolar FFT.
Shearlet Matlab toolboxes are available at ShearLab matlab toolboxes,
local shearlet toolbox by G. Easley and FSST:

Comments:  Potential a hard competitor, for years to come, to the quite oversold curvelets (IMHOlet: In My (little) Humble Opinion). See also Shearlets, after MIA 2012 
In short:  A short name for the wavelet function associated with the cardinal sine (aka sinc function) scaling function 
Etymology:  From sine cardinal function 
Origin:  Unknown, but cited in some papers, such as Mammogram enhancement using a class of smooth wavelets, by Shi, Z. Zhang, D., Wang, H., Kouri, D. and Hoffman, D., (local pdf), submitted to IEEE 33rd Asilomar Conference on Signals, Systems, and Computers, 1999), or Generalized symmetric interpolating wavelets , by Shi, Z., Kouri, D., Wei G. W. and Hoffman, D., Computer Physics Communications, 1999 (local pdf) 
Contributors:  
Some properties:  
Anecdote:  
Usage:  
See also:  
Comments: 
In short:  A singlesided dampled Laplace wavelet transform for modal analysis 
Etymology:  A single mode subsystem related wavelet 
Origin:  Named in RealTime Identification of Flutter Boundaries Using the Discrete Wavelet Transform, J. D. Johnson, Jun Lu, Atam P. Dhawan, Journal of guidance, control and dynamics, Vol. 25, N. 2, 2002, pages 334339. Appeared in Correlation filtering of modal dynamics using the Laplace wavelet, L. Freudinger, R. Lind and M. Brenner, International Modal Analysis Conference, Santa Barbara, CA, February 1998, pp. 868877. See also the NANSA report: 
Contributors:  
Some properties:  Complex, analytic, singlesided damped exponential 
Anecdote:  A singlet is also the name of the attire worn by competitors in the sport of wrestling 
Usage:  Modal analtsis; Flutter identification 
See also:  
Comments: 
In short:  A piecewise linear (but discontinuous) wavelet basis reminiscent of the slant transform 
Etymology:  Using slant, "to strike obliquely" (against something), alteration of slenten "slip sideways" (see etymology and modern meaning clever of superior) 
Origin:  I. W. Selesnick, The slantlet transform, IEEE Trans on Signal Processing, vol 47, no 5, pp 13041313, May 1999 
Contributors:  Ivan Selesnick 
Some properties:  Piecewise linear basis with two zero moments, orthogonal, based on the iteration of different filter banks at each scale 
Anecdote:  Ivan Selesnick's page for slantlet 
Usage:  Image denoising 
See also:  Matlab Source code available at http://taco.poly.edu/selesi/slantlet 
Comments: 
In short:  Continuous generalization of (scond order) wedgelets 
Etymology:  From the smooth, "free from roughness, not harsh" (with interesting etymology and modern meaning clever of superior), and the diminutive let of the wavelet 
Origin:  Agnieszka Lisowska, Smoothlets  Multiscale Functions for
Adaptive Representation of Images, IEEE Trans on Signal Processing, July 2001, Volume: 20 Issue: 7, 17771787 (local copy)
In this paper a special class of functions called smoothlets is presented. They are defined as a generalization of wedgelets and secondorder wedgelets. Unlike all known geometrical methods used in adaptive image approximation, smoothlets are continuous functions. They can adapt to location, size, rotation, curvature, and smoothness of edges. The Mterm approximation of smoothlets is O(M^3) . In this paper, an image compression scheme based on the smoothlet transform is also presented. From the theoretical considerations and experiments, both described in the paper, it follows that smoothlets can assure better image compression than the other known adaptive geometrical methods, namely, wedgelets and secondorder wedgelets. 
Contributors:  Agnieszka Lisowska 
Some properties:  Adaptive geometrical decomposition. Adapt to location, size, rotation, curvature and smoothness of edges. The Mterm approximation of smoothlets is O(M^3) 
Anecdote:  
Usage:  Compression 
See also:  
Comments: 
In short:  A set of mother wavelets, replicated at the different positions and scales of the pyramid and which allow for a translation and scale invariant representation of images 
Etymology:  From the sparse nature of some wavelet representations (and the let) 
Origin:  Laurent Perrinet Dynamical Neural Networks: modeling lowlevel vision at short latencies, The European Physical Journal, 2007 (local copy) 
Contributors:  Laurent Perrinet 
Some properties:  
Anecdote:  
Usage:  
See also:  
Comments:  To be discussed: a seemingly abusive use of sparsity 
In short:  A wavelet transform matching a specified discretetime signal 
Etymology:  From experimental spikes that need to be matched in a signal 
Origin:  Rodrigo Capobianco Guido, Jan Frans Willem Slaets, Roland Kouml;berle, Lírio Onofre Batista Almeida and José Carlos Pereira A new technique to construct a wavelet transform matching a specified signal with applications to digital, real time, spike, and overlap pattern recognition, Digital Signal Processing, 2006 (local copy) 
Contributors:  
Some properties:  
Anecdote:  A spikelet is also a kind of raceme, a small or secondary spike, characteristic of grasses and sedges, having a varying number of reduced flowers each subtended by one or two scalelike bracts.

Usage:  
See also:  
Comments: 
In short:  A not socommon nickname for Bspline wavelets 
Etymology:  From sline+let, obviously 
Origin:  
Contributors:  
Some properties:  
Anecdote:  
Usage:  
See also:  
Comments: 
In short:  Steerable wavelets in 3D 
Etymology:  From steer for the "directional" prefix (as in "cyber" from Greek kubernete) 
Origin:  Papadakis, Azencott and Bodmann at Univ. Houston Three dimensional steerlets: a novel tool for extractiong textural and structural features in 3D images, SPIE Wavelet XIII, August 2009 Azencott, Bodmann, Papadakis at Univ. Houston Steerlets: A novel approach to rigidmotion covariant multiscale transforms, preprint 
Contributors:  Manos Papadakis, Robert Azencott, Bernhard G. Bodmann, 
Some properties:  Steerlets form a new class of wavelets suitable for extracting structural and textural features from 3Dimages. These wavelets extend the framework of Isotropic Multiresolution Analysis and allow a wide variety of design characteristics ranging from isotropy, that is the full insensitivity to orientations, to directional and orientational selectivity. The primary characteristic of steerlets is that any 3Drotation of a steerlet is expressed as a linear combination of other steerlets associated with the same IMRA, yielding 3Drotation covariant fast wavelet transforms. Resulting subband decompositions covariant under the action of rotations. 
Anecdote:  A steer is also a young male of ox type, which is nice from Ol'Texas contributors.

Usage:  
See also:  A WhereIsTheStarlet entry: WITS: Steerlet wavelets from La vertu d'un LA 
Comments: 
In short:  A SURE (Stein's Unbiased Risk Estimate) method for wavelet denoising 
Etymology:  From SURE, acronym for "Stein's Unbiased Risk Estimate" and LET for "Linear Expansion of Thresholds" 
Origin:  Luisier, F. and Blu, T. and Unser, M., A New SURE Approach to Image Denoising: InterScale Orthonormal Wavelet Thresholding, IEEE Transactions on Image Processing, vol. 16, no. 3, pp. 593606, March 2007. [pdf] 
Contributors:  Florian Luisier, Thierry Blu, Michael Unser 
Some properties:  
Anecdote:  
Usage:  Denoising, see bigwww.epfl.ch/demo/suredenoising/ and bigwww.epfl.ch/research/projects/denoising.html for SURE/PURELET and CURELET (A CURE for noisy magnetic resonance images: Chisquare unbiased risk estimation) denoising. On the page Signal and processing (Matlab) codes, a SureLET denoising toolbox for oversampled complex filter banks is offered. 
See also:  
Comments:  SURElet are also related to property rental services, funnily enough related to the word Kingsbury (not Nick)
Surelet  Property Rental Services Surelet 'to let' branding image (top) To Let  Thinking of Letting your Property or ... Gloucester, Hatfield, Hemel Hempstead, Kingsbury, Oldham, Reading ... www.surelet.co.uk/kingsbury/as in the Activelet case. And the PURELET case as well: http://www.purelet.co.uk/ Welcome to Purelet Letting Agency 
In short:  A 3D directional multiresolution analysis, combining a 3D directional filter bank and a Laplacian pyramid 
Etymology:  From surface, obviously 
Origin: 
Lu, Yue and Do, Minh N.
Multidimensional Directional Filter Banks and Surfacelets
IEEE Transactions on Image Processing, , vol. 16, no. 4, April 2007
(pdf)
Lu, Yue and Do, Minh N. 3D directional filter banks and surfacelets Proc. of SPIE Conference on Wavelet Applications in Signal and Image Processing XI, San Diego, USA, Jul. 2005, invited paper (pdf) 
Contributors:  Yue Lu, Minh N. Do 
Some properties:  Redundancy factor up to 24/7 in 3D for the 2005 SPIE version, about 4.05 for the 2006 preprint 
Anecdote:  
Usage:  
See also:  
Comments:  SurfBox: : MATLAB and C++ toolbox implementing the NDFB and the surfacelet transform as described in the paper Multidimensional directional filter banks and surfacelets 
In short:  Representation for approximation and compression of Horizonclass functions containing a C ^{K} smooth discontinuity in N1 dimensions 
Etymology:  From surface 
Origin:  Chandrasekaran, V. Compression of higher dimensional
functions containing smooth discontinuities, 29th Annual
Spring Lecture Series, Recent Developments in Applied Harmonic
Analyis, Multiscale Geometric Analysis, April 1517, 2004
Chandrasekaran, V. and Wakin, M. B. and Baron, D. and Baraniuk, R. G. Representation and Compression of MultiDimensional Piecewise Functions Using Surflets, Preprint (pdf) 
Contributors:  Venkat Chandrasekaran, Mike Wakin, Dror Baron, Richard G. Baraniuk 
Some properties:  
Anecdote:  
Usage:  
See also:  
Comments: 
In short:  Tetrominobased Haar like wavelet 
Etymology:  From the tetrostructured representation and the ubiquitous let 
Origin:  Jens Krommweh, Tetrolet Transform: A New Adaptive Haar Wavelet Algorithm for Sparse Image Representation (local copy), J. Vis. Commun. Image R., Vol. 21, No. 4 (2010) 364374. 
Contributors:  Jens Krommweh 
Some properties:  
Anecdote:  
Usage:  
See also:  A WhereIsTheStarlet entry: WITS: Tetrolet wavelets from La vertu d'un LA 
Comments:  Tetrolet matlab Toolbox by Jens Krommweh 
In short:  An adaptive method combining multiscale representation and eigenanalysis 
Etymology:  From the treestructured representation and the ubiquitous let 
Origin:  Ann B. Lee, Boaz Nadler, and Larry Wasserman Treelets  An Adaptive MultiScale Basis for Sparse Unordered Data (local copy), to appear in Annals of Applied Statistics 
Contributors:  
Some properties:  Dimensionality reduction and feature selection tool; Based on the Jacobi method, it groups together a each level of the tree, the most similar variables and replace them by a coarsegrained "sum variable" and a residual "difference variable" computed by a local PCA 
Anecdote:  The treelet is a small tree 
Usage:  Blocked covariance models; Hyperspectral Analysis and Classification of Biomedical Tissue; Internet Advertisement Data Set 
See also:  Treelet Matlab code 
Comments:  The term was coined before by people at Microsoft: Chris Quirk, Arul Menezes and Colin Cherry, Dependency Treelet Translation: Syntactically Informed Phrasal SMT, July 2005. 
In short:  A "waveletlike" bounded, continuous function which, under
the action of a specific standardization operator (dilation +
translation), satisfies a set of axioms related to

Etymology:  Vague means "wave" in French, as far as liquids are concerned (especially the sea), but also in a more vague sense. Vaguelette could be described as a moderate size ripple, a small wave vanishing on the shore. It could thus be read as "wavelet" or ondelette in a limited sense. Or more precisely, "les vaguelettes sont de vagues ondelettes" 
Origin:  Meyer, Yves, Ondelettes et opérateurs: II. Opérateurs de Calderón Zygmund, 1990, p. 270, Hermann et Cie, Paris 
Contributors:  Yves Meyer 
Some properties:  
Anecdote:  
Usage:  
See also:  WaveletVaguelette 
Comments: 
In short:  Often described as a wavelet analogue to the singular value decomposition. Wavelets and vaguelettes act like "reciprocal" under the action of an linear operator (and its transpose) 
Etymology:  Composition of wavelet and vaguelette. The resulting acronym (WVD for waveletvaguelette decomposition) is reminiscent of that of the SVD (singular value decomposition) 
Origin:  David L. Donoho, Nonlinear solution of linear problems by waveletvaguelette decomposition, 1992, Stanford, Research report (also in App. and Comp. Harmonic Analysis, 2, 1995) 
Contributors:  David L. Donoho 
Some properties:  This decomposition exists for a class of special linear inverse problems of homogeneous type (numerical differentiation, Radon transform, inversion of Abeltype transforms). Improves upon SVD inversion for the recovery of spatially inhomogeneous objets 
Anecdote:  
Usage:  Solution of Nonlinear PDEs via adaptive WaveletVaguelette decomposition, (by J. Fröhlich and K. Schneider, KonradZuseZentrum Berlin, Preprint SC 9528) 
See also:  Vaguelette 
Comments: 
In short:  
Etymology:  let 
Origin:  
Contributors:  
Some properties:  
Anecdote:  
Usage:  
See also:  
Comments: 
In short:  An affine deformation of the Gabor wavelet (aka gaborlet) for Clerc and Mallat, or imagedependent patchlike wavelet representations based on PCA for Bhalerao and Wilson 
Etymology:  From warp, for a twist or distorsion (of a shape) 
Origin:  Clerc, Maureen and Mallat, Stéphane, The Texture
Gradient Equation for recovering Shape from Texture IEEE
Transactions on Pattern Analysis and Matching Intelligence, pp. 536549, vol. 24, no. 4, April 2002 (local copy).
Abstract: Studies the recovery of shape from texture under perspective projection. We regard shape from texture as a statistical estimation problem, the texture being the realization of a stochastic process. We introduce warplets, which generalize wavelets over the 2D affine group. At fine scales, the warpogram of the image obeys a transport equation, called texture gradient equation. In order to recover the 3D shape of the surface, one must estimate the deformation gradient, which measures metric changes in the image. This is made possible by imposing a notion of homogeneity for the original texture, according to which the deformation gradient is equal to the velocity of the texture gradient equation. By measuring the warplet transform of the image at different scales, we obtain a deformation gradient estimator Bhalerao, Abhir and Wilson, Roland, Warplets: An imagedependent wavelet representation, IEEE International Conference on Image Processing (ICIP 2005) (local copy, poster). Abstract: A novel imagedependent representation, warplets, based on selfsimilarity of regions is introduced. The representation is well suited to the description and segmentation of images containing textures and oriented patterns, such as fingerprints. An affine model of an image as a collection of selfsimilar image blocks is developed and it is shown how textured regions can be represented by a single prototype block together with a set of transformation coefficients. Images regions are alligned to a set of dictionary blocks and their variability captured by PCA analysis. The blocktoblock transformations are found by Gaussian mixture modelling of the block spectra and a leastsquares estimation. Clustering in the Warplet domain can be used to determine a warplet dictionary. Experimental results on a variety of images demonstrate the potential of the use of warplets for segmentation and coding, Proc. IEEE International Conference on Image Processing (ICIP) 2005, September 2005. 
Contributors:  Maureen Clerc, Stéphane Mallat Abhir Bhalerao and Roland Wilson 
Some properties:  A four scale operator related to a transport equation called the "texture gradient equation". Addresses the problem known as "shape to texture", i.e. the retrieval of 3D shapes from a textured perspective image 
Anecdote:  For a stochastic process, the variance of the warplets coefficients is called a warpogram 
Usage:  Texture and shape problems 
See also:  Recent works (2004, 2005) on a somewhat different warplets by Abhir Bhalerao and Roland Wilson, thought as imagedependent patchlike wavelet representations based on PCA (principal component analysis, see the following tutorial on PCA) 
Comments:  Also associated with the names of R. Baraniuk and D. L. Jones in a talk by X. Huo, 1999, but no accurate reference found to date 
In short:  Partition based on a recursive, dyadic squares, allowing wedgeshaped final nodes (instead of squares), with piecewise constant value 
Etymology:  
Origin:  David L. Donoho, Wedgelets: Nearlyminimax estimations of edges, Ann. Statist., vol. 27, pp. 353382, 1999 
Contributors:  David Donoho 
Some properties:  NearlyMinimax estimation of edges. The analysis performance is controlled by a key parameter d (the wedgelet resolution), which accounts for the spacing between nodes of the square perimeter 
Anecdote:  
Usage:  A software package for image segmentation is distributed on www.wedgelet.de 
See also:  The platelet generalization 
Comments: 
In short:  A generic name for a wannabee wavelet (before it actually gets its name or waiting to be invented) 
Etymology:  
Origin:  Probably diffuse, but attested in: Do, M. N. and Vetterli, M. The contourlet transform: an efficient directional multiresolution image representation, IEEE Transactions Image Processing, 2005, [pdf] and several other talks by these authors 
Contributors:  Minh N. Do, Martin Vetterli 
Some properties:  
Anecdote:  Man gave names to all the xlets, in the beginning, long time ago (as well to all the animals, long time ago) 
Usage:  
See also:  
Comments: 
Javatype Xlet
An Xlet is very similar to a Java applet and is originally introduced in Sun's Java TV specification to support applications for Digital TV. Though Xlet looks superficially different from other application models in Java such as applet and MIDlet, it is actually meant to be a generalization of such models. 
In short:  (Linear) frame of directional wavelets with variable angular selectivity 
Etymology:  Multiselective wavelet 
Origin:  Jacques, Laurent and Antoine, JeanPierre, Multiselective Pyramidal Decomposition of Images: Wavelets with Adaptive Angular Selectivity, International Journal of Wavelets, Multiresolution and Information Processing, 2007 [pdf][pdf][paper] 
Contributors:  Laurent Jacques, 
Some properties:  
Anecdote:  
Usage:  
See also:  
Comments: 
In short:  The "first spherical Haar wavelet": Orthogonal and Symmetric Haar Wavelets on the Sphere (and extensions) 
Etymology:  Symmetric Orthogonal Haar wavelet 
Origin:  Lessig, Christian
Orthogonal and Symmetric Haar Wavelets on the Sphere, Master of Science thesis 2007,
[pdf][local copy]
Abstract: We propose the SOHO wavelet basis. To our knowledge this is 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. The key to the derivation of the basis is a novel spherical subdivision scheme that defines a partition acting as domain of the basis functions. The construction of the SOHO wavelets refutes earlier claims doubting the existence of such a basis. We also investigate how signals represented in our new basis can be rotated. Experimental results for the representation of spherical signals verify that the superior theoretical properties of the SOHO wavelet basis are also relevant in practice.Lessig, Christian and Fiume, E. Orthogonal and Symmetric Haar Wavelets on the Sphere, ACM Transactions of Graphics, SIGGRAPH 2008, [pdf][local copy] 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 all frequency 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.Chow, Andy. Orthogonal and Symmetric Haar Wavelets on the ThreeDimensional Ball, Master's thesis, 2010, University of Toronto, Toronto, [pdf][local copy] Abstract: 3D SOHO is the first Haar wavelet basis on the threedimensional ball that is both orthogonal and symmetric. These theoretical properties allow for a fast wavelet transform, optimal approximation and perfect reconstruction. 
Contributors:  Christian Lessig
Eugene Fiume Andy Chow 
Some properties:  
Anecdote:  SOHO denotes many things. Among which the SoHo neighborhood in Manhattan (for South of Houston Street), New York and the SOlar and Heliospheric Observatory. The latter may likely be the motivation for wavelets on the sphere. 
Usage:  
See also:  
Comments: 
Domain:  Painting (and music)  
Description:  Cetacean Stills or Shape of the Sound, still paintings based on continuous wavelet transform diagrams of dolphins and whale recording (whalets?).  
Comments: 
Domain:  Painting  
Description:  Art gallery inspired by wavelets (esp. splines), by Annette Unser.  
Comments:  Example for a subset of fractional splines: 
Domain:  Art authentication  
Description:  Le Spy art or ArtSpy, an algorithm to detected the artist of the painting with the discrete wavelet transform, by a team at Rice University, Houston, TX, USA. Tests on Rembrandt, Monet, or Picasso.  
Comments: 