WITS: Where Is The Starlet? (wavelet names in *let)

• Morlet, J., Arens, G., Fourgeau, I. and Giard, D., Wave propagation and sampling theory, Geophysics, 47, pp. 203-236, 1982
  
• Morlet, J., Sampling theory and wave propagation, in NATO ASI Series, vol. 1, Issues in Acoustics signal/Image processing and recognition, C. H. Chen (Ed.), Springer Verlag, pp. 233-261, 1983

Abstract:

Abstract: We propose a new framework to extract the activity-related 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 sparsely-distributed stimuli, as is the case in slow event-related fMRI. We introduce new wavelet bases, termed ``activelets'', which sparsify the activity-related BOLD signal. These wavelets mimic the behavior of the differential operator underlying the hemodynamic system. To recover the sparse representation, we deploy a sparse-solution search algorithm. The feasibility of the method is evaluated using both synthetic and experimental fMRI data. The importance of the activelet basis and the non-linear sparse recovery algorithm is demonstrated by comparison against classical B-spline wavelets and linear regularization, respectively.

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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.

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.

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.

Abstract: 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 beamlet-based algorithms which are able to identify and extract beamlets and chains of beamlets with special properties. In this paper we describe a four-level 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 parent-child dependence between scales; the third level incorporates collinearity and co-curvity 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).

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Due to their abilities to succinctly capture features at different scales and directions, wavelet-based 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 line-based 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 beamlet-based compression method are demonstrated.

The design of an efficient image representation methods using small numbers of features can facilitate image processing tasks such as compression of images and content-based 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 hand-sketches 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, hidden-Markov-model-based method for succinctly describing image texture using a small number of features. This method employs the well known, multi-scale contourlet and steerable-pyramid 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 content-based 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 piece-wise 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 image-representation 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 spatial-histogram-based method. Finally, a new multi-scale curve representation framework, the chordlet, is constructed for succinct curve-based image structure representation. This framework can be viewed as an extension to curves of the well known beamlet transform, a multi-scale 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 chordlet-based 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.

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.

          
Praesepe iam fulget tuum,

lumenque nox spirat suum,

quod nulla nox interpolet

fideque iugi luceat. 

• keyboard aliasing (with typing frequency and proximity of the keys "e" and "r" on Azerty and Qwerty keyboards),
• dilation of the number of people working with wavelets,
• translation (and replication) due to (erroneous) citations.

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.

• ShearLab-1.1 (Implementation based on Cartesian Grids) (direct link)
• ShearLab-PPFT-1.0 (Implementation based on Pseudo-Polar Grids) (direct link)
• The recent (2012) FFST - Fast Finite Shearlet Transform by Sören Häuser

  The FFST package provides a fast implementation of the Finite Shearlet Transform. Following the the path via the continuous shearlet transform, its counterpart on cones and finally its discretization on the full grid we obtain the translation invariant discrete shearlet transform. Our discrete shearlet transform can be efficiently computed by the fast Fourier transform (FFT). The discrete shearlets constitute a Parseval frame of the finite Euclidean space such that the inversion of the shearlet transform can be simply done by applying the adjoint transform.

  with a Fast Finite Shearlet Transform tutorial (local copy)

In this paper a special class of functions called smoothlets is presented. They are defined as a generalization of wedgelets and second-order 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 M-term 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 second-order wedgelets.

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1. fast decay/localization
2. null mean value/oscillation
3. miminal regularity

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 all-frequency 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.

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.

Abstract: 3D SOHO is the first Haar wavelet basis on the three-dimensional ball that is both orthogonal and symmetric. These theoretical properties allow for a fast wavelet transform, optimal approximation and perfect reconstruction.