WITS = Where Is The Starlet? (wavelet names ending with *let)

[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]

Abstract: An arbitrary square integrable real-valued 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 self-reciprocal 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 Lz-theory. They are written in terms of a modified f-function 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 +b-group

• Morlet, J., Arens, G., Fourgeau, I. and Giard, D., Wave propagation and sampling theory (pdf as part I: Complex signal and scattering in multilayered media and part II: Sampling theory and complex waves ), Geophysics, 47, pp. 203-236, 1982

  Abstract: From experimental studies in digital processing of seismic reflection data, geophysicists know that a seismic signal does vary in amplitude, shape, frequency and phase, versus propagation time. To enhance the resolution of the seismic reflection method, we must investigate these variations in more detail.We present quantitative results of theoretical studies on propagation of plane waves for normal incidence, through perfectly elastic multilayered media.As wavelet shapes, we use zero-phase cosine wavelets modulated by a Gaussian envelope and the corresponding complex wavelets. A finite set of such wavelets, for an appropriate sampling of the frequency domain, may be taken as the basic wavelets for a Gabor expansion of any signal or trace in a two-dimensional (2-D) domain (time and frequency). We can then compute the wave propagation using complex functions and thereby obtain quantitative results including energy and phase of the propagating signals. These results appear as complex 2-D functions of time and frequency, i.e., as 'instantaneous frequency spectra.'Choosing a constant sampling rate on the logarithmic scale in the frequency domain leads to an appropriate sampling method for phase preservation of the complex signals or traces. For this purpose, we developed a Gabor expansion involving basic wavelets with a constant time duration/mean period ratio.For layered media, as found in sedimentary basins, we can distinguish two main types of series: (1) progressive series, and (2) cyclic or quasi-cyclic series. The second type is of high interest in hydrocarbon exploration.Progressive series do not involve noticeable distortions of the seismic signal. We studied, therefore, the wave propagation in cyclic series and, first, simple models made up of two components (binary media). Such periodic structures have a spatial period. We present synthetic traces computed in the time domain using the Goupillaud-Kunetz model of propagation for one-dimensional (1-D) synthetic seismograms.Three different cases appear for signal scattering, depending upon the value of the ratio wavelength of the signal/spatial period of the medium.(1) Large wavelengths The composite medium is fully transparent, but phase delaying. It acts like an homogeneous medium, with an 'effective velocity' and an 'effective impedance.'(2) Short wavelengths For wavelengths close to twice the spatial period of the medium, the composite medium strongly attenuates the transmission, and superreflectivity occurs as counterpart.(3) Intermediate wavelengths For intermediate values of the frequency, velocity dispersion versus frequency appears.All these phenomena are studied in the frequency domain, by analytic formulation of the transfer functions of the composite media for transmission and reflection. Such phenomena are similar to Bloch waves in crystal lattices as studied in solid state physics, with only a difference in scale, and we checked their conformity with laboratory measurements.Such models give us an easy way to introduce the use of effective velocities and impedances which are frequency dependent, i.e., complex. They will be helpful for further developments of 'complex deconvolution.'The above results can be extended to quasi-cyclic media made up of a random distribution of double layers. For signal transmission, quasi-cyclic series act as a high cut filter with possible time delay, velocity dispersion, and 'constant Q' type of law for attenuation. For signal reflection they act as a low cut filter, with possible superreflections.These studies could be extended to three-dimensional (3-D) binary models (grains and pores in a porous reservoir), in agreement with well-known acoustic properties of gas reservoirs (theory of bright spots).We present some applications to real well data. Velocity dispersion may explain: mistying between sonic logs and velocity surveys; mistying between synthetic seismograms and seismic sections.Effective velocities lower than mean velocities may explain: very low velocities for P- and S-waves, especially in weathered shallow layers; anomalies on the values of the ratio Vs/Vp ; mistying between P and S seismic sections. Finally, the Gabor expansion provides a tool to obtain sampled instantaneous frequency spectra, and to carry out a suitable recording and processing method in high-resolution seismic, especially to preserve the phase information. Such a processing will involve complex signals, complex traces, complex velocities and complex impedances.For practical purpose, this paper comprises two separate parts. Here, we present some interesting features on the scattering of seismic signals obtained by a simulation method using simple models. We show the usefulness of the notions of complex signals, complex velocities, and complex impedances, and overall of the Gabor expansion, by simulation on very simple models. Morlet et al, (1982, this issue) will be concerned with the development of fundamental notions useful to handle seismic data from the sampling method of recording to processing methods. There we give theoretical and practical tools to sample and handle these data in the time-frequency domain, using complex functions.

  Abstract: Morlet et al (1982, this issue) showed the advantages of using complex values for both waves and characteristics of the media. We simulated the theoretical tools we present here, using the Goupillaud-Kunetz algorithm.Now we present sampling methods for complex signals or traces corresponding to received waves, and sampling methods for complex characterization of multilayered or heterogeneous media.Regarding the complex signals, we present a two-dimensional (2-D) method of sampling in the time-frequency domain using a special or 'extended' Gabor expansion on a set of basic wavelets adapted to phase preservation. Such a 2-D expansion permits us to handle in a proper manner instantaneous frequency spectra. We show the differences between 'wavelet resolution' and 'sampling grid resolution.' We also show the importance of phase preservation in high-resolution seismic.Regarding the media, we show how analytical studies of wave propagation in periodic structured layers could help when trying to characterize the physical properties of the layers and their large scale granularity as a result of complex deconvolution. Analytical studies of wave propagation in periodic structures are well known in solid state physics, and lead to the so-called 'Bloch waves.'The introduction of complex waves leads to replacing the classical wave equation by a Schrodinger equation.Finally, we show that complex wave equations, Gabor expansion, and Bloch waves are three different ways of introducing the tools of quantum mechanics in high-resolution seismic (Gabor, 1946; Kittel, 1976, Morlet, 1975). And conversely, the Goupillaud-Kunetz algorithm and an extended Gabor expansion may be of some use in solid state physics.

• 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

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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.

Abstract: Die vorliegende Arbeit ist, bis auf unwesentliche Änderungen, ein Abdruck meiner im Juli 1909 erschienenen Göttinger Inauguraldissertation.

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);

• Wavelets for Kids, a tutorial introduction by Brani Vidakovic (1991)
• Wavelet Transforms and Their Application to Turbulence by Marie Farge (Ann. Rev. Fluid. Mech., 1992)
• An overview of wavelet based multiresolution analyses by Björn Jawerth and Wim Sweldens (1993)
• Tutorial on Continuous Wavelet Analysis of Experimental Data by Jacques Lewalle, Syracuse University (April 1995)
• An Introduction to Wavelets by Amara Graps (1995)
• A Review of Wavelets in Biomedical Applications by Michael Unser (1996) [CiteSeerX]
• Multiwavelets: Theory and Applications by Vasily Strela and A. Vogan (1996)
• Wavelets in Statistics: A Review by Anestis Antoniadis (1997)
• A Practical Guide to Wavelet Analysis With significance and confidence testing by Christopher Torrence and Gilbert P. Compo (1998)
  • A Practical Guide to Wavelet Analysis by Christopher Torrence and Gilbert P. Compo (1998)
• PolyValens, by Clemens Valens (started 1999):
  • A Really Friendly Wavelet Guide
  • The Fast Lifting Wavelet Transform
  • Embedded Zerotree Wavelet Encoding
• A Tutorial on Wavelets and Their Applications by M. J. Mohlenkamp (Ohio University, 2004)
• Wavelets in industrial applications: a review by Fred Truchetet (2004)
• Wavelet transforms and the ECG: a review by Paul S. Addison (2005) *
• The wavelet tutorial by Robi Polikar (2006)
• Wavelets in industrial applications: a review by Fred Truchetet and Olivier Laligant (2008)
• A review of curvelets and recent applications by J. Ma and G. Plonka (IEEE Signal Processing Magazine, 2009)
• Wavelet Methods in Computational Fluid Dynamics by Kai Schneider and Oleg V. Vasilyev (Annual Reviews, 2010)
• A Tutorial of the Wavelet Transform by Chun-Lin, Liu (2010)
• Everything about 2D directional wavelets: A panorama on Multiscale Geometric Representations by Laurent Jacques, Laurent Duval, Caroline Chaux and Gabriel Peyré (Signal Processing, 2011)

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 non-parametric framework that is based on two main ideas. First, we introduce a problem-specific 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 sparsity-pursuing search techniques to find the most compact representation for the BOLD signal under investigation. The non-linear optimization allows to overcome the sensitivity-specificity trade-off 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.

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.

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 wavelet-based method is then extended to estimate generalized additive models. A primal-dual log-barrier 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.

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.

Abstract: We introduce a sparse image representation that takes advantage of the geometrical regularity of edges in images. A new class of one-dimensional 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 non-zero 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 two-dimensional wavelet basis

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).

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/pel

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 so-called 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.

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.

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 wavelet-based 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 wavelet-based 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 MRA-based 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 MRA-based construction principles.

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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 so-called 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 wavelet-based algorithms in compression, in denoising, and in other applications. On the downside, constructing wavelets with good space-frequency localization properties becomes involved as the spatial dimension grows. An alternative to the sometime-hard-to-construct wavelet representations is the always-easy-to-construct (and slightly older) non-orthogonal 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- Alignment-Prediction (CAP) representations. The CAP representation is based on the selection of three filters: the low-pass decomposition filter, the low-pass prediction filter, and the full-pass alignment filter. Like previous pyrami- dal algorithms, CAP are implemented by a simple, fast, wavelet-like 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 CAP-based 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.

Abstract: We propose a novel transform, an expansion of an arbitrary function onto a basis of multi-scale 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.

Abstract: Dynamic spectra, which exhibit the spectral content of a signal as time elapses, are based on subdivision of the time-frequency plane into minimum-area 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.

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 q-chirps for short), giving rise to a parameter space that includes both the time-frequency plane and the time-scale plane as 2-D subspaces. The parameter space contains a “time-frequency-scale volume” and thus encompasses both the short-time 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 q-chirp) and shear in frequency (obtained through multiplication by a q-chirp). Signals in this multidimensional space can be obtained by a new transform, which we call the “q-chirplet transform” or simply the “chirplet transform”. The proposed chirplets are generalizations of wavelets related to each other by 2-D affine coordinate transformations (translations, dilations, rotations, and shears) in the time-frequency plane, as opposed to wavelets, which are related to each other by 1-D affine coordinate transformations (translations and dilations) in the time domain only

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.

Time-frequency representations are widely used in audio applications involving sound analysis-synthesis. For such applications, obtaining a time-frequency transform that accounts for some aspects of human auditory perception is of high interest. To that end, we exploit the theory of non-stationary Gabor frames to obtain a perception-based, linear, and perfectly invertible time-frequency transform. Our goal is to design a non-stationary Gabor transform (NSGT) whose time-frequency resolution best matches the time-frequency 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 so-called “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.

The ERBlet transform, time-frequency masking and perceptual sparsity Time-frequency (TF) representations are widely used in audio applications involving sound analysis-synthesis. 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 non-stationary signal processing and psychoacoustics. First, we exploit the theory of non-stationary Gabor frames to obtain a linear and perfectly invertible non-stationary 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 so-called “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 long-term 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.

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.

Graphlets are small connected non-isomorphic 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 over-represented in the data when compared with randomized networks, while motifs do.

Abstract: In this work we propose the construction of two-channel wavelet filterbanks for analyzing functions defined on the vertices of any arbitrary finite weighted undirected graph. These graph based functions are referred to as graph-signals 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 two-channel filterbanks, and we propose quadrature mirror filters (referred to as graph-QMF) for bipartite graph which cancel aliasing and lead to perfect reconstruction. For arbitrary graphs we present a bipartite subgraph decomposition which produces an edge-disjoint collection of bipartite subgraphs. Graph-QMFs are then constructed on each bipartite subgraph leading to "multi-dimensional" 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.

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

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.

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 (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 U-statistic estimator of the wavelet ICA contrast, while the previously introduced plug-in estimator C²_j, with moderate computational cost, has a rate in n^-4s/(4s+d).

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 well-known 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 U-statistic 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 block-thresholded version, where knowledge of the regularity s of the underlying multivariate density is useless to obtain an optimal rate. We propose a plug-in type estimator whose convergence rate is sub-optimal but with a numerical complexity linear in n. The plug-in estimator also admits a term by term thresholded version, which dampens the convergence rate but yields an adaptive criteria. In its linear version, the plug-in estimator already seems auto-adaptive 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 U-statistic or a V-statistic. Standard results on U-statistics 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 d-dimensional algorithms) also appear in the document.

          
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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Abstract: Sparse representations has shown to be a very powerful model for real world signals, and has enabled the development of applications with notable performance. Combined with the ability to learn a dictionary from signal examples, sparsity-inspired algorithms are often achieving state-of-the-art results in a wide variety of tasks. Yet, these methods have traditionally been restricted to small dimensions mainly due to the computational constraints that the dictionary learning problem entails. In the context of image processing, this implies handling small image patches. In this work we show how to efficiently handle bigger dimensions and go beyond the small patches in sparsity-based signal and image processing methods. We build our approach based on a new cropped wavelet decomposition, which enables a multi-scale analysis with virtually no border effects. We then employ this as the base dictionary within a double sparsity model to enable the training of adaptive dictionaries. To cope with the increase of training data, while at the same time improving the training performance, we present an Online Sparse Dictionary Learning (OSDL) algorithm to train this model effectively, enabling it to handle millions of examples. This work shows that dictionary learning can be up-scaled to tackle a new level of signal dimensions, obtaining large adaptable atoms that we call trainlets.

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

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

Abstract: A novel image-dependent representation, warplets, based on self-similarity 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 self-similar 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 block-to-block transformations are found by Gaussian mixture modelling of the block spectra and a least-squares 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.

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.

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.