It is well documented that the statistical distribution of wavelet coefficients for natural images is non-Gaussian and that neighboring coefficients are highly dependent. In this paper, we propose a new multivariate non-Gaussian probability model to capture the dependencies among neighboring wavelet coefficients in the same scale. The model can be expressed as K exp(-||w||) where w is a N-element vector of wavelet coefficients and ||w|| is a convex combination of l₂ norms over subspaces of R_N. This model includes the commonly used independent Laplacian model as a special case but it has many more degrees of freedom. Based on this model, the corresponding non-linear threshold (shrinkage) function for denoising is derived using Bayesian estimation theory. Although this function does not have a closed-form solution, a successive substitution method can be used to numerically compute it.

Wavelet thresholding is a powerful tool for denoising images and other signals with sharp discontinuities. Using different wavelet bases gives different results, and since the wavelet transform is not time-invariant, thresholding various shifts of the signal is one way to use different wavelet bases. This paper describes several denoising methods that apply wavelet thresholding or variations on wavelet thresholding recursively. (We previously termed one of these methods "recursive cycle spinning.") These methods are compared experimentally for denoising piecewise polynomial signals. Though similar, the methods differ in computational complexity, convergence speed, and sensitivity to threshold selection.

In this paper, we link concepts from nonuniform sampling, smoothness function spaces, interpolation, and wavelet denoising to derive a new multiscale interpolation algorithm for piecewise smooth signals. We formulate the optimization of finding the signal that balances agreement with the given samples against a wavelet-domain regularization. For signals in the Besov space B^α_p(L_p) p ≥ 1, the optimization corresponds to convex programming in the wavelet domain. The algorithm simultaneously achieves signal interpolation and wavelet denoising, which makes it particularly suitable for noisy sample data, unlike classical approaches such as bandlimited and spline interpolation.

In this paper, we illustrate how a recently proposed wavelet-based estimation scheme for 2-D multichannel signals can utilize an overcomplete wavelet expansion or the BayesShrink adaptive wavelet-domain threshold to improve estimation results. The existing technique approximates the optimal estimator using a DFT and an orthonormal 2-D DWT to efficiently decorrelate the signal in both channel and space, and a wavelet-domain threshold to suppress the noise. Although this technique typically yields signal-to-noise ratio (SNR) gains of over 12 dB, results can be improved 1 to 1.5 dB by replacing the critically-sampled wavelet expansion with an overcomplete wavelet expansion. In addition, provided that the detail subbands of the original signal channels each obey a generalized Gaussian distribution, average channel SNR gains can be improved 3 dB or more using the BayesShrink adaptive wavelet-domain threshold.

A subspace-based method for denoising with a frame works as follows: If a signal is known to have a sparse representation with respect to the frame, the signal can be estimated from a noise-corrupted observation of the signal by finding the best sparse approximation to the observation. The ability to remove noise in this manner depends on the frame being designed to efficiently represent the signal while it inefficiently represents the noise. This paper gives bounds to show how inefficiently white Gaussian noise is represented by sparse linear combinations of frame vectors. The bounds hold for any frame so they are generally loose for frames designed to represent structured signals. Nevertheless, the bounds can be combined with knowledge of the approximation efficiency of a given family of frames for a given signal class to study the merit of frame redundancy for denoising.

Passive localisation and bearing estimation of underwater acoustic sources is a problem of great interest in the area of ocean acoustics. Bearing estimation techniques often perform poorly due to the low signal-to-noise ratio (SNR) at the sensor array. This paper proposes signal enhancement by wavelet denoising to improve the performance of the bearing estimation techniques. Methods have been developed in the recent past to effectively perform wavelet denoising in the multisensor scenario (wavelet array denoising). Following one such approach, the acoustic signal received at the array is spatially decorrelated and then denoised. The denoised and recorrelated signal is then used for bearing estimation employing known bearing estimation techniques (MUSIC and Subspace Intersection). It is shown that wavelet array denoising improves the performance of the bearing estimators significantly. Also the case of perturbed arrays is considered as a special case for application of wavelet array denoising. Simulation results show that the denoising estimator has lower mean square error and higher resolution.

In this paper, we consider classes of not bandlimited signals, namely, streams of Diracs and piecewise polynomial signals, and show that these signals can be sampled and perfectly reconstructed using wavelets as sampling kernel. Due to the multiresolution structure of the wavelet transform, these new sampling theorems naturally lead to the development of a new resolution enhancement algorithm based on wavelet footprints. Preliminary results show that this algorithm is also very resilient to noise.

The contourlet transform is a new extension to the wavelet transform in two dimensions using non-separable and directional filter banks. The contourlet expansion is composed of basis images oriented at varying directions in multiple scales, with flexible aspect ratios. With this rich set of basis images, the contourlet transform can effectively capture the smooth contours that are the dominant features in natural images with only a small number of coefficients. We begin with a detail study of the statistics of the contourlet coefficients of natural images, using histogram estimates of the marginal and joint distributions, and mutual information measurements to characterize the dependencies between coefficients. The study reveals the non-Gaussian marginal statistics and strong intra-subband, cross-scale, and cross-orientation dependencies of contourlet coefficients. It is also found that conditioned on the magnitudes of their generalized neighborhood coefficients, contourlet coefficients can approximately be modeled as Gaussian variables with variances directly related to the generalized neighborhood magnitudes. Based on these statistics, we model contourlet coefficients using a hidden Markov tree (HMT) model that can capture all of their inter-scale, inter-orientation, and intra-subband dependencies. We experiment this model in the image denoising and texture retrieval applications where the results are very promising. In denoising, contourlet HMT outperforms wavelet HMT and other classical methods in terms of both peak signal-to-noise ratio (PSNR) and visual quality. In particular, it preserves edges and oriented features better than other existing methods. In texture retrieval, it shows improvements in performance over wavelet methods for various oriented textures.

In the last few years, it has become apparent that traditional wavelet-based image processing algorithms and models have significant shortcomings in their treatment of edge contours. The standard modeling paradigm exploits the fact that wavelet coefficients representing smooth regions in images tend to have small magnitude, and that the multiscale nature of the wavelet transform implies that these small coefficients will persist across scale (the canonical example is the venerable zero-tree coder). The edge contours in the image, however, cause more and more large magnitude wavelet coefficients as we move down through scale to finer resolutions. But if the contours are smooth, they become simple as we zoom in on them, and are well approximated by straight lines at fine scales. Standard wavelet models exploit the grayscale regularity of the smooth regions of the image, but not the geometric regularity of the contours. In this paper, we build a model that accounts for this geometric regularity by capturing the dependencies between complex wavelet coefficients along a contour. The Geometric Hidden Markov Tree (GHMT) assigns each wavelet coefficient (or spatial cluster of wavelet coefficients) a hidden state corresponding to a linear approximation of the local contour structure. The shift and rotational-invariance properties of the complex wavelet transform allow the GHMT to model the behavior of each coefficient given the presence of a linear edge at a specified orientation --- the behavior of the wavelet coefficient given the state. By connecting the states together in a quadtree, the GHMT ties together wavelet coefficients along a contour, and also models how the contour itself behaves across scale. We demonstrate the effectiveness of the model by applying it to feature extraction.

Wavelet image denoising practice has shown that the performance of simple estimators may be substantially improved by averaging these estimators over a collection of transformations such as translations or rotations. In this paper, we explain and quantify these empirical findings using estimation theory. We consider a general nonlinear observation model, analyze the estimation risk of transformation-averaged estimators, and derive an upper bound on the risk reduction due to transformation averaging. The bound is evaluated for several estimators, using different averaging strategies (including a randomized strategy) and different wavelet bases. The practical usefulness of the bound is established for standard image denoising examples.

The nonparametric multiscale polynomial and platelet methods presented here are powerful new tools for signal and image denoising and reconstruction. Unlike traditional wavelet-based multiscale methods, these methods are both well suited to processing Poisson or multinomial data and capable of preserving image edges. At the heart of these new methods lie multiscale signal decompositions based on polynomials in one dimension and multiscale image decompositions based on what the authors call platelets in two dimensions. Platelets are localized functions at various positions, scales and orientations that can produce highly accurate, piecewise linear approximations to images consisting of smooth regions separated by smooth boundaries. Polynomial and platelet-based maximum penalized likelihood methods for signal and image analysis are both tractable and computationally efficient. Polynomial methods offer near minimax convergence rates for broad classes of functions including Besov spaces. Upper bounds on the estimation error are derived using an information-theoretic risk bound based on squared Hellinger loss. Simulations establish the practical effectiveness of these methods in applications such as density estimation, medical imaging, and astronomy.

Overcomplete wavelet representations have become increasingly popular for their ability to provide highly sparse and robust descriptions of natural signals. We describe a method for incorporating an overcomplete wavelet representation as part of a statistical model of images which includes a sparse prior distribution over the wavelet coefficients. The wavelet basis functions are parameterized by a small set of 2-D functions. These functions are adapted to maximize the average log-likelihood of the model for a large database of natural images. When adapted to natural images, these functions become selective to different spatial orientations, and they achieve a superior degree of sparsity on natural images as compared with traditional wavelet bases. The learned basis is similar to the Steerable Pyramid basis, and yields slightly higher SNR for the same number of active coefficients. Inference with the learned model is demonstrated for applications such as denoising, with results that compare favorably with other methods.

We present here an explicit time-domain representation of any compactly supported dyadic scaling function as a sum of harmonic splines. The leading term in the decomposition corresponds to the fractional splines that have recently been defined by the authors as a continuous-order generalization of the polynomial splines.

We survey some recent results in the theory of multivariate piecewise polynomial approximation. In the univariate case this method is equivalent to Wavelet approximation, but in the multivariate case this is no longer true, since this form of approximation is more adaptive to the geometry of the singularities of the function to be approximated. The theory possibly has applications in image compression.

Functional (time-dependent) Magnetic Resonance Imaging can be used to determine which parts of the brain are active during various limited activities; these parts of the brain are called activation regions. In this preliminary study we describe some experiments that are suggested from the following questions: Does one get improved results by analyzing the complex image data rather than just the real magnitude image data? Does wavelet shrinkage smoothing improve images? Should one smooth in time as well as within and between slices? If so, how should one model the relationship between time smoothness (or correlations) and spatial smoothness (or correlations). The measured data is really the Fourier coefficients of the complex image---should we remove noise in the Fourier domain before computing the complex images? In this preliminary study we describe some experiments related to these questions.

We show that a multi-dimensional scaling function of order γ (possibly fractional) can always be represented as the convolution of a polyharmonic B-spline of order γ and a distribution with a bounded Fourier transform which has neither order nor smoothness. The presence of the B-spline convolution factor explains all key wavelet properties: order of approximation, reproduction of polynomials, vanishing moments, multi-scale differentiation property, and smoothness of the basis functions. The B-spline factorization also gives new insights on the stability of wavelet bases with respect to differentiation. Specifically, we show that there is a direct correspondence between the process of moving a B-spline factor from one side to another in a pair of biorthogonal scaling functions and the exchange of fractional integrals/derivatives on their wavelet counterparts. This result yields two "eigen-relations" for fractional differential operators that map biorthogonal wavelet bases into other stable wavelet bases. This formulation provides a better understanding as to why the Sobolev/Besov norm of a signal can be measured from the ℓ_p-norm of its rescaled wavelet coefficients. Indeed, the key condition for a wavelet basis to be an unconditional basis of the Besov space B_q^s(L_p(R^d)) is that the s-order derivative of the wavelet be in L_p.

We present a simple but generalized interpolation method for digital images that uses multiwavelet-like basis functions. Most of interpolation methods uses only one symmetric basis function; for example, standard and shifted piecewise-linear interpolations use the "hat" function only. The proposed method uses q different multiwavelet-like basis functions. The basis functions can be dissymmetric but should preserve the "partition of unity" property for high-quality signal interpolation. The scheme of decomposition and reconstruction of signals by the proposed basis functions can be implemented in a filterbank form using separable IIR implementation. An important property of the proposed scheme is that the prefilters for decomposition can be implemented by FIR filters. Recall that the shifted-linear interpolation requires IIR prefiltering, but we find a new configuration which reaches almost the same quality with the shifted-linear interpolation, while requiring FIR prefiltering only. Moreover, the present basis functions can be explicitly formulated in time-domain, although most of (multi-)wavelets don’t have a time-domain formula. We specify an optimum configuration of interpolation parameters for image interpolation, and validate the proposed method by computing PSNR of the difference between multi-rotated images and their original version.

In the scalar-valued setting, it is well-known that the two-scale sequences {q_k} of Daubechies orthogonal wavelets can be given explicitly by the two-scale sequences {p_k} of their corresponding orthogonal scaling functions, such as q_k = (-1)^kp_1-k. However, due to the non-commutativity of matrix multiplication, there is little such development in the multi-wavelet literature to express the two-scale matrix sequence {Q_k} of an orthogonal multi-wavelet in terms of the two-scale matrix sequence {P_k} of its corresponding multi-scaling function vector. This paper, in part, is devoted to this study for the setting of orthogonal multi-wavelets of dimension r = 2. We will apply our results to constructing a family of the most recently introduced notion of armlet of order n and a family of the n-balanced orthogonal multi-wavelets.

Several different approaches for joint detection/estimation of amplitude and frequency modulated signals embedded in stationary random noise with prescribed spectral density are considered and compared. Matched filter approaches are compared to time-frequency and time scale based approaches, together with “reassigned” versions. Particular attention is paid to the case of the so-called “power-law chirps”, characterized by monomial and polynomial amplitude and frequency functions. As target application, the problem of gravitational waves at interferometric detectors is considered.

Two-dimensional wavelet analysis with directional frames is well-adapted and efficient for detecting oriented features in images but, for (quasi-)isotropic components, it is unnecessarily redundant. Using wavelets with variable angular selectivity leads to a prohibitive computing cost in the continuous wavelet formalism. We propose here a solution based on a multiresolution analysis in the angular variable (transferred from a biorthogonal analysis on the line), in addition to the usual multiresolution in scale. The resulting scheme is efficient and competitive with traditional methods. Some applications are given to image denoising.

We propose the construction of directional - or Gabor - continuous wavelets on the sphere. We provide a criterion to measure their angular selectivity. We finally discuss implementation issues and potential applications. The code for the spherical wavelet transform is available in the YAWTB Matlab Toolbox, http://www.yawtb.be.tf.

Ten years ago, Mallat and Zhang proposed the Matching Pursuit algorithm : since then, the dictionary approach to signal processing has been a very active field. In this paper, we try to give an overview of a series of recent results in the field of sparse decompositions and nonlinear approximation with redundant dictionaries. We discuss sufficient conditions on a decomposition to be the unique and simultaneous sparsest ℓ^r expansion for all r, 0 ≤ r ≤ 1. In particular, we prove that any decomposition has this nice property if the number of its nonzero coefficients does not exceed a quantity which we call the spread of the dictionary. After a brief discussion of the interplay between sparse decompositions and nonlinear approximation with various families of algorithms, we review several recent results that provide sufficient conditions for the Matching Pursuit, Orthonormal Matching Pursuit, and Basis Pursuit algorithms to have good recovery properties. The most general conditions are not straightforward to check, but weaker estimates based on the notions of coherence of the dictionary are recalled, and we discuss how these results can be applied to approximation and sparse compositions with highly redundant incoherent dictionaries built by taking the union of several orthonormal bases. Eventually, based on Bernstein inequalities, we discuss how much approximation power can be gained by replacing a single basis with such redundant dictionaries.

Very low bit rate image coding is an important problem regarding applications such as storage on low memory devices or streaming data on the internet. The state of the art in image compression is to use 2-D wavelets. The advantages of wavelet bases lie in their multiscale nature and in their ability to sparsely represent functions that are piecewise smooth. Their main problem on the other hand, is that in 2-D wavelets are not able to deal with the natural geometry of images, i.e they cannot sparsely represent objects that are smooth away from regular submanifolds. In this paper we propose an approach based on building a sparse representation of images in a redundant geometrically inspired library of functions, followed by suitable coding techniques. Best N-term non- linear approximations in general dictionaries is, in most cases, a NP-hard problem and sub-optimal approaches have to be followed. In this work we use a greedy strategy, also known as Matching Pursuit to compute the expansion. Finally the last step in our algorithm is an enhancement layer that encodes the residual image: in our simulation we have used a genuine embedded wavelet codec.

Seismic imaging commits itself to locating singularities in the elastic properties of the Earth's subsurface. Using the high-frequency ray-Born approximation for scattering from non-intersecting smooth interfaces, seismic data can be represented by a generalized Radon transform mapping the singularities in the medium to seismic data. Even though seismic data are bandwidth limited, signatures of the singularities in the medium carry through this transform and its inverse and this mapping property presents us with the possibility to develop new imaging techniques that preserve and characterize the singularities from incomplete, bandwidth-limited and noisy data. In this paper we propose a non-adaptive Curvelet/Contourlet technique to image and preserve the singularities and a data-adaptive Matching Pursuit method to characterize these imaged singularities by Multi-fractional Splines. This first technique borrows from the ideas within the Wavelet-Vaguelette/Quasi-SVD approach. We use the almost diagonalization of the scattering operator to approximately compensate for (i) the coloring of the noise and hence facilitate estimation; (ii) the normal operator itself. Results of applying these techniques to seismic imaging are encouraging although many open fundamental questions remain.

This paper introduces a class of wavelet packets based upon a set of biorthogonal basis functions. Using a Kronecker product formulation, we develop a self-similar factorization that obeys a set of perfect reconstruction conditions. This construction is then identified as a wavelet packet decomposition and is applied to the finite field case. Finally, it is demonstrated that the proposed wavelet packets can be applied as a well-known class of error control codes.

This paper describes a new approach for creating compelling virtual acoustic environments by synthesizing and three-dimensionally localizing sounds within the wavelet-domain. A prototype system was developed that combines wavelet-domain convolution for localization with Miner's new method for synthesizing parametrically controlled sounds. Results are presented and discussed, with suggestions as to directions of further interest.

We describe the implementation of a novel adaptive wireless communications waveform for interference avoidance (IA) in re-configurable logic devices. While other transform domain-based IA waveforms have been suggested, the wavelet packet modulation (WPM) system described here is unique in its multiplexing of complex quadrature amplitude modulation symbols onto orthogonal wavelet packets for unrivaled time-frequency agility. We examine the realization aspects of dynamically instantiating the transmit side inverse discrete wavelet packet transform (DWPT) and receive side DWPT filter bank structures, and the WPM symbol timing recovery, in Field Programmable Gate Array (FPGA) devices. This work applies Trenas' re-configurable wavelet packet transform (WPT) architecture to a wireless communications system, draws upon Jones' theoretical foundation for orthogonally multiplexed communications, and utilizes Lindsey's WPM supersymbol tuning and Kjeldsen's WPM symbol synchronization algorithms.

We consider the problem of recovery of a scene recorded through a semirefective medium from its mixture with a virtual reflected image using the blind source separation (BSS) framework. We extend the Sparse ICA (SPICA) approach and apply it to the separation of the desired image from the superimposed images, without having any a priory knowledge about its structure and/or statistics. Advances in the SPICA approach are discussed. Simulations and experimental results illustrate the efficiency of the proposed approach, and of its specific implementation in a simple algorithm of a low computational cost. The approach and the algorithm are generic and can be adapted and applied to a wide range of BSS problems involving one-dimensional signals or images.

We propose a new framework, called piecewise linear separation, for blind source separation of possibly degenerate mixtures, including the extreme case of a single mixture of several sources. Its basic principle is to: 1/ decompose the observations into "components" using some sparse decomposition/nonlinear approximation technique; 2/ perform separation on each component using a "local" separation matrix. It covers many recently proposed techniques for degenerate BSS, as well as several new algorithms that we propose. We discuss two particular methods of multichannel decompositions based on the Best Basis and Matching Pursuit algorithms, as well as several methods to compute the local separation matrices (assuming the mixing matrix is known). Numerical experiments are used to compare the performance of various combinations of the decomposition and local separation methods. On the dataset used for the experiments, it seems that BB with either cosine packets of wavelet packets (Beylkin, Vaidyanathan, Battle3 or Battle 5 filter) are the best choices in terms of overall performance because they introduce a relatively low level of artefacts in the estimation of the sources; MP introduces slightly more artefacts, but can improve the rejection of the unwanted sources.

We review the sparse representation principle for processing speech signals. A transformation for encoding the speech signals is learned such that the resulting coefficients are as independent as possible. We use independent component analysis with an exponential prior to learn a statistical representation for speech signals. This representation leads to extremely sparse priors that can be used for encoding speech signals for a variety of purposes. We review applications of this method for speech feature extraction, automatic speech recognition and speaker identification. Furthermore, this method is also suited for tackling the difficult problem of separating two sounds given only a single microphone.

In this paper a constrained non-negative matrix factorization (cNMF) algorithm for recovering constituent spectra is described together with experiments demonstrating the broad utility of the approach. The algorithm is based on the NMF algorithm of Lee and Seung, extending it to include a constraint on the minimum amplitude of the recovered spectra. This constraint enables the algorithm to deal with observations having negative values by assuming they arise from the noise distribution. The cNMF algorithm does not explicitly enforce independence or sparsity, instead only requiring the source and mixing matrices to be non-negative. The algorithm is very fast compared to other "blind" methods for recovering spectra. cNMF can be viewed as a maximum likelihood approach for finding basis vectors in a bounded subspace. In this case the optimal basis vectors are the ones that envelope the observed data with a minimum deviation from the boundaries. Results for Raman spectral data, hyperspectral images, and ³¹P human brain data are provided to illustrate the algorithm's performance.

It was previously shown that sparse representations can improve and simplify the estimation of an unknown mixing matrix of a set of images and thereby improve the quality of separation of source images. Here we propose a multiscale approach to the problem of blind separation of images from a set of their mixtures. We take advantage of the properties of multiscale transforms such as wavelet packets and decompose signals and images according to sets of local features. The resulting partial representations on a tree of data structure depict various degrees of sparsity. We show how the separation error is affected by the sparsity of the decomposition coefficients, and by the misfit between the prior, formulated in accordance with the probabilistic model of the coefficients' distribution, and the actual distribution of the coefficients. Our error estimator, based on the Taylor expansion of the quasi Log-Likelihood function, is used in selection of the best subsets of coefficients, utilized in turn for further separation. The performance of the proposed method is assessed by separation of noise-free and noisy data. Experiments with simulated and real signals and images demonstrate significant improvement of separation quality over previously reported results.

Using the time-frequency (or -scale) diversity of the source processes allows the blind source separation problem to be tackled within Gaussian models. In this work, we show that this approach amounts to minimizing a certain sparseness criterion for the energy distribution of the source over the time-frequency (or -scale) plane. We also explore the link between independence and sparsity and shows that other sparsity criteria (some examples of which are provided) can be used. Further, we introduce an adaptive method which tries to find the best sparse representation of the source energy in order to exploit the sparsity in a most efficient way. An algorithm, adapted from that of Coifman and Wickerhauser has been developed for this end. Finally a simulation example has been given.

We study a relative optimization framework for the quasi-maximum likelihood blind source separation and relative Newton method as its particular instance. Convergence of the Newton method is stabilized by the line search and by the modification of the Hessian, which forces its positive definiteness. The structure of the Hessian allows fast approximate inversion. In order to separate sparse sources, we use a non-linearity based on smooth approximation to the absolute value function. Sequential optimization with the gradual reduction of the smoothing parameter leads to the super-efficient separation.

In this work, we consider the problem of blind source separation in the wavelet domain via a Bayesian estimation framework. We use the sparsity and multiresolution properties of the wavelet coefficients to model their distribution by heavy tailed prior probability laws: the generalized exponential family and the Gaussian mixture family. Appropriate MCMC algorithms are developed in each case for the estimation purposes and simulation results are presented for comparaison.

We give a physical interpretation for finite tight frames along the lines of Columb's Law in Physics. This allows us to use results from classical mechanics to anticipate results in frame theory. As a consequence, we are able to classify those frames for an N-dimensional Hilbert space which are the closest to being tight (in the sense of minimizing potential energy) while having the norms of the frame vectors prescribed in advance. This also yields a fundamental inequality that all finite tight frames must satisfy.

In the early days of Gabor analysis it was a common to say that Gabor expansions of signals are interesting due to the natural interpretation of Gabor coefficients, but unfortunately the computation of Gabor coefficients is costly. Nowadays a large variety of efficient numerical algorithms exists and it has been recognized that stable and robust Gabor expansions can be achieved at low redundancy, e.g., by using a Gaussian atom and any time-frequency lattice of the form (see formula in paper). Consequently Gabor multipliers, i.e., linear operators obtained by applying a pointwise multiplication of the Gabor coefficients, become an important class of time-variant filters. It is the purpose of this paper to describe that fact that - provided one uses Gabor atoms from a suitable subspace (formula in paper)one has the expected continuous dependence of Gabor multipliers on the ingredients. In particular, we will provide new results which show that a small change of lattice parameters implies only a small change of the corresponding Gabor multiplier (e.g., in the Hilbert-Schmidt norm).

Density conditions have turned out to be a powerful tool for deriving necessary conditions for weighted wavelet systems to possess an upper or lower frame bound. In this paper we study different definitions of density and compare them with respect to their appropriateness and practicality.

Wavelet-based methods for multiple hypothesis testing are described and their potential for activation mapping of human functional magnetic resonance imaging (fMRI) data is investigated. In this approach, we emphasize convergence between methods of wavelet thresholding or shrinkage and the problem of multiple hypothesis testing in both classical and Bayesian contexts. Specifically, our interest will be focused on ensuring a trade off between type I probability error control and power dissipation. We describe a technique for controlling the false discovery rate at an arbitrary level of type 1 error in testing multiple wavelet coefficients generated by a 2D discrete wavelet transform (DWT) of spatial maps of {fMRI} time series statistics. We also describe and apply recursive testing methods that can be used to define a threshold unique to each level and orientation of the 2D-DWT. Bayesian methods, incorporating a formal model for the anticipated sparseness of wavelet coefficients representing the signal or true image, are also tractable. These methods are comparatively evaluated by analysis of "null" images (acquired with the subject at rest), in which case the number of positive tests should be exactly as predicted under the hull hypothesis, and an experimental dataset acquired from 5 normal volunteers during an event-related finger movement task. We show that all three wavelet-based methods of multiple hypothesis testing have good type 1 error control (the FDR method being most conservative) and generate plausible brain activation maps.

Statistical Parametric Mapping (SPM) is a widely deployed tool for detecting and analyzing brain activity from fMRI data. One of SPM's main features is smoothing the data by a Gaussian filter to increase the SNR. The subsequent statistical inference is based on the continuous Gaussian random field theory. Since the remaining spatial resolution has deteriorated due to smoothing, SPM introduces the concept of "resels" (resolution elements) or spatial information-containing cells. The number of resels turns out to be inversely proportional to the size of the Gaussian smoother. Detection the activation signal in fMRI data can also be done by a wavelet approach: after computing the spatial wavelet transform, a straightforward coefficient-wise statistical test is applied to detect activated wavelet coefficients. In this paper, we establish the link between SPM and the wavelet approach based on two observations. First, the (iterated) lowpass analysis filter of the discrete wavelet transform can be chosen to closely resemble SPM's Gaussian filter. Second, the subsampling scheme provides us with a natural way to define the number of resels; i.e., the number of coefficients in the lowpass subband of the wavelet decomposition. Using this connection, we can obtain the degree of the splines of the wavelet transform that makes it equivalent to SPM's method. We show results for two particularly attractive biorthogonal wavelet transforms for this task; i.e., 3D fractional-spline wavelets and 2D+Z fractional quincunx wavelets. The activation patterns are comparable to SPM's.

We address the problem of the analysis of event-related functional Magnetic Resonance Images (fMRI). We propose to separate the fMRI time series into "activated" and "non-activated" clusters. The fMRI time series are projected onto a basis, and the clustering is performed using the coefficients in that basis. We developed a new algorithm to select that basis which provides the optimal clustering of the time series. Our approach does not require any training datasets or any model of the hemodynamic response. The basis is constructed using a dictionary of wavelet packets. We search for the optimal basis in this dictionary using a new cost function that measures the clustering power of a set of wavelet packets. Our approach can be easily extended to classification problems. We have conducted several experiments with synthetic and in-vivo event-related fMRI data. Our method is capable of discovering the structures of the synthetic data. The method also successfully detected activated voxels in the in-vivo fMRI.

Morphometric analysis of medical images is playing an increasingly important role in understanding brain structure and function, as well as in understanding the way in which these change during development, aging and pathology. This paper presents three wavelet-based methods with related applications in morphometric analysis of magnetic resonance (MR) brain images. The first method handles cases where very limited datasets are available for the training of statistical shape models in the deformable segmentation. The method is capable of capturing a larger range of shape variability than the standard active shape models (ASMs) can, by using the elegant spatial-frequency decomposition of the shape contours provided by wavelet transforms. The second method addresses the difficulty of finding correspondences in anatomical images, which is a key step in shape analysis and deformable registration. The detection of anatomical correspondences is completed by using wavelet-based attribute vectors as morphological signatures of voxels. The third method uses wavelets to characterize the morphological measurements obtained from all voxels in a brain image, and the entire set of wavelet coefficients is further used to build a brain classifier. Since the classification scheme operates in a very-high-dimensional space, it can determine subtle population differences with complex spatial patterns. Experimental results are provided to demonstrate the performance of the proposed methods.

Spherical filters have recently been introduced in order to avoid the spherical harmonic transform. Spherical filtering can be used in a variety of applications, such as climate modelling, electromagnetic and acoustic scattering, and several other areas. However, up to now these methods have been restricted to special grids on the sphere. The main reason for this was to enable the use of FFT techniques. In this paper we extend the spherical filter to arbitrary grids by using the the Nonequispaced Fast Fourier Transform (NFFT). The new algorithm can be applied to a variety of different distributions on the sphere, equidistributions on the sphere being an important example. The algorithm's performance is illustrated with several numerical examples.

Integrated wavelets are a new method for discretizing the continuous wavelet transform (CWT). Independent of the choice of discrete scale and orientation parameters they yield tight families of convolution operators. Thus these families can easily be adapted to specific problems. After presenting the fundamental ideas, we focus primarily on the construction of directional integrated wavelets and their application to medical images. We state an exact algorithm for implementing this transform and present applications from the field of digital mammography. The first application covers the enhancement of microcalcifications in digital mammograms. Further, we exploit the directional information provided by integrated wavelets for better separation of microcalcifications from similar structures.

The problem of image denoising has received more attention than the problem of image sharpening. In the paper, we propose that wavelet-based algorithms for image denoising can be used to perform image sharpening. Consequently, a variety of new image sharpening techniques becomes available. We examine the sharpening of natural images using an algorithm for image denoising with oriented complex 2D wavelets.

We present complex rotation-covariant multiresolution families aimed for image analysis. Since they are complex-valued functions, they provide the important phase information, which is missing in the discrete wavelet transform with real wavelets. Our basis elements have nice properties in Hilbert space such as smoothness of fractional order α ε R⁺. The corresponding filters allow a FFT-based implementation and thus provide a fast algorithm for the wavelet transform.

We propose to model satellite and aerial images using a probabilistic approach. We show how the properties of these images, such as scale invariance, rotational invariance and spatial adaptivity lead to a new general model which aims to describe a broad range of natural images. The complex wavelet transform initially proposed by Kingsbury is a simple way of taking into account all these characteristics. We build a statistical model around this transform, by defining an adaptive Gaussian model with interscale dependencies, global parameters, and hyperpriors controlling the behaviour of these parameters. This model has been successfully applied to denoising and deconvolution, for real images and simulations provided by the French Space Agency.

We introduce a new local sine transform that can completely localize image information in both the space and spatial frequency domains. Instead of constructing a basis, we first segment an image into local pieces using the characteristic functions, then decompose each piece into two component: the polyharmonic component and the residual. The polyharmonic component is obtained by solving the elliptic boundary value problem associated with the so-called polyharmonic equation (e.g., Laplace equation, biharmonic equation, etc.) given the boundary values (the pixel values along the borders created by the characteristic functions) possibly with the estimates of normal derivatives at the boundaries. Once this component is obtained this is subtracted from the original local piece to obtain the residual, whose Fourier sine series expansion has quickly decaying coefficients since the boundary values of the residual (possibly with their normal derivatives) vanish. Using this transform, we can distinguish intrinsic singularities in the data from the artificial discontinuities created by the local windowing. We will demonstrate the superior performance of this new transform in terms of image compression to some of the conventional methods such as JPEG/DCT and the lapped orthogonal transform using actual examples.

Natural images can be viewed as combinations of smooth regions, textures, and geometry. Wavelet-based image coders, such as the space-frequency quantization (SFQ) algorithm, provide reasonably efficient representations for smooth regions (using zerotrees, for example) and textures (using scalar quantization) but do not properly exploit the geometric regularity imposed on wavelet coefficients by features such as edges. In this paper, we develop a representation for wavelet coefficients in geometric regions based on the wedgelet dictionary, a collection of geometric atoms that construct piecewise-linear approximations to contours. Our wedgeprint representation implicitly models the coherency among geometric wavelet coefficients. We demonstrate that a simple compression algorithm combining wedgeprints with zerotrees and scalar quantization can achieve near-optimal rate-distortion performance D(R) ~ (log R)²/R² for the class of piecewise-smooth images containing smooth C² regions separated by smooth C² discontinuities. Finally, we extend this simple algorithm and propose a complete compression framework for natural images using a rate-distortion criterion to balance the three representations. Our Wedgelet-SFQ (WSFQ) coder outperforms SFQ in terms of visual quality and mean-square error.

Aim of this paper is investigating the use of overcomplete bases for the representation of hyperspectral image data. The idea is building an overcomplete basis starting from several orthogonal or non-orthogonal bases and picking the subset of such vectors best matching pixel spectra. A common technique to select the most representative elements of a signal is Matching Pursuit (MP). An iterative approach is used to find the coefficients of the linear combination of vectors, so that the residual function has minimum energy. The computational cost is extremely high when a large set of data is to be processed. Therefore, a reduced data set (RDS) is produced by applying the projection pursuit (PP) technique to each of the segments in which the hyperspectral image is partitioned based on a spatial homogeneity criterion of pixel spectra. Then MP is applied to the RDS to find a non-orthogonal frame capable to represent such data through waveforms selected to best match spectral features. Experimental results carried out on the hyperspectral data AVIRIS Moffett Field '97 compare a dictionary of wavelet functions with a dictionary of endmembers spectra. Although the former is preferable in terms of energy compaction, the latter is superior for physical significance of the resulting components.

The binary-tree best base (BTBB) searching method developed by Coifman and Wickerhauser is well known and widely used in wavelet packet applications. However, the requirement that the base vectors be chosen from either a parent or its directly related children in the binary-tree structure is a limitation because it doesn't search all possible orthogonal bases and therefore may not provide a optimal result. We have recently found that the set of all possible orthogonal bases in a wavelet packet is much larger than the set searched by the BTBB method. Based on this observation, we have developed the true best base (TBB) searching method - a new way to search the best base among a much larger set of orthogonal bases. In this paper, we show that considerable improvements in signal compression, de-noising, and time-frequency analysis can be achieved using the new TBB method. Furthermore, we show that the TBB method can be used as a searching engine to extract the local discriminant base (LDB) for feature extraction and signal/object classification, and we compare the performances of the LDBs extracted by the TBB and BTBB.

We are reviewing scalar quantizers with deadzone and overload in the high-bitrate approximation, give sharp bounds on the quality of this approximation and present results on the optimal quantizer in this setting. Some recent results by Hui and Neuhoff are reproduced under weaker conditions. We conclude by comparing the mathematical results with experimental data. For details about the results of this article we refer to an article in prepeparation.

We present a zero-order and twin image elimination algorithm for digital Fresnel holograms that were acquired in an off-axis geometry. These interference terms arise when the digital hologram is reconstructed and corrupt the result. Our algorithm is based on the Fresnelet transform, a wavelet-like transform that uses basis functions tailor-made for digital holography. We show that in the Fresnelet domain, the coefficients associated to the interference terms are separated both spatially and with respect to the frequency bands. We propose a method to suppress them by selectively thresholding the Fresnelet coefficients. Unlike other methods that operate in the Fourier domain and affect the whole spacial domain, our method operates locally in both space and frequency, allowing for a more targeted processing.

Recently, the contourlet transform has been developed as a true two-dimensional representation that can capture the geometrical structure in pictorial information. Unlike other transforms that were initially constructed in the continuous-domain and then discretized for sampled data, the contourlet construction starts from the discrete-domain using filter banks, and then convergences to a continuous-domain expansion via a multiresolution analysis framework. In this paper we study the approximation behavior of the contourlet expansion for two-dimensional piecewise smooth functions resembling natural images. Inspired by the vanishing moment property which is the key for the good approximation behavior of wavelets, we introduce the directional vanishing moment condition for contourlets. We show that with anisotropic scaling and sufficient directional vanishing moments, contourlets essentially achieve the optimal approximation rate, O((log M)³ M^-2) square error with a best M-term approximation, for 2-D piecewise smooth functions with C² contours. Finally, we show some numerical experiments demonstrating the potential of contourlets in several image processing applications.

This paper presents a novel method for separating images into texture and piecewise smooth parts. The proposed approach is based on a combination of the Basis Pursuit Denoising (BPDN) algorithm and the Total-Variation (TV) regularization scheme. The basic idea promoted in this paper is the use of two appropriate dictionaries, one for the representation of textures, and the other for the natural scene parts. Each dictionary is designed for sparse representation of a particular type of image-content (either texture or piecewise smooth). The use of BPDN with the two augmented dictionaries leads to the desired separation, along with noise removal as a by-product. As the need to choose a proper dictionary for natural scene is very hard, a TV regularization is employed to better direct the separation process. Experimental results validate the algorithm's performance.

The application of the wavelet transform in image processing is most frequently based on a separable construction. Lines and columns in an image are treated independently and the basis functions are simply products of the corresponding one dimensional functions. Such method keeps simplicity in design and computation, but is not capable of capturing properly all the properties of an image. In this paper, a new truly separable discrete multi-directional transform is proposed with a subsampling method based on lattice theory. Alternatively, the subsampling can be omitted and this leads to a multi-directional frame. This transform can be applied in many areas like denoising, non-linear approximation and compression. The results on non-linear approximation and denoising show interesting gains compared to the standard two-dimensional analysis.

Taking advantage of the new developments in mathematical statistics, a multiscale approach is designed to detect filament or filament-like features in noisy images. The major contribution is to introduce a general framework in cases when the data is digital. Our detection method can detect the presence of an underlying curvilinear feature with the lowest possible strength that are still detectible in theory. Simulation results on synthetic data will be reported to illustrate its effectiveness in finite digital situations.

The denoising of video data should take into account both temporal and spatial dimensions, however, true 3D transforms are rarely used for video denoising. Separable 3-D transforms have artifacts that degrade their performance in applications. This paper describes the design and application of the non-separable oriented 3-D dual-tree wavelet transform for video denoising. This transform gives a motion-based multi-scale decomposition for video - it isolates in its subbands motion along different directions. In addition, we investigate the denoising of video using the 2-D and 3-D dual-tree oriented wavelet transforms, where the 2-D transform is applied to each frame individually.

We propose a new subspace decomposition scheme called anisotropic wavelet packets which broadens the existing definition of 2-D 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 double-tree algorithm are then presented and experimental results are shown.

In this paper we present a non-separable multiresolution structure based on frames which is defined by radial scaling functions of the form of the Shannon scaling function. We also construct the resulting frame multiwavelets, which can be isotropic as well. Our construction can be carried out in any number of dimensions and for a great variety of dilation matrices.

It is a challenging task to design orthogonal filter banks, especially multidimensional (MD) ones. In the one-dimensional (1D) two-channel finite impulse response (FIR) filter bank case, several design methods exist. Among them, designs based on spectral factorizations (by Smith and Barnwell) and designs based on lattice factorizations (by Vaidynanathan and Hoang) are the most effective and widely used. The 1D two-channel infinite impulse response (IIR) filter banks and associated wavelets were considered by Herley and Vetterli. All of these design methods are based on spectral factorization. Since in multiple dimensions, there is no factorization theorem, traditional 1D design methods fail to generalize. Tensor products can be used to construct MD orthogonal filter banks from 1D orthogonal filter banks, yielding separable filter banks. In contrast to separable filter banks, nonseparable filter banks are designed directly, and result in more freedom and better frequency selectivity. In the FIR case, Kovacevic and Vetterli designed specific two-dimensional and three-dimensional nonseparable FIR orthogonal filter banks. In the IIR case, there are few design results (if any) for MD orthogonal IIR filter banks. To design orthogonal filter banks, we must design paraunitary matrices, which leads to solving sets of nonlinear equations. The Cayley transform establishes a one-to-one mapping between paraunitary matrices and para-skew-Hermitian matrices. In contrast to nonlinear equations, the para-skew-Hermitian condition amounts to linear constraints on the matrix entries which are much easier to solve. We present the complete characterization of both paraunitary FIR matrices and paraunitary IIR matrices in the Cayley domain. We also propose efficient design methods for MD orthogonal filter banks and corresponding methods to impose the vanishing-moment condition.

Directional multiresolution image representations have lately attracted much attention. A number of new systems, such as the curvelet transform and the more recent contourlet transform, have been proposed. A common issue of these transforms is the redundancy in representation, an undesirable feature for certain applications (e.g. compression). Though some critically sampled transforms have also been proposed in the past, they can only provide limited directionality or limited flexibility in the frequency decomposition. In this paper, we propose a filter bank structure achieving a nonredundant multiresolution and multidirectional expansion of images. It can be seen as a critically sampled version of the original contourlet transform (hence the name CRISP-contourets) in the sense that the corresponding frequency decomposition is similar to that of contourlets, which divides the whole spectrum both angularly and radially. However, instead of performing the multiscale and directional decomposition steps separately as is done in contourlets, the key idea here is to use a combined iterated nonseparable filter bank for both steps. Aside from critical sampling, the proposed transform possesses other useful properties including perfect reconstruction, flexible configuration of the number of directions at each scale, and an efficient tree-structured implementation.

Accurate geometric registration is an important step that precedes various tasks of processing of remotely sensed imagery. Assuming that, after radiometric and systematic correction, images are registered to within a few pixels, our goal is to develop fast and reliable automatic registration methods for multi-sensor data that would yield sub-pixel accuracy. This paper compares two gradient-based algorithms for sub-pixel image registration developed by Thevenaz et al. One of them optimizes intensity difference while the other maximizes mutual information between two images. The algorithms were combined with three invariant wavelet pyramids, a centered cubic spline pyramid as well as both low-pass and band-pass Simoncelli Steerable pyramids. This paper compared the different variations of the two algorithms on both synthetic and real satellite imagery. We found that for single-sensor data, the intensity-based algorithm combined with a band-pass wavelet pyramid produces the best results, while for multi-sensor images, the best choice is the mutual-information-based method combined with a steerable low-pass pyramid.

This paper addresses the modeling of wavelet coefficients for multispectral (MS) band sharpening based on undecimated multiresolution analysis (MRA). The coarse MS bands are sharpened by injecting highpass details taken from a high resolution panchromatic (Pan) image. Besides the MRA, crucial point is modeling the relationships between detail coefficients of a generic MS band and the Pan image at the same resolution. The goal is that the merged MS images are most similar to what the MS sensor would collect if it had the same resolution as the broadband Pan imager. Three injection models embedded in an "a trous" wavelet decomposition will be described and compared on a test set of very high resolution QuickBird MS+Pan data. Two models work on approximation and detail coefficients, respectively, and provide a certain degree of unmixing of coarse MS pixels. The third model is based on spectral fidelity of the merged image to the (resampled) original MS data, that is, no unmixing is attempted. Fusion comparisons on spatially degraded data, of which higher-resolution true MS data are available for reference, show that the former two models yield better results than the latter, in terms of both radiometric and spectral fidelity. The latter, however, yields a reliable sharpening unaffected by local artifacts, regardless of landscape complexity. When local statistics of wavelet coefficients are utilized, the model estimated on approximation yields slightly better but considerably stabler results than that calculated starting from bandpass details.

An algorithm for multidimensional nonlinear registration is proposed. The deformation field between two elastic bodies is represented by a multi-resolution separable wavelet. Using a progressive approach that reduces algorithm complexity the registration parameters are recovered in a coarse to fine order. A custom wavelet that approximates threefold orthogonality is developed. The performance of the algorithm is evaluated by the alignment of sections from mouse brains. The wavelet registration algorithm demonstrated on average fourfold improvement in section alignment over linear alignment.

The performance of a balanced, nonseparable orthogonal multiwavelet for edge detection is analized. We present two alternative methods to meet this objective: in the first one the normed fine detail coefficients of the multiwavelet transform are thresholded, in the other, we adapt the modulus maxima algorithm to nonseparable multiwavelets. Results are highly satisfactory.

In this paper, we study unsupervised image segmentation using wavelet-domain hidden Markov models (HMMs), where three clustering methods are used to obtain the initial segmentation results. We first review recent supervised Bayesian image segmentation algorithms using wavelet-domain HMMs. Then, a new unsupervised segmentation approach is developed by capturing the likelihood disparity of different texture features with respect to wavelet-domain HMMs. Three clustering methods, i.e., K-mean, soft clustering and multiscale clustering, are studied to convert the unsupervised segmentation problem into the self-supervised process by identifying the reliable training samples. The simulation results on synthetic mosaics and real images show that the proposed unsupervised segmentation algorithms can achieve high classification accuracy.

Amplitude deviation (AD), frequency deviation (FD) and phase deviation (PD) are the important compositions of power quality disturbance (PQD). To analyze PQD deeply, this paper introduces a wavelet-based method for separating slight AD, FD and PD from a combined PQD, then quantifying and identifying them. The method is based on the fact that a linear-phase complex wavelet is certainly with an even real part and an odd imaginary part, or inversely. The distinctive characteristics of the method are: complex biorthogonal wavelet with the shortest smoothing filter (Haar filter), shift-invariant wavelet transform (WT) at a few scales, simple relationships between the WT coefficients and the magnitudes of AD, FD and PD, simple binary feature victor and binary-decimal conversion identifying process. These make the method simple, correct and fast.

A wavelet-based pavement distress detection and evaluation method is proposed. This method consists of two main parts, real-time processing for distress detection and offline processing for distress evaluation. The real-time processing part includes wavelet transform, distress detection and isolation, and image compression and noise reduction. When a pavement image is decomposed into different frequency subbands by wavelet transform, the distresses, which are usually irregular in shape, appear as high-amplitude wavelet coefficients in the high-frequency details subbands, while the background appears in the low-frequency approximation subband. Two statistical parameters, high-amplitude wavelet coefficient percentage (HAWCP) and high-frequency energy percentage (HFEP), are established and used as criteria for real-time distress detection and distress image isolation. For compression of isolated distress images, a modified EZW (Embedded Zerotrees of Wavelet coding) is developed, which can simultaneously compress the images and reduce the noise. The compressed data are saved to the hard drive for further analysis and evaluation. The offline processing includes distress classification, distress quantification, and reconstruction of the original image for distress segmentation, distress mapping, and maintenance decision-making. The compressed data are first loaded and decoded to obtain wavelet coefficients. Then Radon transform is then applied and the parameters related to the peaks in the Radon domain are used for distress classification. For distress quantification, a norm is defined that can be used as an index for evaluating the severity and extent of the distress. Compared to visual or manual inspection, the proposed method has the advantages of being objective, high-speed, safe, automated, and applicable to different types of pavements and distresses.

Z pinches produce an X ray rich plasma environment where backlighting imaging of imploding targets can be quite challenging to analyze. What is required is a detailed understanding of the implosion dynamics by studying snapshot images of its in flight deformations away from a spherical shell. We have used wavelets, curvelets and multiresolution analysis techniques to address some of these difficulties and to establish the Shell Thickness Averaged Radius (STAR) of maximum density, r*(N,θ) where N is the percentage of the shell thickness over which we average. The non-uniformities of r*(N,θ) are quantified by a Legendre polynomial decomposition in angle, θ, and the identification of its largest coefficients. Undecimated wavelet decompositions outperform decimated ones in denoising and both are surpassed by the curvelet transform. In each case, hard thresholding based on noise modeling is used.

In this paper we introduce a biologically-inspired spatial video filtering chip and discuss its application in wavelet filter banks. Two types of spatial filtering chips have been developed - the thin film analog image processor (TAIP) and the switched-capacitor analog image processor (SCAIP). Each chip can filter video at high frame rates with Gaussian-like filters having adjustable widths. By linearly combining the outputs of a bank of spatial filter chips we can create a large variety of filters. Effective use of the filtering chips requires two things. First, an assessment of the filters that can be realized within the constraints of the hardware is required. Although any function within reasonable constraints can be decomposed into a combination of Gaussian functions, an efficient method to do so is an open problem. We have restricted ourselves to a simpler problem - given a limited number of Gaussian-like functions, what useful classes of filters can be generated? Second, given an image-processing application, a method to organize a choice of filters is needed. We are currently investigating these problems in the context of feature analysis/discrimination, and have found a useful organizing principle in the continuous wavelet transform (CWT).

The wavelet transform is a powerful tool for capturing the joint time-frequency characteristics of a signal. However, the resulting wavelet coefficients are typically high-dimensional, since at each time sample the wavelet transform is evaluated at a number of distinct scales. Unfortunately, modelling these coefficients can be problematic because of the large number of parameters needed to capture the dependencies between different scales. In this paper we investigate the use of algorithms from the field of dimensionality reduction to extract informative and compact descriptions of shape from wavelet coefficients. These low-dimensional shape descriptors lead to models that are governed by only a small number of parameters and can be learnt successfully from limited amounts of data. The validity of our approach is demonstrated on the task of automatically segmenting an electrocardiogram signal into its constituent waveform features.

Wavelet packets are well-known for their ability to compactly represent textures consiting of oscillatory patterns such as fingerprints or striped cloth. In this paper, we report recent work on representing both periodic and granular types of texture using adaptive wavelet basis functions. The discrimination power of a wavelet packet subband can be defined as its ability to differentiate between any two texture classes in the transform domain, consequently leading to better classification results. The problem of adaptive wavelet basis selection for texture analysis can, therefore, be solved by using a dynamic programming approach to find the best basis from a library of orthonormal basis functions with respect to a discriminant measure. We present a basis selection algorithm which extends the concept of 'Local Discrminant Basis' (Saito and Coifman, 1994) to two dimensions. The problem of feature selection is addressed by sorting the features according to their relevance as described by the discriminant measure, which has a significant advantage over other feature selection methods that both basis selection and reduction of dimensionality of the feature space can be done simultaneously. We show that wavelet packets are good at representing not only oscillatory patterns but also granular textures. Comparative results are presented for four different distance metrics: Kullback-Leibler (KL) divergence, Jensen-Shannon (JS) divergence, Euclidean distance, and Hellinger distance. Initial experimental results show that Hellinger and Euclidean distance metrics may perform better as compared to other cost functions.

Conventional techniques for signal analysis and processing in the time-frequency domain are not well adapted to digital processing of music signals. This restricts the features and quality of applications. We present the current status of a research initiative on this problem. A novel family of wavelet-like bases allows a tiling of the time-frequency plane that is better adapted to digital music signals. This will allow performance enhancements in all kinds of digital audio applications.

In previous works, Umesh et al, demonstrated that phonetically similar vowels spoken by different individuals are related by a simple translation in a universal warped spectral representation. They experimentally derived this function and called it the “speech-scale”. We present further experimental evidence, based on a large data set, validating the speech-scale. We also estimate speaker-specific scale factors based on the speech-scale, and we present a vowel classification experiment, which demonstrates a significant performance improvement through a normalization based on the speech-scale. The results we present are based on formant estimates of vowels in a Western Michigan vowel database.

We propose formal analytical models for identification of tumors in medical images based on the hypothesis that the tumors have a fractal (self-similar) growth behavior. Therefore, the images of these tumors may be characterized as Fractional Brownian motion (fBm) processes with a fractal dimension (D) that is distinctly different than that of the image of the surrounding tissue. In order to extract the desired features that delineate different tissues in a MR image, we study multiresolution signal decomposition and its relation to fBm. The fBm has proven successful to modeling a variety of physical phenomena and non-stationary processes, such as medical images, that share essential properties such as self-similarity, scale invariance and fractal dimension (D). We have developed the theoretical framework that combines wavelet analysis with multiresolution fBm to compute D.

A class of adaptive wavelet transforms that map integers to integers based on the adaptive update lifting scheme is presented. The main feature in the adaptive update lifting scheme is that the update lifting step, which is considered as an averaging operator and is performed prior to the prediction step, is adapted to the underlying signal content and the adaptivity decisions can be recovered at the synthesis transform without bookkeeping of the adaptivity decisions. The perfect reconstruction criterion for the integer realisation of such transforms are presented in this paper. These adaptive integer-to-integer wavelet transforms can be used in scalable lossless image coding applications. The lossless image coding and spatially scalable decoding performances are demonstrated.

Wavelet transform is a powerful and useful mathematical tool for signal processing. In this paper detailed description of procedures for numerical integration and derivation in Haar domain has been done. These procedures are necessary both in control systems analysis, and, especially, for control design and development. A detailed comparison between classical methods of evaluation and the Haar way is presented and critically discussed.

In order to compress 3-D shaped data, this paper presents an adaptive coding method using 2-D signal compression technique for the structured data on a 2-D plane, maintaining the adjacent relation of each vertex of the 3-D model. This structured data on a 2-D plane is not rectangular, so we apply Shape Adaptive Wavelet Transform to compress the structured data on a 2-D plane. And we change the weight for wavelet coefficients adaptively for each frequency band, considering the normal vectors on the surface model rendering. The experiments show that the rendering characteristics can be improved by using the proposed method.

Data decorrelation and energy compaction are the two fundamental characteristics of wavelets that led to wavelet based image compression models. Wavelet transform is not a perfect whitening transform; but it is viewed as an approximation to Karhunen-Loeve transform (KLT). In general, decorrelation does not imply statistical independence. Thus, a wavelet transform results in coefficients which exhibit inter and intra band dependencies. The energy compaction property of a wavelet is reflected in the coding performance, which can be measured by its coding gain. This paper investigates the above two important aspects of bi-orthogonal wavelets in the context of lossy compression. This investigation suggests that simple predictive models are sufficient to capture the dependencies exhibited by the wavelet coefficients. This paper also compares the metrics that measure the performance of bi-orthogonal wavelets in lossy coding schemes.

Typical neuroimaging studies place great emphasis on not only the estimation but also the standard error estimates of underlying parameters derived from a temporal model. This is principally done to facilitate the use of t-statistics. Due to the spatial correlations in the data, it can often be more advantageous to interrogate models in the wavelet domain than in the image domain. However, widespread acceptance of these wavelet techniques has been hampered due to the limited ability to generate both parametric and error estimates in the image domain from these temporal models in the wavelet domain, without which comparison to current standard non-wavelet methods can prove difficult. This paper introduces a derivation of these estimates and an implementation for their calculation from these models for a class of thresholding estimators which have been shown to be useful for neuroimaging studies. This work stems from a consideration of the wavelet operator as a multidimensional linear operator and builds on work from the image processing community.

This paper considers the extraction of information from a locally stationary process modelled by wavelet packets. A method is presented to select subprocesses that characterize the key aspects of the nonstationary process for pattern analysis. The estimated parameters of the selected subprocesses are used to infer the process' time varying behavior. The estimated parameters can be used as features in the attempt to distinguish changing states within a process or differentiate two different locally stationary processes.

This work is devoted to the construction, the analysis and some applications of multiresolution analyses involving non translation invariant bases. It uses the very elementary tools of the Harten's multiresolution framework and its connections to non uniform, stationary subdivision schemes described for instance by Dyn. The applications deal with the analysis of the Gibbs phenomenon and the compression property of the corresponding multiscale process in one dimension as well as with compression of images.

We treat nonparametric estimation of a regression function defined on a 'tensor product irregular grid,' that is, a grid constructed as the Cartesian product of two irregular one-dimensional grids. Our wavelet-type estimator is based on a wavelet transform which is the tensor product of two one-dimensional design-adapted wavelet transforms. We propose a denoising scheme and show the performance of the resulting estimator through a simulation study.

Wavelet packet transform analyzes signals more finely than wavelet transform does. This advantage can be utilized in optical wavelet transform. To introduce wavelet packet transform into optics, mother wavelets that have scaling functions must be used. If the scaling function does not have analytical formula, its approximation can be computed using the cascade algorithm. With the refinement relationship, its wavelet function can by calculated. After the 1-D wavelet packet bases are obtained, 2-D separable wavelet packet bases can be constructed for optical wavelet packet transform. As an example, a volume holographic opto-electronic system is proposed to fulfill joint best basis selection for a face image bank with the mother Db3.

Adaptive systems are useful when the signals or images are changing with time. For example, with adaptive wavelets, different filters are used for different parts of the signal: the signal itself should indicate whether a high or low order filter should be used. With adaptive optics, rapidly varying atmospheric wavefront distortions in a medium changing with time is reduced using optics: i.e. in astronomical adaptive optical systems, a system of control-driven deformable mirrors eliminates distortion produced by a medium changing with time. Adaptive wavelets has the potential for achieving the same objective while reducing cost. Adaptive optics provides real-time compensation for aberrations produced by atmospheric turbulence, jitter, and the optics. The adaptive optics subsystem consists of a Wavefront Sensor, Real-Time Reconstructor and Server Compensator, Deformable Mirror, Tilt Correction, Optical Assembly, and Adaptive Optics Control. The Wavefront Sensor senses phase difference and wavefront tilts. The Real-Time Reconstructor and Servo Compensation system computes the Deformable Mirror actuator. The Tilt Correction system corrects wavefront tilt errors and angle of arrival jitter caused by atmospheric turbulence, mount vibration, wobble dynamics lag and system vibration. In summation, adaptive optics systems are highly complex and both assembly and maintenance very expensive. Adaptive wavelets offers the potential of simplifying the system and reducing the cost. The ultimate goal is higher image resolution. Adaptive systems are important when the signals or environments are changing in time. With adaptive lifting, the prediction/update filters or wavelet/scaling functions are chosen in a fixed fashion. They can be chosen in such a way that a signal is approximated with very high accuracy using only a limited number of coefficients. Different prediction filters can be used for different parts of the signal. A high or low order prediction filter is chosen based on the signal itself. For example, the space-adaptive approach, the prediction filter depends on local information of the image pixels of one of two complementary groups. For applying adaptive wavelet lifting to optical images modulated by atmospheric turbulence, certain assumptions can be made: (1) the image is radially symmetric, and (2) the atmospheric turbulence is to some degree periodic. After that, choice of the prediction filter will take into account the characteristics of the optical image being investigated. Phase, for example, is never an easy problem.

In this paper we present a technique for multiwavelet analysis using a phase-encoded optoelectronic joint transfer correlator (JTC). The optoelectronic joint transform correlator (JTC) is presented and analyzed as an effective optoelectronic correlator for ultra-fast wavelet analysis. By multiplexing reference wavelets in phase, mutiwavelet analysis is performed in a single correlation. The architecture of the proposed system and computer simulations are presented.

We propose an approximation scheme on representation spaces which elements are piecewise polynomial functions, with deficient regularity on a pre-established grid of knots. We characterize these spaces, expose relevant properties and define appropriate bases. We design approximations methods based on orthogonal projections of a given signal, restricted to certain conditions, according with the regularity that is wished. We suggest applications for this procedure in the context of signal processing.

In this work we will generalize results linking multiresolution analysis structures and vectorial spaces generated from integer shifts of self-similar or radial basis functions. This connection results of a remarkable relation between causal scaling and causal radial functions, recently exposed by T. Blu and M. Unser for the unidimensional case. Here, we will detail some definitions and will enunciate the main theorems for the r dimensional case.