Sparse and redundant representations − an emerging and powerful model for signals − suggests that a data source could be described as a linear combination of few atoms from a pre-specified and over-complete dictionary. This model has drawn a considerable attention in the past decade, due to its appealing theoretical foundations, and promising practical results it leads to. Many of the applications that use this model are formulated as a mixture of l₂-l_p (p ≤ 1) optimization expressions. Iterated Shrinkage algorithms are a new family of highly effective numerical techniques for handling these optimization tasks, surpassing traditional optimization techniques. In this paper we aim to give a broad view of this group of methods, motivate their need, present their derivation, show their comparative performance, and most important of all, discuss their potential in various applications.

This paper investigates the performance of randomly dithered first and higher-order sigma-delta quantization applied to the frame coefficients of a vector in a infinite-dimensional Hilbert space. We compute the mean square error resulting from linear reconstruction with the quantized frame coefficients. When properly dithered, this computation simplifies in the same way as under the assumption of the white-noise hypothesis. The results presented here are valid for a uniform mid-tread quantizer operating in the no-overload regime. We estimate the large-redundancy asymptotics of the error for each family of tight frames obtained from regular sampling of a bounded, differentiable path in the Hilbert space. In order to achieve error asymptotics that are comparable to the quantization of oversampled band-limited functions, we require the use of smoothly terminated frame paths.

Compressive sensing is a new data acquisition technique that aims to measure sparse and compressible signals at close to their intrinsic information rate rather than their Nyquist rate. Recent results in compressive sensing show that a sparse or compressible signal can be reconstructed from very few measurements with an incoherent, and even randomly generated, dictionary. To date the hardware implementation of compressive sensing analog-to-digital systems has not been straightforward. This paper explores the use of Sigma-Delta quantizer architecture to implement such a system. After examining the challenges of using Sigma-Delta with a randomly generated compressive sensing dictionary, we present efficient algorithms to compute the coefficients of the feedback loop. The experimental results demonstrate that Sigma-Delta relaxes the required analog filter order and quantizer precision. We further demonstrate that restrictions on the feedback coefficient values and stability constraints impose a small penalty on the performance of the Sigma-Delta loop, while they make hardware implementations significantly simpler.

ΣΔ-modulation is an A/D-conversion method which represents a bandlimited signal by sequences of ±1 whose local averages approximate the function values. The best bounds for the decay rate of the script-l^∞-error arising from such quantization schemes have been given by Güntürk.1 He constructs an infinite family of schemes which lead to an algorithm that establishes exponential error decay with decay rate 0.077. In this paper we improve his construction introducing an additional symmetry, which is suggested by numerical experiments. To show that the modified schemes are still stable, we use the asymptotics of the Γ-function. This leads to a bound of 0.088 for the error decay rate.

Currently, the most efficient technique to coarsely quantize the coefficients of redundant shift invariant signal expansions is a recursive method called ΣΔ modulation. However, the error of approximation resulting from this type of quantization is difficult to analyze rigorously. This is because a ΣΔ modulator is basically a nonlinear feedback system. With constant inputs, it was previously shown that the state vectors of ΣΔ modulators of a certain class appear to remain in a tile. In this paper, we show experimentally that this property remains valid with a class of time-varying inputs. We explain the importance of the tiling property for the error analysis of ΣΔ modulation.

The β-encoder, introduced as an alternative to binary encoding in A/D conversion, creates a quantization scheme robust with respect to quantizer imperfections by the use of a β-expansion, where 1 < β < 2. In this paper we introduce a more general encoder called the βα-encoder, that can offer more flexibility in design and robustness without any significant drawback on the exponential rate of convergence of the obtained expansion. Although an extra multiplication is introduced, it needs not be very accurate. Mathematically, the βα-encoder gives rise to a dynamical system that is both very interesting and challenging.

In this note we will show that the so called Sobolev dual is the minimizer over all linear reconstructions using dual frames for stable r^th order ΣΔ quantization schemes under the so called White Noise Hypothesis (WNH) design criteria. We compute some Sobolev duals for common frames and apply them to audio clips to test their performance against canonical duals and another alternate dual corresponding to the well known Blackman filter.

We extend the notion of complex B-splines to a multivariate setting by employing the relationship between ordinary B-splines and multivariate B-splines by means of ridge functions. In order to obtain properties of complex B-splines in R^s, 1 < s ∈ N, the Dirichlet average has to be generalized to include infinite dimensional simplices. Based on this generalization several identities of multivariate complex B-splines are exhibited.

We develop a wavelet transform on the sphere, based on the spherical HEALPix coordinate system (Hierarchical Equal Area iso-Latitude Pixelization). HEALPix is heavily used for astronomical data processing applications; it is intrinsically multiscale and locally euclidean, hence appealing for building multiscale system. Furthermore, the equal-area pixelization enables us to employ average-interpolating refinement, giving wavelets of local support. HEALPix wavelets have numerous applications in geopotential modeling. A statistical analysis demonstrates wavelet compressibility of the geopotential field and shows that geopotential wavelet coefficients have many of the statistical properties that were previously observed with wavelet coefficients of natural images. The HEALPix wavelet expansion allows to evaluate a gravimetric quantity over a local region far more rapidly than the classic approach based on spherical harmonics. Our software tools are specifically tailored to demonstrate these advantages.

We present multiresolution spaces of complex rotation-covariant functions, deployed on the 2-D hexagonal lattice. The designed wavelets, which are complex-valued, provide an important phase information for image analysis, which is missing in the discrete wavelet transform with real wavelets. Moreover, the hexagonal lattice allows to build wavelets having a more isotropic magnitude than on the Cartesian lattice. The associated filters, defined in the Fourier domain, yield an efficient FFT-based implementation.

In this work, we present new methods for creating M-channel directional filters to construct multiresolution and multidirectional orthogonal/biorthogonal transforms. A key feature of these methods is the ability to solve the polynomial Bezout equation in higher dimensions by taking advantage of solutions that have been proposed for solving a related equation known as the analytic Bezout equation. These new techniques are capable of creating directional filters that yield spatial-frequency tilings equivalent to those of the contourlet and the shearlet transforms. Such directional filter banks can create sparse representations for a large class of images and can be used for various restoration problems, compression schemes, and image enhancements.

We present an iterative deconvolution algorithm that minimizes a functional with a non-quadratic wavelet-domain regularization term. Our approach is to introduce subband-dependent parameters into the bound optimization framework of Daubechies et al.; it is sufficiently general to cover arbitrary choices of wavelet bases (non-orthonormal or redundant). The resulting procedure alternates between the following two steps: 1. a wavelet-domain Landweber iteration with subband-dependent step-sizes; 2. a denoising operation with subband-dependent thresholding functions. The subband-dependent parameters allow for a substantial convergence acceleration compared to the existing optimization method. Numerical experiments demonstrate a potential speed increase of more than one order of magnitude. This makes our "fast thresholded Landweber algorithm" a viable alternative for the deconvolution of large data sets. In particular, we present one of the first applications of wavelet-regularized deconvolution to 3D fluorescence microscopy.

We propose in this paper an iterative algorithm for 3D confocal microcopy image restoration. The image quality is limited by the diffraction-limited nature of the optical system which causes blur and the reduced amount of light detected by the photomultiplier leading to a noise having Poisson statistics. Wavelets have proved to be very effective in image processing and have gained much popularity. Indeed, they allow to denoise efficiently images by applying a thresholding on coefficients. Moreover, they are used in algorithms as a regularization term and seem to be well adapted to preserve textures and small objects. In this work, we propose a 3D iterative wavelet-based algorithm and make some comparisons with state-of-the-art methods for restoration.

A variety of image and signal processing algorithms based on wavelet filtering tools have been developed during the last few decades, that are well adapted to the experimental variability typically encountered in live biological microscopy. A number of processing tools are reviewed, that use wavelets for adaptive image restoration and for motion or brightness variation analysis by optical flow computation. The usefulness of these tools for biological imaging is illustrated in the context of the restoration of images of the inner ear and the analysis of cochlear motion patterns in two and three dimensions. I also report on recent work that aims at capturing fluorescence intensity changes associated with vesicle dynamics at synaptic zones of sensory hair cells. This latest application requires one to separate the intensity variations associated with the physiological process under study from the variations caused by motion of the observed structures. A wavelet optical flow algorithm for doing this is presented, and its effectiveness is demonstrated on artificial and experimental image sequences.

We survey our work on adaptive multiresolution (MR) approaches to the classification of biological and fingerprint images. The system adds MR decomposition in front of a generic classifier consisting of feature computation and classification in each MR subspace, yielding local decisions, which are then combined into a global decision using a weighting algorithm. The system is tested on four different datasets, subcellular protein location images, drosophila embryo images, histological images and fingerprint images. Given the very high accuracies obtained for all four datasets, we demonstrate that the space-frequency localized information in the multiresolution subspaces adds significantly to the discriminative power of the system. Moreover, we show that a vastly reduced set of features is sufficient. Finally, we prove that frames are the class of MR techniques that performs the best in this context. This leads us to consider the construction of a new family of frames for classification, which we term lapped tight frame transforms.

This paper presents a new method for computing the Feature-adapted Radon and Beamlet transforms [1] in a fast and accurate way. These two transforms can be used for detecting features running along lines or piecewise constant curves. The main contribution of this paper is to unify the Fast Slant Stack method, introduced in [2], with linear filtering technique in order to define what we call the Feature-adapted Fast Slant Stack. If the desired feature detector is chosen to belong to the class of steerable filters, our method can be achieved in O(N log(N)), where N = n² is the number of pixels. This new method leads to an efficient implementation of both Feature-adapted Radon and Beamlet transforms, that outperforms our previous works [1] both in terms of accuracy and speed. Our method has been developed in the context of biological imaging to detect DNA filaments in fluorescent microscopy.

In recent years, the focus in biological science has shifted to understanding complex systems at the cellular and molecular levels, a task greatly facilitated by fluorescence microscopy. Segmentation, a fundamental yet difficult problem, is often the first processing step following acquisition. We have previously demonstrated that a stochastic active contour based algorithm together with the concept of topology preservation (TPSTACS) successfully segments single cells from multicell images. In this paper we demonstrate that TPSTACS successfully segments images from other imaging modalities such as DIC microscopy, MRI and fMRI. While this method is a viable alternative to hand segmentation, it is not yet ready to be used for high-throughput applications due to its large run time. Thus, we highlight some of the benefits of combining TPSTACS with the multiresolution approach for the segmentation of fluorescence microscope images. Here we propose a multiscale active contour (MSAC) transformation framework for developing a family of modular algorithms for the segmentation of fluorescence microscope images in particular, and biomedical images in general. While this framework retains the flexibility and the high quality of the segmentation provided by active contour-based algorithms, it offers a boost in the efficiency as well as a framework to compute new features that further enhance the segmentation.

We present variants of both the digital curvelet transform, and the digital wave atom transform, which handle the image boundaries by mirror extension. Previous versions of these transforms treated image boundaries by periodization. The main ideas of the modifications are 1) to tile the discrete cosine domain instead of the discrete Fourier domain, and 2) to adequately reorganize the in-tile data. In their shift-invariant versions, the new constructions come with no penalty on the redundancy or computational complexity. For shift-variant wave atoms, the penalty is a factor 2 instead of the naive factor 4. These various modifications have been included in the CurveLab and WaveAtom toolboxes, and extend the range of applicability of curvelets (good for edges and bandlimited wavefronts) and wave atoms (good for oscillatory patterns and textures) to situations where periodization at the boundaries is uncalled for. The new variants are dubbed ME-curvelets and ME-wave atoms, where ME stands for mirror-extended.

This article presents the first adaptive quasi minimax estimator for geometrically regular images in the white noise model. This estimator is computed using a thresholding in an adapted orthogonal bandlet basis optimized for the noisy observed image. In order to analyze the quadratic risk of this best basis denoising, the thresholding in an orthogonal bandlets basis is recasted as a model selection process. The resulting estimator is computed with a fast algorithm whose theoretical performance can be derived. This efficiency is confirmed through numerical experiments on natural images.

The standard separable two-dimensional (2-D) wavelet transform (WT) has recently achieved a great success in image processing because it provides a sparse representation of smooth images. However, it fails to capture efficiently one-dimensional (1-D) discontinuities, like edges or contours. These features, being elongated and characterized by geometrical regularity along different directions, intersect and generate many large magnitude wavelet coefficients. Since contours are very important elements in visual perception of images, to provide a good visual quality of compressed images, it is fundamental to preserve good reconstruction of these directional features. We propose a construction of critically sampled perfect reconstruction transforms with directional vanishing moments (DVMs) imposed in the corresponding basis functions along different directions, called directionlets. We also demonstrate the outperforming non-linear approximation (NLA) results achieved by our transforms and we show how to design and implement a novel efficient space-frequency quantization (SFQ) compression algorithm using directionlets. Our new compression method beats the standard SFQ both in terms of mean-square-error (MSE) and visual quality, especially in the low-rate compression regime. We also show that our compression method, does not increase the order of computational complexity as compared to the standard SFQ algorithm.

In this paper we present a brief account of the use of the spectral theory of slanted matrices in frame and sampling theory. Some abstract results on slanted matrices are also presented.

We introduce the notion of operators localized with respect to frames and prove the boundedness of such operators on families of Banach spaces. This generalizes the method used in proving boundedness of pseudodifferential operators on various spaces. We also use this notion to provide sufficient conditions for the construction of frames which have the localization property.

Probably the most important property of wavelets for signal processing is their multiscale derivative-like behavior when applied to functions. In order to extend the class of problems that can profit of wavelet-based techniques, we propose to build new families of wavelets that behave like an arbitrary scale-covariant operator. Our extension is general and includes many known wavelet bases. At the same time, the method takes advantage a fast filterbank decomposition-reconstruction algorithm. We give necessary conditions for the scale-covariant operator to admit our wavelet construction, and we provide examples of new wavelets that can be obtained with our method.

The classical sampling theorem, attributed to Whittaker, Shannon, Nyquist, and Kotelnikov, states that a bandlimited function can be recovered from its samples, as long as we use a sufficiently dense sampling grid. Here, we review the recent development of an operator sampling theory which allows for a "widening" of the classical sampling theorem. In this realm, bandlimited functions are replaced by "bandlimited operators". that is, by pseudodifferential operators which have bandlimited Kohn-Nirenberg symbols. Similar to the Nyquist sampling density condition alluded to above, we discuss sufficient and necessary conditions on the bandlimitation of pseudodifferential operators to ensure that they can be recovered by their action on a single distribution. In fact, we show that an operator with Kohn-Nirenberg symbol bandlimited to a Jordan domain of measure less than one can be recovered through its action on a distribution defined on a appropriately chosen sampling grid. Further, an operator with bandlimitation to a Jordan domain of measure larger than one cannot be recovered through its action on any tempered distribution whatsoever, pointing towards a fundamental difference to the classical sampling theorem where a large bandwidth could always be compensated through a sufficiently fine sampling grid. The dichotomy depending on the size of the bandlimitation is related to Heisenberg's uncertainty principle. Further, we discuss an application of this theory to the channel measurement problem for Multiple-Input Multiple-Output (MIMO) channels.

The Rangan-Goyal (RG) algorithm is a recursive method for constructing an estimate x_N ∈ R^d of a signal x ∈ R^d, given N ≥ d frame coefficient measurements of x that have been corrupted by uniform noise. Rangan and Goyal proved that the RG-algorithm is constrained by the Bayesian lower bound: lim inf_N→∞N² E||x − x_N||² > 0. As a positive counterpart to this, they also proved that for every p < 1 and x ∈ R^d, the RG-algorithm satisfies lim_N→∞ N^p||x − x_N|| = 0 almost surely. One consequence of the existing results is that one "almost" has mean square error E||x − x_N||² of order 1/N² for random choices of frames. It is proven here that the RG-algorithm achieves mean square error of the optimal order 1/N², and the applicability of such error estimates is also extended to deterministic frames where ordering issues play an important role. Approximation error estimates for consistent reconstruction are also proven.

Traditional neuroimaging experiments, dictated by the dogma of functional specialization, aim at identifying regions of the brain that are maximally correlated with a simple cognitive or sensory stimulus. Very recently, functional MRI (fMRI) has been used to infer subjective experience and brain states of subjects immersed in natural environments. These environments are rich with uncontrolled stimuli and resemble real life experiences. Conventional methods of analysis of neuroimaging data fail to unravel the complex activity that natural environments elicit. The contribution of this work is a novel method to predict action and sensory experiences of a subject from fMRI. This method relies on an embedding that provides an optimal coordinate system to reduce the dimensionality of the fMRI dataset while preserving its intrinsic dynamics. We learn a set of time series that are implicit functions of the fMRI data, and predict the values of these times series in the future from the knowledge of the fMRI data only. We conducted several experiments with the datasets of the 2007 Pittsburgh Experience Based Cognition competition.

This article presents a statistical framework to analyse brain functional Magnetic Resonance Imaging (fMRI) data. A particular emphasis is made on spatial correlation, which, contrary to the usual preprocessing step of spatial smoothing, is now part of the probabilistic model. The characterisation of regionally specific effects is done via the General Linear Model (GLM) using Posterior Probability Maps (PPMs). The spatial regularisation is defined over regression coefficients by specifying a spatial prior using Sparse Spatial Basis Functions (SSBFs), such as Wavelets. These are embedded in a hierarchical probabilistic model which, when inverted, automatically selects an appropriate subset of basis functions. The inversion of the model is done using Variational Bayes. We present results on synthetic data and on data from an event-related fMRI experiment. We conclude that SSBFs allow for spatial variations in signal smoothness, provide an increased sensitivity and are more computationally efficient than previously presented work.

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.

In this paper, a general framework for the inversion of a linear operator in the case where one seeks several components from several observations is presented. The estimation is done by minimizing a functional balancing discrepancy terms by regularization terms. The regularization terms are adapted norms that enforce the desired properties of each component. The main focus of this paper is the definition of the discrepancy terms. Classically, these are quadratic. We present novel discrepancy terms adapt to the observations. They rely on adaptive projections that emphasize important information in the observations. Iterative algorithms to minimize the functionals with adaptive discrepancy terms are derived and their convergence and stability is studied. The methods obtained are compared for the problem of reconstruction of astrophysical maps from multifrequency observations of the Cosmic Microwave Background. We show the added flexibility provided by the adaptive discrepancy terms.

We address the statistical modeling of solar images provided by the Extreme ultraviolet Imaging Telescope (EIT) onboard the Solar and Heliospheric Observatory (SoHO, a joint ESA/NASA mission). We focus in particular on the less structured regions, the "Quiet Sun". We first review on a brief historical viewpoint on multifractal processes for physical modeling. Then we present a multifractal analysis of Quiet Sun images. Our aim is to identify a model that would permit to simulate images that are similar to real ones, and to use the scale invariance property to obtain artificial images at any finer resolution. We compare various families of models including infinitely divisible cascades and fractional stable fields that permit to synthesize images that are statistically similar to Quiet Sun images. This modeling will assist in promoting forthcoming high resolution observations by analysing sub-pixel variability in today's solar corona images.

A worldwide collaboration attempts to confirm the existence of gravitational waves predicted by Einstein's theory of General Relativity, through direct observation with a network of large-scale laser interferometric antennas. This paper is a contribution to the methodologies used to scrutinize the data in order to reveal the tiny signature of a gravitational wave from rare cataclysmic events of astrophysical origin. More specifically, we are interested in the detection of short frequency modulated transients or gravitational wave chirps. The amount of information about the frequency vs. time evolution is limited: we only know that it is smooth. The detection problem is thus non-parametric. We introduce a finite family of "template waveforms" which accurately samples the set of admissible chirps. The templates are constructed as a puzzle, by assembling elementary bricks (the chirplets) taken a dictionary. The detection amounts to testing the correlation between the data and the template family. With an adequate time-frequency mapping, we establish a connection between this correlation measurement and combinatorial optimization problems of graph theory, from which we obtain efficient algorithms to perform the calculation. We present two variants. A first one addresses the case of amplitude modulated chirps and the second allows the joint analysis of the data from several antennas. Those methods are not limited to the specific context for which they have been developed. We pay a particular attention to the aspects that can be source of inspiration for other applications.

We present a method for analyzing and classifying 2d-pure-point (pp) diffraction spectra (i.e. set of Bragg peaks) of certain self-similar structures with scaling factor β > 1, like quasicrystals. The 2d-pp diffraction spectrum is viewed as a point set in the complex plane in which each point is assigned a positive number, its Bragg intensity. Then, by using a nested sequence of self-similar subsets called beta-lattices, a multiresolution analysis is carried out on the spectrum, leading to a partition of it at once in geometry, in scale, and in intensity ("fingerprint" of the spectrum). As an illustration of our approach, the method is experimented on pp diffraction spectra of a few mathematical structures.

In this paper we summarize the basic formulas of wavelet analysis with the help of Poisson wavelets on the sphere. These wavelets have the nice property that all basic formulas of wavelet analysis as reproducing kernels, etc. may be expressed simply with the help of higher degree Poisson wavelets. This makes them numerically attractive for applications in geophysical modeling. We do not give any proofs and we refer to "M. Holschneider, I. Iglewska-Nowak, JFAA, 2007", where all proofs are published.

Dark energy dominates the energy density of our Universe, yet we know very little about its nature and origin. Although strong evidence in support of dark energy is provided by the cosmic microwave background, the relic radiation of the Big Bang, in conjunction with either observations of supernovae or of the large scale structure of the Universe, the verification of dark energy by independent physical phenomena is of considerable interest. We review works that, through a wavelet analysis on the sphere, independently verify the existence of dark energy by detecting the integrated Sachs-Wolfe effect. The effectiveness of a wavelet analysis on the sphere is demonstrated by the highly statistically significant detections of dark energy that are made. Moreover, the detection is used to constrain properties of dark energy. A coherent picture of dark energy is obtained, adding further support to the now well established cosmological concordance model that describes our Universe.

The statistics of the temperature anisotropies in the primordial Cosmic Microwave Background radiation field provide a wealth of information for cosmology and the estimation of cosmological parameters. An even more acute inference should stem from the study of maps of the polarization state of the CMB radiation. Measuring the latter extremely weak CMB polarization signal requires very sensitive instruments. The full-sky maps of both temperature and polarization anisotropies of the CMB to be delivered by the upcoming Planck Surveyor satellite experiment are hence awaited with excitement. Still, analyzing CMB data requires tackling a number of practical difficulties, notably that several other astrophysical sources emit radiation in the frequency range of CMB observations. Separating the different astrophysical foreground components and the CMB proper from available multichannel data is a problem that has drawn much attention in the community. Nevertheless, some level of residual contributions, most significantly in the galactic region and at the locations of strong radio point sources will unavoidably contaminate the estimated spherical CMB map. Masking out these regions is common practice but the gaps in the data need proper handling. In order to restore the stationarity of a partly incomplete CMB map and thus lower the impact of the gaps on non-local statistical tests, we developed an inpainting algorithm on the sphere to fill in the gaps, based on an iterative thresholding scheme in a sparse representation of the data. This algorithm relies on the variety of recently developed transforms on the sphere among which several multiscale transforms which we will review. We also contribute to enlarging the set of available transforms for polarized data on the sphere. We describe new multiscale decompositions namely the isotropic undecimated wavelet and curvelet transforms for polarized data on the sphere. The proposed transforms are invertible and so allow for applications in image restoration and denoising.

Satellites mapping the spatial variations of the gravitational or magnetic fields of the Earth or other planets ideally fly on polar orbits, uniformly covering the entire globe. Thus, potential fields on the sphere are usually expressed in spherical harmonics, basis functions with global support. For various reasons, however, inclined orbits are favorable. These leave a "polar gap": an antipodal pair of axisymmetric polar caps without any data coverage, typically smaller than 10° in diameter for terrestrial gravitational problems, but 20° or more in some planetary magnetic configurations. The estimation of spherical harmonic field coefficients from an incompletely sampled sphere is prone to error, since the spherical harmonics are not orthogonal over the partial domain of the cut sphere. Although approaches based on wavelets have gained in popularity in the last decade, we present a method for localized spherical analysis that is firmly rooted in spherical harmonics. We construct a basis of bandlimited spherical functions that have the majority of their energy concentrated in a subdomain of the unit sphere by solving Slepian's (1960) concentration problem in spherical geometry, and use them for the geodetic problem at hand. Most of this work has been published by us elsewhere. Here, we highlight the connection of the "spherical Slepian basis" to wavelets by showing their asymptotic self-similarity, and focus on the computational considerations of calculating concentrated basis functions on irregularly shaped domains.

Time-frequency analysis and wavelet analysis are generally used for providing signal expansions that are suitable for various further tasks such as signal analysis, de-noising, compression, source separation, ... However, time-frequency analysis and wavelet analysis also provide efficient ways for constructing signals' transformations. They are modelled as linear operators that can be designed directly in the transformed domain, i.e. the time-frequency plane, or the time-scale half plane. Among these linear operators, transformations that are diagonal in the time-frequency or time scale spaces, i.e. that may be expressed by multiplications in these domains, deserve particular attention, as they are extremely simple to implement, even though their properties are not necessarily easy to control. This work is a first attempt for exploring such approaches in the context of the analysis and the design of sound signals. We study more specifically the transformations that may be interpreted as linear time-varying (LTV) systems (often called time-varying filters). It is known that under certain assumptions, the latter may be conveniently represented by pointwise multiplication with a certain time frequency transfer function in the time-frequency domain. The purpose of this work is to examine such representations in practical situations, and investigate generalizations. The originality of this approach for sound synthesis lies in the design of practical operators that can be optimized to morph a given sound into another one, at a very high sound quality.

This article focuses on the study of the statistical properties of the cosmic microwave background (CMB) temperature fluctuations. This study helps to define a coherent framework for the origin of the Universe, its evolution and the structure formation. The current standard model is based in the Big-Bang theory, in the context of the Cosmic Inflation scenario, and predicts that the CMB temperature fluctuations can be understood as the realization of a statistical isotropic and Gaussian random field. To probe whether these statistical properties are satisfied or not is capital, since deviations from these hypotheses would indicate that non-standard models might play an important role in the dynamics of the Universe. But that is not all. There are alternative sources of anisotropy or non-Gaussianity that could contaminate the CMB signal as well. Hence, sophisticated techniques must be used to carry out such a probe. Among all the methodologies that one can face in the literature, those based on wavelets are providing one of the most interesting insights to the problem. Their ability to explore several scales keeping, at the same time, spatial information of the CMB, is a very useful property to discriminate among possible sources of anisotropy and/or non-Gaussianity.

Wavelet decomposition of signals using the classical Daubechies-Wavelets could also be considered as a decomposition using a filter bank with two channels, a low pass and a high pass channel, represented by the father and mother wavelet, respectively. By generalizing this two channel approach filter banks with N ≥ 2 channels can be constructed. They possess one scaling function or father wavelet representing the low pass filter and one, two or more mother wavelets representing band pass filters. The resulting band pass filters do not show a satisfactory selective behavior, in general. Hence, a modification of the generalized design seems appropriate. Based on Vaidyanathan's procedure we developed a method to modify the modulation matrix under the condition that the low pass is unchanged and the degree of the band pass filters is not increased. This can be achieved by introducing one or more additional elementary building blocks under certain orthogonality constraints with respect to their generating vectors. While the (polynomial) degree of the modulation matrix remains unchanged, its complexity increases due to its increased McMillan degree.

Wavelets based on Hilbert pairs have appealing properties when applied to image denoising and feature detection due to their directional sensitivity. In this paper we propose dual-tree tight frame wavelets and scaling functions {φh, ψh1, ψh2, ψh3} and {φg, ψg1, ψg2, ψg3} based on FIR filterbanks of four filters, and downsampling by 2. Such wavelets closely approximate shift invariance. Moreover, the resulting complex wavelets are smooth and lead to exactly symmetric envelope. The filters in this paper enjoy vanishing moments property.

This paper proposes a new approach to distributed video coding. Distributed video coding is a new paradigm in video coding, which is based on the concept of decoding with side information at the decoder. Such a coding scheme employs a low-complexity encoder, making it well suited for low-power devices such as mobile video cameras. The uniqueness of our work lies in the combined use of discrete wavelet transform (DWT) and the concept of sampling of signals with finite rate of innnovation (FRI). This enables the decoder to retrieve the motion parameters and reconstruct the video sequence from the low-resolution version of each transmitted frame. Unlike the currently existing practical coders, we do not employ traditional channel coding techniqe. For a simple video sequence with a fixed background, Our preliminary results show that the proposed coding scheme can achieve a better PSNR than JPEG2000-intraframe coding at low bit rates.

The quality of video sequences (e.g. old movies, webcam, TV broadcast) is often reduced by noise, usually assumed white and Gaussian, being superimposed on the sequence. When denoising image sequences, rather than a single image, the temporal dimension can be used for gaining in better denoising performance, as well as in the algorithms' speed. This paper extends single image denoising method reported in to sequences. This algorithm relies on sparse and redundant representations of small patches in the images. Three different extensions are offered, and all are tested and found to lead to substantial benefits both in denoising quality and algorithm complexity, compared to running the single image algorithm sequentially. After these modifications, the proposed algorithm displays state-of-the-art denoising performance, while not relying on motion estimation.

In this paper we propose a prediction method that is geared toward forming successful estimates of a signal based on a correlated anchor signal contaminated with complex interference. The interference model is based on real-life, and it involves intensity modulations, linear distortions, structured clutter, and white noise just to name a few. The proposed method first transforms signals to an over-complete domain where we assume sparse decompositions. In this sparse domain, we show that very simple predictors can be designed to perform efficient prediction. The parameters of these predictors are derived from causal information, enabling completely automated and blind operation. The utilized over-complete representation allows multiple predictions for each sample in signal domain, which are averaged and combined into a single prediction. Experimental results on images and video frames show that the proposed method can provide successful predictions under a variety of complex transitions, such as cross-fades, brightness changes, focus variations, and other complex distortions. The proposed prediction method is also implemented to operate inside a state-of-the-art video compression codec and results show significant improvements on scenes that are hard to encode using traditional prediction techniques.

We describe here two ways to improve on recent results in image restoration using Bayes least squares estimation with local Gaussian scale mixtures (BLS-GSM) in overcomplete oriented pyramids. First one consists of allowing for a spatial adaptation of the covariance matrix defining the GSM model at each pyramid subband. This can be implemented in practice by dividing the subbands into spatial blocks. The other, more powerful, method is to generalize the GSM model to include more than one covariance matrices for each subband. The advantage of the latter method is its flexibility, as it allows for mixing Gaussian densities with different covariance matrices at every spatial location in every subband. It also allows for non-local selective processing, taking advantage of the repetition in the scene of image features that are not necessarily spatially grouped. We also describe an empirical method to adapt denoising algorithms for doing image restoration, with the only constraint on the denoising method of being applicable to non-white noise sources. Here we present mature results of the spatially adaptive method applied to denoising and deblurring, plus some estimation techniques and encouraging preliminary results of the multi-GSM concept.

We propose a new orthonormal wavelet thresholding algorithm for denoising color images that are assumed to be corrupted by additive Gaussian white noise of known intercolor covariance matrix. The proposed wavelet denoiser consists of a linear expansion of thresholding (LET) functions, integrating both the interscale and intercolor dependencies. The linear parameters of the combination are then solved for by minimizing Stein's unbiased risk estimate (SURE), which is nothing but a robust unbiased estimate of the mean squared error (MSE) between the (unknown) noise-free data and the denoised one. Thanks to the quadratic form of this MSE estimate, the parameters optimization simply amounts to solve a linear system of equations. The experimentations we made over a wide range of noise levels and for a representative set of standard color images have shown that our algorithm yields even slightly better peak signal-to-noise ratios than most state-of-the-art wavelet thresholding procedures, even when the latters are executed in an undecimated wavelet representation.

Owing to the properties of joint time-frequency analysis that compress energy and approximately decorrelate temporal redundancies in sequential data, filterbank and wavelets are popular and convenient platforms for statistical signal modeling. Motivated by the prior knowledge and empirical studies, much of the emphasis in signal processing has been placed on the choice of the prior distribution for these transform coefficients. In this paradigm however, the issues pertaining to the loss of information due to measurement noise are difficult to reconcile because the effects of point-wise signal-dependent noise permeate across scale and through multiple coefficients. In this work, we show how a general class of signal-dependent noise can be characterized to an arbitrary precision in a Haar filterbank representation, and the corresponding maximum a posteriori estimate for the underlying signal is developed. Moreover, the structure of noise in the transform domain admits a variant of Stein's unbiased estimate of risk conducive to processing the corrupted signal in the transform domain. We discuss estimators involving Poisson process, a situation that arises often in real-world applications such as communication, signal processing, and imaging.

Both the Kingsbury dual-tree and the subsequent Selesnick double-density dual-tree complex wavelet transform approximate an analytic function. The classification of the phase dependency across scales is largely unexplored except by Romberg et al.. Here we characterize the sub-band dependency of the orientation of phase gradients by applying the Helmholtz principle to bivariate histograms to locate meaningful modes. A further characterization using the Earth Mover's Distance with the fundamental Rudin-Osher-Meyer Banach space decomposition into cartoon and texture elements is presented. Possible applications include image compression and invariant descriptor selection for image matching.

In this paper, we are interested in modeling groups of wavelet coefficients using a zero-mean, elliptically-contoured multivariate Laplace probability distribution function (pdf). Specifically, we are interested in the problem of estimating a d-point Laplace vector, s, in additive white Gaussian noise (AWGN), n, from an observation, y = s + n. In the scalar case (d = 1), the MAP and MMSE estimators are already known; and in the vector case (d > 1), the MAP estimator can be obtained by an iterative successive substitution algorithm. For the special case where the contour of the Laplace pdf is spherical, the MMSE estimators for the vector case (d > 1) have been derived in our previous work; we have shown that the MMSE estimator can be expressed in terms of the generalized incomplete Gamma function. For the general elliptically-contoured case, the MMSE estimator can not be expressed as such. In this paper, we therefore investigate approximations to the MMSE estimator of a Laplace vector in AWGN.

We derive fast algorithms for doing signal reconstruction without phase. This type of problem is important in signal processing, especially speech recognition technology, and has relevance for state tomography in quantum theory. We show that a generic frame gives reconstruction from the absolute value of the frame coefficients in polynomial time. An improved efficiency of reconstruction is obtained with a family of sparse frames or frames associated with complex projective 2-designs.

The new notion of fusion frames will be presented in this article. Fusion frames provide an extensive framework not only to model sensor networks, but also to serve as a means to improve robustness or develop efficient and feasible reconstruction algorithms. Fusion frames can be regarded as sets of redundant subspaces each of which contains a spanning set of local frame vectors, where the subspaces have to satisfy special overlapping properties. Main aspects of the theory of fusion frames will be presented with a particular focus on the design of sensor networks. New results on the construction of Parseval fusion frames will also be discussed.

Multiscale moment-based transformations, such as the multiscale Harris corner point detector, have been used in image processing applications for several years. Typically, such transforms are used to identify objects-of-interest in a given image, which, in turn, facilitate target tracking and registration. Though these transforms are usually applied to digitally sampled images, many of their properties were previously only known to hold for images over continuous domains. We prove that many of these properties indeed generalize to images over discrete domains. In particular, after introducing a mathematically well-behaved method for rotating an image over the two-dimensional integer lattice, we show that this rotation commutes with the moment-based transform in the expected manner.

The general objective in this paper is the loss-insensitive transmission of vectors by linear, redundant encoding with the help of frames. Our specific goal is to find frames which minimize the mean-square reconstruction error for cyclic burst erasures with known burst-length statistics. Finding the best frames among the cyclic ones is reduced to a discrete optimization problem. We provide an upper and lower bound for the mean-square error and discuss a family of frames for which both bounds coincide while the upper bound is minimized.

The main goal of this paper is to introduce formally the concept of texture segmentation/identification in three dimensional images. A major problem in texture texture segmentation/identification is the lack of robustness to both translations and rotations. This problem is more difficult to overcome in 3D-images, such as those generated by modalities such as x-ray CT and MRI. To facilitate 3D-texture segmentation/identification which is robust to 3D rigid motions we formally introduce the concept of steerable feature maps and of appropriate metrics in the feature space. We also introduce a new multiscale representation giving rise to a steerable feature map used in our exploratory project in cardiovascular imaging and we propose a 3D-texture segmentation algorithm utilizing this steerable feature map.

This article proposes a new method for image separation into a linear combination of morphological components. This method is applied to decompose an image into meaningful cartoon and textural layers and is used to solve more general inverse problems such as image inpainting. For each of these components, a dictionary is learned from a set of exemplar images. Each layer is characterized by a sparse expansion in the corresponding dictionary. The separation inverse problem is formalized within a variational framework as the optimization of an energy functional. The morphological component analysis algorithm allows to solve iteratively this optimization problem under sparsity-promoting penalties. Using adapted dictionaries learned from data allows to circumvent some difficulties faced by fixed dictionaries. Numerical results demonstrate that this adaptivity is indeed crucial to capture complex texture patterns.

Over the last few years, the development of multi-channel sensors motivated interest in methods for the coherent processing of multivariate data. From blind source separation (BSS) to multi/hyper-spectral data restoration, an extensive work has already been dedicated to multivariate data processing. Previous work has emphasized on the fundamental role played by sparsity and morphological diversity to enhance multichannel signal processing. Morphological diversity has been first introduced in the mono-channel case to deal with contour/texture extraction. The morphological diversity concept states that the data are the linear combination of several so-called morphological components which are sparse in different incoherent representations. In that setting, piecewise smooth features (contours) and oscillating components (textures) are separated based on their morphological differences assuming that contours (respectively textures) are sparse in the Curvelet representation (respectively Local Discrete Cosine representation). In the present paper, we define a multichannel-based framework for sparse multivariate data representation. We introduce an extension of morphological diversity to the multichannel case which boils down to assuming that each multichannel morphological component is diversely sparse spectrally and/or spatially. We propose the Generalized Morphological Component Analysis algorithm (GMCA) which aims at recovering the so-called multichannel morphological components. Hereafter, we apply the GMCA framework to two distinct multivariate inverse problems : blind source separation (BSS) and multichannel data restoration. In the two aforementioned applications, we show that GMCA provides new and essential insights into the use of morphological diversity and sparsity for multivariate data processing. Further details and numerical results in multivariate image and signal processing will be given illustrating the good performance of GMCA in those distinct applications.

We present a new method for discrimination of data classes or data sets in a high-dimensional space. Our approach combines two important relatively new concepts in high-dimensional data analysis, i.e., Diffusion Maps and Earth Mover's Distance, in a novel manner so that it is more tolerant to noise and honors the characteristic geometry of the data. We also illustrate that this method can be used for a variety of applications in high dimensional data analysis and pattern classification, such as quantifying shape deformations and discrimination of acoustic waveforms.

Seismic data and their complexity still challenge signal processing algorithms in several applications. The advent of wavelet transforms has allowed improvements in tackling denoising problems. We propose here coherent noise filtering in seismic data with the dual-tree M-band wavelet transform. They offer the possibility to decompose data locally with improved multiscale directions and frequency bands. Denoising is performed in a deterministic fashion in the directional subbands, depending of the coherent noise properties. Preliminary results show that they consistently better preserve seismic signal of interest embedded in highly energetic directional noises than discrete critically sampled and redundant separable wavelet transforms.

This paper introduces p-thresholding, an algorithm to compute simultaneous sparse approximations of multichannel signals over redundant dictionaries. We work out both worst case and average case recovery analyses of this algorithm and show that the latter results in much weaker conditions on the dictionary. Numerical simulations confirm our theoretical findings and show that p-thresholding is an interesting low complexity alternative to simultaneous greedy or convex relaxation algorithms for processing sparse multichannel signals with balanced coefficients.

Inverse problems play an important role in engineering. A problem that often occurs in electromagnetics (e.g. EEG) is the estimation of the locations and strengths of point sources from boundary data. We propose a new technique, for which we coin the term "analytic sensing". First, generalized measures are obtained by applying Green's theorem to selected functions that are analytic in a given domain and at the same time localized to "sense" the sources. Second, we use the finite-rate-of-innovation framework to determine the locations of the sources. Hence, we construct a polynomial whose roots are the sources' locations. Finally, the strengths of the sources are found by solving a linear system of equations. Preliminary results, using synthetic data, demonstrate the feasibility of the proposed method.

We propose two methods for sparse approximation of images under l₂ error metric. First one performs an approximation error minimization given a l_p-norm of the representation through alternated orthogonal projections onto two sets. We study the cases p = 0 (sub-optimal) and p = 1 (optimal), and find that the l₀-AP method is neatly superior, for typical images and overcomplete oriented pyramids. Given that l₁-AP is optimal, this shows that it is not equivalent in practical image processing conditions to minimize one or the other norm, contrarily to what is often assumed. The second method is more powerful, and it performs gradient descent onto decreasingly smoothed versions of the sparse approximation cost function, yielding a method previously proposed as a heuristic. We adapt these techniques for being applied to image restoration, with very positive results.

The theory of compressive sensing enables accurate and robust signal reconstruction from a number of measurements dictated by the signal's structure rather than its Fourier bandwidth. A key element of the theory is the role played by randomization. In particular, signals that are compressible in the time or space domain can be recovered from just a few randomly chosen Fourier coefficients. However, in some scenarios we can only observe the magnitude of the Fourier coefficients and not their phase. In this paper, we study the magnitude-only compressive sensing problem and in parallel with the existing theory derive sufficient conditions for accurate recovery. We also propose a new iterative recovery algorithm and study its performance. In the process, we develop a new algorithm for the phase retrieval problem that exploits a signal's compressibility rather than its support to recover it from Fourier transform magnitude measurements.

Recent results in compressed sensing or compressive sampling suggest that a relatively small set of measurements taken as the inner product with universal random measurement vectors can well represent a source that is sparse in some fixed basis. By adapting a deterministic, non-universal and structured sensing device, this paper presents results on using the annihilating filter to decode the information taken in this new compressed sensing environment. The information is the minimum amount of nonadaptive knowledge that makes it possible to go back to the original object. We will show that for a k-sparse signal of dimension n, the proposed decoder needs 2k measurements and its complexity is of O(k²) whereas for the decoding based on the l₁ minimization, the number of measurements needs to be of O(k log(n)) and the complexity is of O(n³). In the case of noisy measurements, we first denoise the signal using an iterative algorithm that finds the closest rank k and Toeplitz matrix to the measurements matrix (in Frobenius norm) before applying the annihilating filter method. Furthermore, for a k-sparse vector with known equal coefficients, we propose an algebraic decoder which needs only k measurements for the signal reconstruction. Finally, we provide simulation results that demonstrate the performance of our algorithm.

We study a sampling problem where sampled signals come from a known union of shift-invariant subspaces and the sampling operator is a linear projection of the sampled signals into a fixed shift-invariant subspace. In practice, the sampling operator can be easily implemented by a multichannel uniform sampling procedure. We present necessary and sufficient conditions for invertible and stable sampling operators in this framework, and provide the corresponding minimum sampling rate. As an application of the proposed general sampling framework, we study the specific problem of spectrum-blind sampling of multiband signals. We extend the previous results of Bresler et al. by showing that a large class of sampling kernels can be used in this sampling problem, all of which lead to stable sampling at the minimum sampling rate.

Denoising is always a challenge problem in natural image and geophysical data processing. From the point of view of data assimilation, in this paper we consider the denoising of texture images in a scale space using a geometric wavelet based nonlinear reaction-diffusion equation, in which a curvelet shrinkage is used as a regularization of the diffusivity to preserve important features in the diffusion smoothing and a wave atom shrinkage is used as a pseudo-observation in the reaction for enhancement of interesting oriented textures. We named the general framework as image assimilation. The goal of the image assimilation is to link together these rich information such as sparse constraint in multiscale geometric space in order to retrieve the state of images. As a byproduct, we proposed a 2D wavelet-inspired numerical scheme for solving of the nonlinear diffusion. Experimental results show the performance of the proposed model for texture-preserving denoising and enhancement.

A common frustration in signal processing and, more generally, information recovery is the presence of irregularities in the data. At best, the standard software or methods will no longer be directly applicable when data are missing, incomplete or irregularly spaced (e.g., as with wavelets). Self-consistency is a very general and powerful statistical principle for dealing with such problems. Conceptually it is extremely appealing, for it is essentially a mathematical formalization of iterating common-sense "trial-and-error" methods until no more improvement is possible. Mathematically it is elegant, with one fixed-point equation to solve and a general projection theorem to establish optimality. Practically it is straightforward to program because it directly uses the regular/complete-data method for iteration. Its major disadvantage is that it can be computationally intensive. However, increasingly efficient (approximate) implementations are being discovered, such as for wavelet de-noising with hard and soft thresholding. This brief overview summarizes the author's keynote presentation on those points, based on joint work with Thomas Lee on wavelet applications and with Zhan Li on the theoretical properties of the self-consistent estimators.

This paper describes a new class of discrete heap transforms which are unitary energy-preserving transforms and induced by input signals. These transforms have a simple form of composition and fast algorithms for any size of processed signals. We consider the heap transforms, defined by two-dimensional elementary rotations, as satisfying the given decision equations. The main feature of each heap transform is the corresponding system of basis functions, which represent themselves a family of interactive waves which are moving in the field generated by the input signal. Properties and examples of heap transforms, which we also call discrete signal-induced heap transforms, are described in detail.

This paper introduces a new class of discrete unitary transformations, the so-called discrete Haar-type heap transformations (DHHT) which are induced by input signals and use the path similar to the traditional Haar transformation. These transformations are fast and performed by simple rotations, can be composed for any order, and their complete systems of basis functions represent themselves variable waves that are generated by signals. The 2^r-point discrete Haar transform is the particular case of the proposed transformations, when the generator is the constant sequence {1, 1, 1, ..., 1}. These transformations can be used in many applications and improve the results of the Haar transformation. As an example, the approximation of signals in the simple compression process, when truncating the coefficients of the discrete Haar-type heap transform is illustrated.

The polyharmonic local cosine transform (PHLCT), presented by Yamatani and Saito in 2006, is a new tool for local image analysis and synthesis. It can compress and decompress images with better visual fidelity, less blocking artifacts, and better PSNR than those processed by the JPEG-DCT algorithm. Now, we generalize PHLCT to the high-dimensional case and apply it to compress the high-dimensional data. For this purpose, we give the solution of the high-dimensional Poisson equation with the Neumann boundary condition. In order to reduce the number of coefficients of PHLCT, we use not only d-dimensional PHLCT decomposition, but also d-1, d-2, . . . , 1 dimensional PHLCT decompositions. We find that our algorithm can more efficiently compress the high-dimensional data than the block DCT algorithm. We will demonstrate our claim using both synthetic and real 3D datasets.

In this work we use α-bi-Lipschitz transformation of signals both from empirical and theoretical sources to obtain new tests for the accomplishment of the multifractal formalisms associated with many methods (Wavelet Leaders, Wavelet Transform Modulus Maxima, Multifractal Detrended Fluctuation Analysis, Box Counting, and other) and we give improvements of the present algorithms that result numerically more trustworthy. Moreover the multifractal spectrum does not change in the theory, but as the numeric implementation of the computations may differ for discrete series so we can analyze its variation to study the stability of the proposed algorithms to compute it. In addition some single coefficients that have been proposed to quantify the whole irregularity of the signal are preserved by enough high α-bi-Lipschitz transformations. We exhibit the performance of the tests and the improvements of this methods not only in signals generated from deterministic (or sometimes random) numerical processes performed with the computer but also against series from empirical sources in which the multifractal spectrum and the irregularity coefficient were proven of utility both from the analysis and the segmentation of the signal in significant parts as series of Longwave outgoing radiation of tropical regions (and the consequent forecasting applications of precipitations) and certain series of EEG (from patients with crisis of brain absences for instance) and the ability to distinguish (and perhaps to predict) the beginning of the consecutive stages.

Often, in signal processing applications, it is useful and convenient to have on the hand flexible tools providing time-frequency information from the wavelet coefficients. This means the well known wavelet packets techniques. The main purpose of this work is to explore and discuss several alternatives in this field, related to orthogonal spline wavelets.

It is believed by many that three-dimensional (3D) television will be the next logical development toward a more natural and vivid home entertaiment experience. While classical 3D approach requires the transmission of two video streams, one for each view, 3D TV systems based on depth image rendering (DIBR) require a single stream of monoscopic images and a second stream of associated images usually termed depth images or depth maps, that contain per-pixel depth information. Depth map is a two-dimensional function that contains information about distance from camera to a certain point of the object as a function of the image coordinates. By using this depth information and the original image it is possible to reconstruct a virtual image of a nearby viewpoint by projecting the pixels of available image to their locations in 3D space and finding their position in the desired view plane. One of the most significant advantages of the DIBR is that depth maps can be coded more efficiently than two streams corresponding to left and right view of the scene, thereby reducing the bandwidth required for transmission, which makes it possible to reuse existing transmission channels for the transmission of 3D TV. This technique can also be applied for other 3D technologies such as multimedia systems. In this paper we propose an advanced wavelet domain scheme for the reconstruction of stereoscopic images, which solves some of the shortcommings of the existing methods discussed above. We perform the wavelet transform of both the luminance and depth images in order to obtain significant geometric features, which enable more sensible reconstruction of the virtual view. Motion estimation employed in our approach uses Markov random field smoothness prior for regularization of the estimated motion field. The evaluation of the proposed reconstruction method is done on two video sequences which are typically used for comparison of stereo reconstruction algorithms. The results demonstrate advantages of the proposed approach with respect to the state-of-the-art methods, in terms of both objective and subjective performance measures.

We introduce a modified Matching Pursuit algorithm for estimating frequency and frequency slope of FM-modulated music signals. The use of Matching Pursuit with constant frequency atoms provides coarse estimates which could be improved with chirped atoms, more suited in principle to this kind of signals. Application of the reassignment method is suggested by its good localization properties for chirps. We start considering a family of atoms generated by modulation and scaling of a prolate spheroidal wave function. These functions are concentrated in frequency on intervals of a semitone centered at the frequencies of the well-tempered scale. At each stage of the pursuit, we search the atom most correlated with the signal. We then consider the spectral peaks at each frame of the spectrogram and calculate a modified frequency and frequency slope using the derivatives of the reassignment operators; this is then used to estimate the parameters of a cubic interpolation polynomial that models local pitch fluctuations. We apply the method both to synthetic and music signals.

In this paper we investigate the use of the Affine Scaling Transformation (AST) family of algorithms in solving the sparse signal recovery problem of harmonic retrieval for the DFT-grid frequencies case. We present the problem in the more general Compressive Sampling/Sensing (CS) framework where any set of incomplete, linearly independent measurements can be used to recover or approximate a sparse signal. The compressive sampling problem has been approached mostly as a problem of l₁ norm minimization, which can be solved via an associated linear programming problem. More recently, attention has shifted to the random linear projection measurements case. For the harmonic retrieval problem, we focus on linear measurements in the form of: consecutively located time samples, randomly located time samples, and (Gaussian) random linear projections. We use the AST family of algorithms which is applicable to the more general problem of minimization of the l_p p-norm-like diversity measure that includes the numerosity (p=0), and the l₁ norm (p=1). Of particular interest in this paper is to experimentally find a relationship between the minimum number M of measurements needed for perfect recovery and the number of components K of the sparse signal, which is N samples long. Of further interest is the number of AST iterations required to converge to its solution for various values of the parameter p. In addition, we quantify the reconstruction error to assess the closeness of the AST solution to the original signal. Results show that the AST for p=1 requires 3-5 times more iterations to converge to its solution than AST for p=0. The minimum number of data measurements needed for perfect recovery is approximately the same on the average for all values of p, however, there is an increasing spread as p is reduced from p=1 to p=0. Finally, we briefly contrast the AST results with those obtained using another l₁ minimization algorithm solver.

The Short Term Fourier Transform (STFT) is a classical linear time-frequency (T-F) representation. Despites its relative simplicity, it has become a standard tool for the analysis of non-stationary signals. Since it provides a redundant representation, it raises some issues such as (i) "optimal" window choice for analysis, (ii) existence and determination of an inverse transformation, (iii) performance of analysis-modification-synthesis, or reconstruction of selected components of the time-frequency plane and (iv) redundancy controllability for low-cost applications, e.g. real-time computations. We address some of these issues, as well as the less often mentioned problem of transform symmetry in the inverse, through oversampled FBs and their optimized inverse(s) in a slightly more general setting than the discrete windowed Fourier transform.

The paper presents new tight frame dyadic limit functions with dense time-frequency grid. The filterbank consists of one lowpass filter and three bandpass and/or highpass filters. We add the requirement that the bandpass and highpass filters be all of equal norms. All the filters in the paper are FIR and enjoy vanishing moments.

Although wavelet-based image denoising is a powerful tool for image processing applications, relatively few publications have addressed so far wavelet-based video denoising. The main reason is that the standard 3-D data transforms do not provide useful representations with good energy compaction property, for most video data. For example, the multi-dimensional standard separable discrete wavelet transform (M-D DWT) mixes orientations and motions in its subbands, and produces the checkerboard artifacts. So, instead of M-D DWT, usually oriented transforms suchas multi-dimensional complex wavelet transform (M-D DCWT) are proposed for video processing. In this paper we use a Laplace distribution with local variance to model the statistical properties of noise-free wavelet coefficients. This distribution is able to simultaneously model the heavy-tailed and intrascale dependency properties of wavelets. Using this model, simple shrinkage functions are obtained employing maximum a posteriori (MAP) and minimum mean squared error (MMSE) estimators. These shrinkage functions are proposed for video denoising in DCWT domain. The simulation results shows that this simple denoising method has impressive performance visually and quantitatively.

In this paper, we design a bivariate maximum a posteriori (MAP) estimator that supposes the prior of wavelet coefficients as a mixture of bivariate Cauchy distributions. This model not only is a mixture but is also bivariate. Since mixture models are able to capture the heavy-tailed property of wavelets and bivaraite distributions can model the intrascale dependences of wavelet coefficients, this bivariate mixture probability density function (pdf) can better capture statistical properties of wavelet coefficients. The simulation results show that our proposed technique achieves better performance than other methods employing non mixture pdfs such as bivariate Cauchy pdf and circular symmetric Laplacian pdf visually and in terms of peak signal-to-noise ratio (PSNR). We also compare our algorithm with several recently published denoising methods and see that it is among the best reported in the literature.

This paper presents a refining estimation to control the process of adaptive mesh refinement (AMR) based on the curvelet transform. The curvelet is a recently developed geometric multiscale system that could provide optimal approximation to curve-singularity functions and sparse representation of wavefront phenomena. Utilizing these advantages, we attempt to introduce the curvelet transform into AMR as a criterion estimate of refinement for adaptive solving of the wave equation. Numerical simulations show that the proposed method could optimally capture interesting areas where refinement is needed, so that a high accuracy result is obtained.