Multi-modal microscopy, such as combined bright-fiel and multi-color fluorescenc imaging, allows capturing a sample's anatomical structure, cell dynamics, and molecular activity in distinct imaging channels. However, only a limited number of channels can be acquired simultaneously and acquiring each channel sequentially at every time-point drastically reduces the achievable frame rate. Multi-modal imaging of rapidly moving objects (such as the beating embryonic heart), which requires high frame-rates, has therefore remained a challenge. We have developed a method to temporally register multimodal, high-speed image sequences of the beating heart that were sequentially acquired. Here we describe how maximizing the mutual information of time-shifted wavelet coefficien sequences leads to an implementation that is both accurate and fast. Specificall , we validate our technique on synthetically generated image sequences and show its effectiveness on experimental bright-fiel and fluorescenc image sequences of the beating embryonic zebrafis heart. This method opens the prospect of cardiac imaging in multiple channels at high speed without the need for multiple physical detectors.

We propose an active mask segmentation framework that combines the advantages of statistical modeling, smoothing, speed and flexibility offered by the traditional methods of region-growing, multiscale, multiresolution and active contours respectively. At the crux of this framework is a paradigm shift from evolving contours in the continuous domain to evolving multiple masks in the discrete domain. Thus, the active mask framework is particularly suited to segment digital images. We demonstrate the use of the framework in practice through the segmentation of punctate patterns in fluorescence microscope images. Experiments reveal that statistical modeling helps the multiple masks converge from a random initial configuration to a meaningful one. This obviates the need for an involved initialization procedure germane to most of the traditional methods used to segment fluorescence microscope images. While we provide the mathematical details of the functions used to segment fluorescence microscope images, this is only an instantiation of the active mask framework. We suggest some other instantiations of the framework to segment different types of images.

In this paper, we explore the use of anatomical information as a guide in the image formation process of fluorescence molecular tomography (FMT). Namely, anatomical knowledge obtained from high resolution computed tomography (micro-CT) is used to construct a model for the diffusion of light and to constrain the reconstruction to areas candidate to contain fluorescent volumes. Moreover, a sparse regularization term is added to the state-of-the-art least square solution to contribute to the sparsity of the localization. We present results showing the increase in accuracy of the combined system over conventional FMT, for a simulated experiment of lung cancer detection in mice.

Noise level and photobleaching are cross-dependent problems in biological fluorescence microscopy. Indeed, observation of fluorescent molecules is challenged by photobleaching, a phenomenon whereby the fluorophores are degraded by the excitation light. One way to control this process is by reducing the intensity of the light or the time exposure, but it comes at the price of decreasing the signal-to-noise ratio (SNR). Although a host of denoising methods have been developed to increase the SNR, most are post-processing techniques and require full data acquisition. In this paper we propose a novel technique, based on Compressed Sensing (CS) that simultaneously enables reduction of exposure time or excitation light level and improvement of image SNR. Our CS-based method can simultaneously acquire and denoise data, based on statistical properties of the CS optimality, noise reconstruction characteristics and signal modeling applied to microscopy images with low SNR. The proposed approach is an experimental optimization combining sequential CS reconstructions in a multiscale framework to perform image denoising. Simulated and practical experiments on fluorescence image data demonstrate that thanks to CS denoising we obtain images with similar or increased SNR while still being able to reduce exposure times. Such results open the gate to new mathematical imaging protocols, offering the opportunity to reduce photobleaching and help biological applications based on fluorescence microscopy.

We consider the problem of estimating the channel response between multiple source receiver pairs all sensing the same medium. A different pulse is sent from each source, and the response is measured at each receiver. If each source sends its pulse while the others are silent, estimating the channel is a classical deconvolution problem. If the sources transmit simultaneously, estimating the channel requires "inverting" an underdetermined system of equations. In this paper, we show how this second scenario relates to the theory of compressed sensing. In particular, if the pulses are long and random, then the channel matrix will be a restricted isometry, and we can apply the tools of compressed sensing to simultaneously recover the channels from each source to a single receiver.

This paper considers a simple on-off random multiple access channel, where n users communicate simultaneously to a single receiver over m degrees of freedom. Each user transmits with probability λ, where typically λn<m(symbol)n, and the receiver must detect which users transmitted. We show that when the codebook has i.i.d. Gaussian entries, detecting which users transmitted is mathematically equivalent to a certain sparsity detection problem considered in compressed sensing. Using recent sparsity results, we derive upper and lower bounds on the capacities of these channels. We show that common sparsity detection algorithms, such as lasso and orthogonal matching pursuit (OMP), can be used as tractable multiuser detection schemes and have significantly better performance than single-user detection. These methods do achieve some near-far resistance but-at high signal-to-noise ratios (SNRs) - may achieve capacities far below optimal maximum likelihood detection. We then present a new algorithm, called sequential OMP, that illustrates that iterative detection combined with power ordering or power shaping can significantly improve the high SNR performance. Sequential OMP is analogous to successive interference cancellation in the classic multiple access channel. Our results thereby provide insight into the roles of power control and multiuser detection on random-access signaling.

Suppose the signal x ∈ Rⁿ is realized by driving a d-sparse signal z ∈ Rⁿ through an arbitrary unknown stable discrete-linear time invariant system H, namely, x(t) = (h * z)(t), where h(·) is the impulse response of the operator H. Is x(·) compressible in the conventional sense of compressed sensing? Namely, can x(t) be reconstructed from sparse set of measurements. For the case when the unknown system H is auto-regressive (i.e. all pole) of a known order it turns out that x can indeed be reconstructed from O(k log(n)) measurements. The main idea is to pass x through a linear time invariant system G and collect O(k log(n)) sequential measurements. The filter G is chosen suitably, namely, its associated Toeplitz matrix satisfies the RIP property. We develop a novel LP optimization algorithm and show that both the unknown filter H and the sparse input z can be reliably estimated. These types of processes arise naturally in Reflection Seismology.

We present steerlets, a new class of wavelets which allow us to define wavelet transforms that are covariant with respect to rigid motions in d dimensions. The construction of steerlets is derived from an Isotropic Multiresolution Analysis, a variant of a Multiresolution Analysis whose core subspace is closed under translations by integers and under all rotations. Steerlets admit a wide variety of design characteristics ranging from isotropy, that is the full insensitivity to orientations, to directional and orientational selectivity for local oscillations and singularities. The associated 2D or 3D-steerlet transforms are fast MRA-type of transforms suitable for processing of discrete data. The subband decompositions obtained with 2D or 3D-steerlets behave covariantly under the action of the respective rotation group on an image, so that each rotated steerlet is the linear combination of other steerlets in the same subband.

Shearlab is a Matlab toolbox for digital shearlet transformation of two-D (image) data we developed following a rational design process. The Pseudo-Polar FFT fits very naturally with the continuum theory of the Shearlet transform and allows us to translate Shearlet ideas naturally into a digital framework. However, there are still windows and weights which must be chosen. We developed more than a dozen performance measures quantifying precision of the reconstruction, tightness of the frame, directional and spatial localization and other properties. Such quantitative performance metrics allow us to: (a) tune parameters and objectively improve our implementation; and (b) compare different directional transform implementations. We present and interpret the most important performance measures for our current implementation.

In this paper, we present a new approach for inverse halftoning of error diffused halftones using a shearlet representation. We formulate inverse halftoning as a deconvolution problem using Kite et al.'s linear approximation model for error diffusion halftoning. Our method is based on a new M-channel implementation of the shearlet transform. By formulating the problem as a linear inverse problem and taking advantage of unique properties of an implementation of the shearlet transform, we project the halftoned image onto a shearlet representation. We then adaptively estimate a gray-scaled image from these shearlet-toned or shear-tone basis elements in a multi-scale and anisotropic fashion. Experiments show that, the performance of our method improves upon many of the state-of-the-art inverse halftoning routines, including a wavelet-based method and a method that shares some similarities to a shearlet-type decomposition known as the local polynomial approximation (LPA) technique.

The variational approach to signal restoration calls for the minimization of a cost function that is the sum of a data fidelity term and a regularization term, the latter term constituting a 'prior'. A synthesis prior represents the sought signal as a weighted sum of 'atoms'. On the other hand, an analysis prior models the coefficients obtained by applying the forward transform to the signal. For orthonormal transforms, the synthesis prior and analysis prior are equivalent; however, for overcomplete transforms the two formulations are different. We compare analysis and synthesis ℓ₁-norm regularization with overcomplete transforms for denoising and deconvolution.

One of the main challenges of high level analysis of human behavior is the high dimension of the feature space. To overcome the curse of dimensionality, we propose in this paper, a space curve representation of the high dimensional behavior features. The features of interest here, are restricted to sequences of shapes of the human body such as those extracted from a video sequence. This evolution is a one dimensional sub-manifold in shape space. The central idea of the proposed representation takes root in the Whitney embedding theorem which guarantees an embedding of a one dimensional manifold in as a space curve. The resulting of such dimension reduction, is a simplification of comparing two behaviors to that of comparing two curves in R³. This comparison is additionally theoretically and numerically easier to implement for statistical analysis. By exploiting sampling theory, we are moreover able to achieve a computationally efficient embedding that is invertible. Specifically, we first construct a global coordinates expression for the one dimension manifold and sampled along a generating curve.As experiment result, we provide substantiating modeling examples and illustrations of behavior classification.

It is a well known fact that the time-frequency domain is very well adapted for representing audio signals. The main two features of time-frequency representations of many classes of audio signals are sparsity (signals are generally well approximated using a small number of coefficients) and persistence (significant coefficients are not isolated, and tend to form clusters). This contribution presents signal approximation algorithms that exploit these properties, in the framework of hierarchical probabilistic models. Given a time-frequency frame (i.e. a Gabor frame, or a union of several Gabor frames or time-frequency bases), coefficients are first gathered into groups. A group of coefficients is then modeled as a random vector, whose distribution is governed by a hidden state associated with the group. Algorithms for parameter inference and hidden state estimation from analysis coefficients are described. The role of the chosen dictionary, and more particularly its structure, is also investigated. The proposed approach bears some resemblance with variational approaches previously proposed by the authors (in particular the variational approach exploiting mixed norms based regularization terms). In the framework of audio signal applications, the time-frequency frame under consideration is a union of two MDCT bases or two Gabor frames, in order to generate estimates for tonal and transient layers. Groups corresponding to tonal (resp. transient) coefficients are constant frequency (resp. constant time) time-frequency coefficients of a frequency-selective (resp. time-selective) MDCT basis or Gabor frame.

While many geological and geophysical processes such as the melting of icecaps, the magnetic expression of bodies emplaced in the Earth's crust, or the surface displacement remaining after large earthquakes are spatially localized, many of these naturally admit spectral representations, or they may need to be extracted from data collected globally, e.g. by satellites that circumnavigate the Earth. Wavelets are often used to study such nonstationary processes. On the sphere, however, many of the known constructions are somewhat limited. And in particular, the notion of 'dilation' is hard to reconcile with the concept of a geological region with fixed boundaries being responsible for generating the signals to be analyzed. Here, we build on our previous work on localized spherical analysis using an approach that is firmly rooted in spherical harmonics. We construct, by quadratic optimization, a set of bandlimited functions that have the majority of their energy concentrated in an arbitrary subdomain of the unit sphere. The 'spherical Slepian basis' that results provides a convenient way for the analysis and representation of geophysical signals, as we show by example. We highlight the connections to sparsity by showing that many geophysical processes are sparse in the Slepian basis.

Recent advances in signal processing have focused on the use of sparse representations in various applications. A new field of interest based on sparsity has recently emerged: compressed sensing. This theory is a new sampling framework that provides an alternative to the well-known Shannon sampling theory. In this paper we investigate how compressed sensing (CS) can provide new insights into astronomical data compression. In a previous study1 we gave new insights into the use of Compressed Sensing (CS) in the scope of astronomical data analysis. More specifically, we showed how CS is flexible enough to account for particular observational strategies such as raster scans. This kind of CS data fusion concept led to an elegant and effective way to solve the problem ESA is faced with, for the transmission to the earth of the data collected by PACS, one of the instruments onboard the Herschel spacecraft which will launched in late 2008/early 2009. In this paper, we extend this work by showing how CS can be effectively used to jointly decode multiple observations at the level of map making. This allows us to directly estimate large areas of the sky from one or several raster scans. Beyond the particular but important Herschel example, we strongly believe that CS can be applied to a wider range of applications such as in earth science and remote sensing where dealing with multiple redundant observations is common place. Simple but illustrative examples are given that show the effectiveness of CS when decoding is made from multiple redundant observations.

We consider the problem of reconstruction of astrophysical signals probed by radio interferometers with baselines bearing a non-negligible component in the pointing direction. The visibilities measured essentially identify with a noisy and incomplete Fourier coverage of the product of the planar signals with a linear chirp modulation. We analyze the related spread spectrum phenomenon and suggest its universality relative to the sparsity dictionary, in terms of the achievable quality of reconstruction through the Basis Pursuit problem. The present manuscript represents a summary of recent work.

This paper surveys recent results on frame sequences. The first group of results characterizes the relationships that hold among various types of dual frame sequences. The second group of results characterizes the relationships that hold among the major Paley-Wiener perturbation theorems for frame sequences, and some of the properties that remain invariant under such perturbations.

We propose a design procedure for the real, equal-norm, lapped tight frame transforms (LTFTs). These transforms have been recently proposed as both a redundant counterpart to lapped orthogonal transforms and an infinite-dimensional counterpart to harmonic tight frames. In addition, LTFTs can be efficiently implemented with filter banks. The procedure consists of two steps. First, we construct new lapped orthogonal transforms designed from submatrices of the DFT matrix. Then we specify the seeding procedure that yields real equal-norm LTFTs. Among them we identify the subclass of maximally robust to erasures LTFTs.

Redundant systems such as frames are often used to represent a signal for error correction, denoising and general robustness. In the digital domain quantization needs to be performed. Given the redundancy, the distribution of quantization errors can be rather complex. In this paper we study quantization error for a signal X in R^d represented by a frame using a lattice quantizer. We characterize the asymptotic distribution of the quantization error as the cell size of the lattice goes to zero. We apply these results to get the necessary and sufficient conditions for the White Noise Hypothesis to hold asymptotically in the case of the pulse-code modulation scheme. This is an abbreviated version of a paper that will appear elsewhere in a regular refereed journal.

One key property of frames is their resilience against erasures due to the possibility of generating stable, yet over-complete expansions. Blind reconstruction is one common methodology to reconstruct a signal when frame coefficients have been erased. In this paper we introduce several novel low complexity replacement schemes which can be applied to the set of faulty frame coefficients before blind reconstruction is performed, thus serving as a preconditioning of the received set of frame coefficients. One main idea is that frame coefficients associated with frame vectors close to the one erased should have approximately the same value as the lost one. It is shown that injecting such low complexity replacement schemes into blind reconstruction significantly reduce the worst-case reconstruction error. We then apply our results to the circle frames. If we allow linear combinations of different neighboring coefficients for the reconstruction of missing coefficients, we can even obtain perfect reconstruction for the circle frames under certain weak conditions on the set of erasures.

In this paper we analyze the use of frames for the transmission and error-correction of analog signals via a memoryless erasure-channel. We measure performance in terms of the mean-square error remaining after error correction and reconstruction. Our results continue earlier works on frames as codes which were mostly concerned with the smallest number of erased coefficients. To extend these works we borrow some ideas from binary coding theory and realize them with a novel class of frames, which carry a particular fusion frame architecture. We show that a family of frames from this class achieves a mean-square reconstruction error remaining after corrections which decays faster than any inverse power in the number of frame coefficients.

We lay a philosophical framework for the design of overcomplete multidimensional signal decompositions based on the union of two or more orthonormal bases. By combining orthonormal bases in this way, tight (energy preserving) frames are automatically produced. The advantage of an overcomplete (tight) frame over a single orthonormal decomposition is that a signal is likely to have a more sparse representation among the overcomplete set than by using any single orthonormal basis. We discuss the question of the relationship between pairs of bases and the various criteria that can be used to measure the goodness of a particular pair of bases. A particular case considered is the dual-tree Hilbert-pair of wavelet bases. Several definitions of optimality are presented along with conjectures about the subjective characteristics of the ensembles where the optimality applies. We also consider relationships between sparseness and approximate representations.

We show how an overcomplete dictionary may be adapted to the statistics of natural images so as to provide a sparse representation of image content. When the degree of overcompleteness is low, the basis functions that emerge resemble those of Gabor wavelet transforms. As the degree of overcompleteness is increased, new families of basis functions emerge, including multiscale blobs, ridge-like functions, and gratings. When the basis functions and coefficients are allowed to be complex, they provide a description of image content in terms of local amplitude (contrast) and phase (position) of features. These complex, overcomplete transforms may be adapted to the statistics of natural movies by imposing both sparseness and temporal smoothness on the amplitudes. The basis functions that emerge form Hilbert pairs such that shifting the phase of the coefficient shifts the phase of the corresponding basis function. This type of representation is advantageous because it makes explicit the structural and dynamic content of images, which in turn allows later stages of processing to discover higher-order properties indicative of image content. We demonstrate this point by showing that it is possible to learn the higher-order structure of dynamic phase - i.e., motion - from the statistics of natural image sequences.

We propose an amplitude-phase representation of the dual-tree complex wavelet transform (DT-CWT) which provides an intuitive interpretation of the associated complex wavelet coefficients. The representation, in particular, is based on the shifting action of the group of fractional Hilbert transforms (fHT) which allow us to extend the notion of arbitrary phase-shifts beyond pure sinusoids. We explicitly characterize this shifting action for a particular family of Gabor-like wavelets which, in effect, links the corresponding dual-tree transform with the framework of windowed-Fourier analysis. We then extend these ideas to the bivariate DT-CWT based on certain directional extensions of the fHT. In particular, we derive a signal representation involving the superposition of direction-selective wavelets affected with appropriate phase-shifts.

Many algorithms have been proposed during the last decade in order to deal with inverse problems. Of particular interest are convex optimization approaches that consist of minimizing a criteria generally composed of two terms: a data fidelity (linked to noise) term and a prior (regularization) term. As image properties are often easier to extract in a transform domain, frame representations may be fruitful. Potential functions are then chosen as priors to fit as well as possible empirical coefficient distributions. As a consequence, the minimization problem can be considered from two viewpoints: a minimization along the coefficients or along the image pixels directly. Some recently proposed iterative optimization algorithms can be easily implemented when the frame representation reduces to an orthonormal basis. Furthermore, it can be noticed that in this particular case, it is equivalent to minimize the criterion in the transform domain or in the image domain. However, much attention should be paid when an overcomplete representation is considered. In that case, there is no longer equivalence between coefficient and image domain minimization. This point will be developed throughout this paper. Moreover, we will discuss how the choice of the transform may influence parameters and operators necessary to implement algorithms.

This paper describes an approach for decomposing a signal into the sum of an oscillatory component and a transient component. The method uses a newly developed rational-dilation wavelet transform (WT), a self-inverting constant-Q transform with an adjustable Q-factor (quality-factor). We propose that the oscillatory component be modeled as signal that can be sparsely represented using a high Q-factor WT; likewise, we propose that the transient component be modeled as a piecewise smooth signal that can be sparsely represented using a low Q-factor WT. Because the low and high Q-factor wavelet transforms are highly distinct (having low coherence), morphological component analysis (MCA) successfully yields the desired decomposition of a signal into an oscillatory and non-oscillatory component. The method, being non-linear, is not constrained by the limits of conventional LTI filtering.

Based on the class of complex gradient-Laplace operators, we show the design of a non-separable two-dimensional wavelet basis from a single and analytically defined generator wavelet function. The wavelet decomposition is implemented by an efficient FFT-based filterbank. By allowing for slight redundancy, we obtain the Marr wavelet pyramid decomposition that features improved translation-invariance and steerability. The link with Marr's theory of early vision is due to the replication of the essential processing steps (Gaussian smoothing, Laplacian, orientation detection). Finally, we show how to find a compact multiscale primal sketch of the image, and how to reconstruct an image from it.

One very influential method for texture synthesis is based on the steerable pyramid by alternately imposing marginal statistics on the image and the pyramid's subbands. In this work, we investigate two extensions to this framework. First, we exploit the steerability of the transform to obtain histograms of the subbands independent of the local orientation; i.e., we select the direction of maximal response as the reference orientation. Second, we explore the option of multidimensional histogram matching. The distribution of the responses to various orientations is expected to capture better the local geometric structure. Experimental results show how the proposed approach improves the performance of the original pyramid-based synthesis method.

This paper is concerned with the mathematical characterization and wavelet analysis of self-similar random vector fields. The study consists of two main parts: the construction of random vector models on the basis of their invariance under coordinate transformations, and a study of the consequences of conducting a wavelet analysis of such random models. In the latter part, after briefly examining the effects of standard wavelets on the proposed random fields, we go on to introduce a new family of Laplacian-like vector wavelets that in a way duplicate the covariant-structure and whitening relations governing our random models.

We consider an extension of the 1-D concept of analytical wavelets to n-D, which is by construction compatible with rotations. This extension, called the monogenic wavelet, yields a decomposition of the wavelet coefficients into amplitude, phase, and phase direction, analogous to the decomposition of an analytical wavelet coefficient into amplitude and phase. We demonstrate the usefulness of this decomposition with two applications.

In this paper we explore the design of 5-band dual frame (overcomplete) wavelets with a dilation factor M = 4. The resulting limit functions are significantly smoother than their orthogonal counterparts at the same dilation factor. An advantage of the proposed filters over the dyadic filterbanks (M = 2) is that the proposed filterbanks result in a reduced redundancy when compared with dyadic frames, while maintaining smoothness. The proposed filterbanks are symmetric and generate four wavelets and a scaling function for each the synthesis and analysis limit functions. All wavelets are equipped with at least one vanishing moment each.

The concept of stable space splittings has been introduced in the early 1990ies, as a convenient framework for developing a unified theory of iterative methods for variational problems in Hilbert spaces, especially for solving large-scale discretizations of elliptic problems in Sobolev spaces. The more recently introduced notions of frames of subspaces and fusion frames turn out to be concrete instances of stable space splittings. However, driven by applications to robust distributed signal processing, their study has focused so far on different aspects. The paper surveys the existing results on stable space splittings and iterative methods, outlines the connection to fusion frames, and discusses the investigation of quarkonial or multilevel partition-of-unity frames as an example of current interest.

A fusion frame is a frame-like collection of subspaces in a Hilbert space. It generalizes the concept of a frame system for signal representation. In this paper, we study the existence and construction of fusion frames. We first introduce two general methods, namely the spatial complement and the Naimark complement, for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given M, N, m ∈ N and {λ_j}^M_j =1, does there exist a fusion frame in R^M with N subspaces of dimension m for which {λ_j}^M_j =1 are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters M, N, m ∈ N and {λ_j}^M_j =1. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become tight.

Fusion frames are an emerging topic of frame theory, with applications to communications and distributed processing. However, until recently, little was known about the existence of tight fusion frames, much less how to construct them. We discuss a new method for constructing tight fusion frames which is akin to playing Tetris with the spectrum of the frame operator. When combined with some easily obtained necessary conditions, these Spectral Tetris constructions provide a near complete characterization of the existence of tight fusion frames.

Compressed Sensing (CS) is a new signal acquisition technique that allows sampling of sparse signals using significantly fewer measurements than previously thought possible. On the other hand, a fusion frame is a new signal representation method that uses collections of subspaces instead of vectors to represent signals. This work combines these exciting new fields to introduce a new sparsity model for fusion frames. Signals that are sparse under the new model can be compressively sampled and uniquely reconstructed in ways similar to sparse signals using standard CS. The combination provides a promising new set of mathematical tools and signal models useful in a variety of applications. With the new model, a sparse signal has energy in very few of the subspaces of the fusion frame, although it needs not be sparse within each of the subspaces it occupies. We define a mixed ℓ₁/ℓ₂ norm for fusion frames. A signal sparse in the subspaces of the fusion frame can thus be sampled using very few random projections and exactly reconstructed using a convex optimization that minimizes this mixed ℓ₁/ℓ₂ norm. The sampling conditions we derive are very similar to the coherence and RIP conditions used in standard CS theory.

In this work, a modification of the k q-flats framework for pattern classification introduced in [9] is used for pixelwise object detection. We include a preliminary discussion of augmenting this method is with a Chan-Vese-like geometric regularization.

We derive lower and upper bounds for the distance between a frame and the set of equal-norm Parseval frames. The lower bound results from variational inequalities. The upper bound is obtained with a technique that uses a family of ordinary differential equations for Parseval frames which can be shown to converge to an equal-norm Parseval frame, if the number of vectors in a frame and the dimension of the Hilbert space they span are relatively prime, and if the initial frame consists of vectors having sufficiently nearly equal norms.

Image-based medical diagnosis typically relies on the (poorly reproducible) subjective classification of textures in order to differentiate between diseased and healthy pathology. Clinicians claim that significant benefits would arise from quantitative measures to inform clinical decision making. The first step in generating such measures is to extract local image descriptors - from noise corrupted and often spatially and temporally coarse resolution medical signals - that are invariant to illumination, translation, scale and rotation of the features. The Dual-Tree Complex Wavelet Transform (DT-CWT) provides a wavelet multiresolution analysis (WMRA) tool e.g. in 2D with good properties, but has limited rotational selectivity. Also, it requires computationally-intensive steering due to the inherently 1D operations performed. The monogenic signal, which is defined in n >= 2D with the Riesz transform gives excellent orientation information without the need for steering. Recent work has suggested the Monogenic Riesz-Laplace wavelet transform as a possible tool for integrating these two concepts into a coherent mathematical framework. We have found that the proposed construction suffers from a lack of rotational invariance and is not optimal for retrieving local image descriptors. In this paper we show: 1. Local frequency and local phase from the monogenic signal are not equivalent, especially in the phase congruency model of a "feature", and so they are not interchangeable for medical image applications. 2. The accuracy of local phase computation may be improved by estimating the denoising parameters while maximizing a new measure of "featureness".

In the scope of the Fermi mission, Poisson noise removal should improve data quality and make source detection easier. This paper presents a method for Poisson data denoising on sphere, called Multi-Scale Variance Stabilizing Transform on Sphere (MS-VSTS). This method is based on a Variance Stabilizing Transform (VST), a transform which aims to stabilize a Poisson data set such that each stabilized sample has an (asymptotically) constant variance. In addition, for the VST used in the method, the transformed data are asymptotically Gaussian. Thus, MS-VSTS consists in decomposing the data into a sparse multi-scale dictionary (wavelets, curvelets, ridgelets...), and then applying a VST on the coefficients in order to get quasi-Gaussian stabilized coefficients. In this present article, the used multi-scale transform is the Isotropic Undecimated Wavelet Transform. Then, hypothesis tests are made to detect significant coefficients, and the denoised image is reconstructed with an iterative method based on Hybrid Steepest Descent (HST). The method is tested on simulated Fermi data.

The Multi-scale Variance Stabilization Transform (MSVST) has recently been proposed for 2D Poisson data denoising.¹ In this work, we present an extension of the MSVST with the wavelet transform to multivariate data-each pixel is vector-valued-, where the vector field dimension may be the wavelength, the energy, or the time. Such data can be viewed naively as 3D data where the third dimension may be time, wavelength or energy (e.g. hyperspectral imaging). But this naive analysis using a 3D MSVST would be awkward as the data dimensions have different physical meanings. A more appropriate approach would be to use a wavelet transform, where the time or energy scale is not connected to the spatial scale. We show that our multivalued extension of MSVST can be used advantageously for approximately Gaussianizing and stabilizing the variance of a sequence of independent Poisson random vectors. This approach is shown to be fast and very well adapted to extremely low-count situations. We use a hypothesis testing framework in the wavelet domain to denoise the Gaussianized and stabilized coefficients, and then apply an iterative reconstruction algorithm to recover the estimated vector field of intensities underlying the Poisson data. Our approach is illustrated for the detection and characterization of astrophysical sources of high-energy gamma rays, using realistic simulated observations. We show that the multivariate MSVST permits efficient estimation across the time/energy dimension and immediate recovery of spectral properties.

During data acquisition, the loss of data is usual. It can be due to malfunctioning sensors of a CCD camera or any other acquiring system, or because we can only observe a part of the system we want to analyze. This problem has been addressed using diffusion through the use of partial differential equations in 2D and in 3D, and recently using sparse representations in 2D in a process called inpainting which uses sparsity to get a solution (in the masked/unknown part) which is statistically similar to the known data, in the sense of the transformations used, so that one cannot tell the inpainted part from the real one. It can be applied on any kind of 3D data, whether it is 3D spatial data, 2D and time (video) or 2D and wavelength (multi-spectral imaging). We present inpainting results on 3D data using sparse representations. These representations may include the wavetet transforms, the discrete cosine transform, and 3D curvelet transforms.

The recent development of multi-channel sensors has motivated interest in devising new methods for the coherent processing of multivariate data. An extensive work has already been dedicated to multivariate data processing ranging from blind source separation (BSS) to multi/hyper-spectral data restoration. Previous work has emphasized on the fundamental role played by sparsity and morphological diversity to enhance multichannel signal processing. GMCA is a recent algorithm for multichannel data analysis which was used successfully in a variety of applications including multichannel sparse decomposition, blind source separation (BSS), color image restoration and inpainting. Inspired by GMCA, a recently introduced algorithm coined HypGMCA is described for BSS applications in hyperspectral data processing. It assumes the collected data is a linear instantaneous mixture of components exhibiting sparse spectral signatures as well as sparse spatial morphologies, each in specified dictionaries of spectral and spatial waveforms. We report on numerical experiments with synthetic data and application to real observations which demonstrate the validity of the proposed method.

Patch based methods give some of the best denoising results. Their theoretical performances are still unexplained mathematically. We propose a novel insight of NL-Means based on an aggregation point of view. More precisely, we describe the framework of PAC-Bayesian aggregation, show how it allows to derive some new patch based methods and to characterize their theoretical performances, and present some numerical experiments.

Distributed compressed sensing is the extension of compressed sampling (CS) to sensor networks. The idea is to design a CS joint decoding scheme at a central decoder (base station) that exploits the inter-sensor correlations, in order to recover the whole observations from very few number of random measurements per node. In this paper, we focus on modeling the correlations and on the design and analysis of efficient joint recovery algorithms. We show, by extending earlier results of Baron et al.,¹ that a simple thresholding algorithm can exploit the full diversity offered by all channels to identify a common sparse support using a near optimal number of measurements.