Image Decomposition and Separation Using Sparse Representations: An Overview
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paper_title: Modeling Textures with Total Variation Minimization and Oscillating Patterns in Image Processing

paper_content:

This paper is devoted to the modeling of real textured images by functional minimization and partial differential equations. Following the ideas of Yves Meyer in a total variation minimization framework of L. Rudin, S. Osher, and E. Fatemi, we decompose a given (possible textured) image f into a sum of two functions u+v, where u∈BV is a function of bounded variation (a cartoon or sketchy approximation of f), while v is a function representing the texture or noise. To model v we use the space of oscillating functions introduced by Yves Meyer, which is in some sense the dual of the BV space. The new algorithm is very simple, making use of differential equations and is easily solved in practice. Finally, we implement the method by finite differences, and we present various numerical results on real textured images, showing the obtained decomposition u+v, but we also show how the method can be used for texture discrimination and texture segmentation.
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paper_title: Blind Source Separation: the Sparsity Revolution

paper_content:

Over the last few years, the development of multi-channel sensors motivated interest in methods for the coherent processing of multivariate data. Some specific issues have already been addressed as testified by the wide literature on the so-called blind source separation (BSS) problem. In this context, as clearly emphasized by previous work, it is fundamental that the sources to be retrieved present some quantitatively measurable diversity. Recently, sparsity and morphological diversity have emerged as a novel and effective source of diversity for BSS. We give here some essential insights into the use of sparsity in source separation and we outline the essential role of morphological diversity as being a source of diversity or contrast between the sources. This paper overviews a sparsity-based BSS method coined Generalized Morphological Component Analysis (GMCA) that takes advantages of both morphological diversity and sparsity, using recent sparse overcomplete or redundant signal representations. GMCA is a fast and efficient blind source separation method. In remote sensing applications, the specificity of hyperspectral data should be accounted for. We extend the proposed GMCA framework to deal with hyperspectral data. In a general framework, GMCA provides a basis for multivariate data analysis in the scope of a wide range of classical multivariate data restorate. Numerical results are given in color image denoising and inpainting. Finally, GMCA is applied to the simulated ESA/Planck data. It is shown to give effective astrophysical component separation.
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paper_title: Underdetermined blind source separation using sparse representations

paper_content:

The scope of this work is the separation of N sources from M linear mixtures when the underlying system is underdetermined, that is, when Mi N. If the input distribution is sparse the mixing matrix can be estimated either by external optimization or by clustering and, given the mixing matrix, a minimal l1 norm representation of the sources can be obtained by solving a low-dimensional linear programming problem for each of the data points. Yet, when the signals per se do not satisfy this assumption, sparsity can still be achieved by realizing the separation in a sparser transformed domain. The approach is illustrated here for M = 2. In this case we estimate both the number of sources and the mixing matrix by the maxima of a potential function along the circle of unit length, and we obtain the minimal l1 norm representation of each data point by a linear combination of the pair of basis vectors that enclose it. Several experiments with music and speech signals show that their time-domain representation is not sparse enough. Yet, excellent results were obtained using their short-time Fourier transform, including the separation of up to six sources from two mixtures. ? 2001 Elsevier Science B.V. All rights reserved.
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paper_title: Morphological Diversity and Sparsity for Multichannel Data Restoration

paper_content:

Over the last decade, overcomplete dictionaries and the very sparse signal representations they make possible, have raised an intense interest from signal processing theory. In a wide range of signal processing problems, sparsity has been a crucial property leading to high performance. As multichannel data are of growing interest, it seems essential to devise sparsity-based tools accounting for such specific multichannel data. Sparsity has proved its efficiency in a wide range of inverse problems. Hereafter, we address some multichannel inverse problems issues such as multichannel morphological component separation and inpainting from the perspective of sparse representation. In this paper, we introduce a new sparsity-based multichannel analysis tool coined multichannel Morphological Component Analysis (mMCA). This new framework focuses on multichannel morphological diversity to better represent multichannel data. This paper presents conditions under which the mMCA converges and recovers the sparse multichannel representation. Several experiments are presented to demonstrate the applicability of our approach on a set of multichannel inverse problems such as morphological component decomposition and inpainting.
---
paper_title: Sparsity and Morphological Diversity in Blind Source Separation

paper_content:

Over the last few years, the development of multichannel sensors motivated interest in methods for the coherent processing of multivariate data. Some specific issues have already been addressed as testified by the wide literature on the so-called blind source separation (BSS) problem. In this context, as clearly emphasized by previous work, it is fundamental that the sources to be retrieved present some quantitatively measurable diversity. Recently, sparsity and morphological diversity have emerged as a novel and effective source of diversity for BSS. Here, we give some new and essential insights into the use of sparsity in source separation, and we outline the essential role of morphological diversity as being a source of diversity or contrast between the sources. This paper introduces a new BSS method coined generalized morphological component analysis (GMCA) that takes advantages of both morphological diversity and sparsity, using recent sparse overcomplete or redundant signal representations. GMCA is a fast and efficient BSS method. We present arguments and a discussion supporting the convergence of the GMCA algorithm. Numerical results in multivariate image and signal processing are given illustrating the good performance of GMCA and its robustness to noise.
---
paper_title: Blind source separation by sparse decomposition

paper_content:

The blind source separation problem is to extract the underlying source signals from a set of their linear mixtures, where the mixing matrix is unknown. This situation is common, eg in acoustics, radio, and medical signal processing. We exploit the property of the sources to have a sparse representation in a corresponding signal dictionary. Such a dictionary may consist of wavelets, wavelet packets, etc., or be obtained by learning from a given family of signals. Starting from the maximum a posteriori framework, which is applicable to the case of more sources than mixtures, we derive a few other categories of objective functions, which provide faster and more robust computations, when there are an equal number of sources and mixtures. Our experiments with artificial signals and with musical sounds demonstrate significantly better separation than other known techniques.
---
paper_title: Blind Source Separation: the Sparsity Revolution

paper_content:

Over the last few years, the development of multi-channel sensors motivated interest in methods for the coherent processing of multivariate data. Some specific issues have already been addressed as testified by the wide literature on the so-called blind source separation (BSS) problem. In this context, as clearly emphasized by previous work, it is fundamental that the sources to be retrieved present some quantitatively measurable diversity. Recently, sparsity and morphological diversity have emerged as a novel and effective source of diversity for BSS. We give here some essential insights into the use of sparsity in source separation and we outline the essential role of morphological diversity as being a source of diversity or contrast between the sources. This paper overviews a sparsity-based BSS method coined Generalized Morphological Component Analysis (GMCA) that takes advantages of both morphological diversity and sparsity, using recent sparse overcomplete or redundant signal representations. GMCA is a fast and efficient blind source separation method. In remote sensing applications, the specificity of hyperspectral data should be accounted for. We extend the proposed GMCA framework to deal with hyperspectral data. In a general framework, GMCA provides a basis for multivariate data analysis in the scope of a wide range of classical multivariate data restorate. Numerical results are given in color image denoising and inpainting. Finally, GMCA is applied to the simulated ESA/Planck data. It is shown to give effective astrophysical component separation.
---
paper_title: From sparse solutions of systems of equations to sparse modeling of signals and images

paper_content:

A full-rank matrix ${\bf A}\in \mathbb{R}^{n\times m}$ with $n<m$ generates an underdetermined system of linear equations ${\bf Ax} = {\bf b}$ having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries. Can it ever be unique? If so, when? As optimization of sparsity is combinatorial in nature, are there efficient methods for finding the sparsest solution? These questions have been answered positively and constructively in recent years, exposing a wide variety of surprising phenomena, in particular the existence of easily verifiable conditions under which optimally sparse solutions can be found by concrete, effective computational methods. ::: ::: Such theoretical results inspire a bold perspective on some important practical problems in signal and image processing. Several well-known signal and image processing problems can be cast as demanding solutions of undetermined systems of equations. Such problems have previously seemed, to many, intractable, but there is considerable evidence that these problems often have sparse solutions. Hence, advances in finding sparse solutions to underdetermined systems have energized research on such signal and image processing problems—to striking effect. ::: ::: In this paper we review the theoretical results on sparse solutions of linear systems, empirical results on sparse modeling of signals and images, and recent applications in inverse problems and compression in image processing. This work lies at the intersection of signal processing and applied mathematics, and arose initially from the wavelets and harmonic analysis research communities. The aim of this paper is to introduce a few key notions and applications connected to sparsity, targeting newcomers interested in either the mathematical aspects of this area or its applications.
---
paper_title: Morphological Diversity and Sparsity for Multichannel Data Restoration

paper_content:

Over the last decade, overcomplete dictionaries and the very sparse signal representations they make possible, have raised an intense interest from signal processing theory. In a wide range of signal processing problems, sparsity has been a crucial property leading to high performance. As multichannel data are of growing interest, it seems essential to devise sparsity-based tools accounting for such specific multichannel data. Sparsity has proved its efficiency in a wide range of inverse problems. Hereafter, we address some multichannel inverse problems issues such as multichannel morphological component separation and inpainting from the perspective of sparse representation. In this paper, we introduce a new sparsity-based multichannel analysis tool coined multichannel Morphological Component Analysis (mMCA). This new framework focuses on multichannel morphological diversity to better represent multichannel data. This paper presents conditions under which the mMCA converges and recovers the sparse multichannel representation. Several experiments are presented to demonstrate the applicability of our approach on a set of multichannel inverse problems such as morphological component decomposition and inpainting.
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paper_title: Matching pursuits with time-frequency dictionaries

paper_content:

The authors introduce an algorithm, called matching pursuit, that decomposes any signal into a linear expansion of waveforms that are selected from a redundant dictionary of functions. These waveforms are chosen in order to best match the signal structures. Matching pursuits are general procedures to compute adaptive signal representations. With a dictionary of Gabor functions a matching pursuit defines an adaptive time-frequency transform. They derive a signal energy distribution in the time-frequency plane, which does not include interference terms, unlike Wigner and Cohen class distributions. A matching pursuit isolates the signal structures that are coherent with respect to a given dictionary. An application to pattern extraction from noisy signals is described. They compare a matching pursuit decomposition with a signal expansion over an optimized wavepacket orthonormal basis, selected with the algorithm of Coifman and Wickerhauser see (IEEE Trans. Informat. Theory, vol. 38, Mar. 1992). >
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paper_title: Morphological Component Analysis: An Adaptive Thresholding Strategy

paper_content:

In a recent paper, a method called morphological component analysis (MCA) has been proposed to separate the texture from the natural part in images. MCA relies on an iterative thresholding algorithm, using a threshold which decreases linearly towards zero along the iterations. This paper shows how the MCA convergence can be drastically improved using the mutual incoherence of the dictionaries associated to the different components. This modified MCA algorithm is then compared to basis pursuit, and experiments show that MCA and BP solutions are similar in terms of sparsity, as measured by the lscr1 norm, but MCA is much faster and gives us the possibility of handling large scale data sets.
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paper_title: Block Coordinate Relaxation Methods for Nonparametric Wavelet Denoising

paper_content:

Abstract An important class of nonparametric signal processing methods entails forming a set of predictors from an overcomplete set of basis functions associated with a fast transform (e.g., wavelet packets). In these methods, the number of basis functions can far exceed the number of sample values in the signal, leading to an ill-posed prediction problem. The “basis pursuit” denoising method of Chen, Donoho, and Saunders regularizes the prediction problem by adding an l 1 penalty term on the coefficients for the basis functions. Use of an l 1 penalty instead of l 2 has significant benefits, including higher resolution of signals close in time/frequency and a more parsimonious representation. The l 1 penalty, however, poses a challenging optimization problem that was solved by Chen, Donoho and Saunders using a novel application of interior-point algorithms (IP). This article investigates an alternative optimization approach based on block coordinate relaxation (BCR) for sets of basis functions that are th...
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paper_title: Wave atoms and sparsity of oscillatory patterns

paper_content:

We introduce \wave atoms" as a variant of 2D wavelet packets obeying the parabolic scaling wavelength (diameter) 2 . We prove that warped oscillatory functions, a toy model for texture, have a signicantly sparser expansion in wave atoms than in other xed standard representations like wavelets, Gabor atoms, or curvelets. We propose a novel algorithm for a tight frame of wave atoms with redundancy two, directly in the frequency plane, by the \wrapping" technique. We also propose variants of the basic transform for applications in image processing, including an orthonormal basis, and a shift-invariant tight frame with redundancy four. Sparsity and denoising experiments on both seismic and ngerprint images demonstrate the potential of the tool introduced.
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paper_title: Image processing and data analysis: the multiscale approach

paper_content:

Foreword 1. The wavelet transform 2. Multiresolution support and filtering 3. Deconvolution 4. 1-D signals and Euclidean-based data analysis 5. Geometric registration 6. Disparity analysis and applications in remote sensing 7. Image compression 8. Object detection and point pattern clustering 9. Multiscale vision models Appendix I. Variance stabilisation Appendix II. Software information Acronyms References Index.
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paper_title: Brushlets: A Tool for Directional Image Analysis and Image Compression

paper_content:

Abstract We construct a new adaptive basis of functions which is reasonably well localized with only one peak in frequency. We develop a compression algorithm that exploits this basis to obtain the most economical representation of the image in terms of textured patterns with different orientations, frequencies, sizes, and positions. The technique directly works in the Fourier domain and has potential applications for compression of highly textured images, texture analysis, etc.
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paper_title: Learning adapted dictionaries for geometry and texture separation

paper_content:

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.
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paper_title: A Wavelet Tour of Signal Processing

paper_content:

Introduction to a Transient World. Fourier Kingdom. Discrete Revolution. Time Meets Frequency. Frames. Wavelet Zoom. Wavelet Bases. Wavelet Packet and Local Cosine Bases. An Approximation Tour. Estimations are Approximations. Transform Coding. Appendix A: Mathematical Complements. Appendix B: Software Toolboxes.
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paper_title: Morphological Component Analysis: An Adaptive Thresholding Strategy

paper_content:

In a recent paper, a method called morphological component analysis (MCA) has been proposed to separate the texture from the natural part in images. MCA relies on an iterative thresholding algorithm, using a threshold which decreases linearly towards zero along the iterations. This paper shows how the MCA convergence can be drastically improved using the mutual incoherence of the dictionaries associated to the different components. This modified MCA algorithm is then compared to basis pursuit, and experiments show that MCA and BP solutions are similar in terms of sparsity, as measured by the lscr1 norm, but MCA is much faster and gives us the possibility of handling large scale data sets.
---
paper_title: MCALab: Reproducible Research in Signal and Image Decomposition and Inpainting

paper_content:

Morphological component analysis of signals and images has far-reaching applications in science and technology, but some consider it problematic and even intractable. Reproducible research is essential to give MCA a firm scientific foundation. Researchers developed MCALab to demonstrate key MCA concepts and make them available to interested researchers.
---
paper_title: Sparse component analysis and blind source separation of underdetermined mixtures

paper_content:

In this letter, we solve the problem of identifying matrices S /spl isin/ /spl Ropf//sup n/spl times/N/ and A /spl isin/ /spl Ropf//sup m/spl times/n/ knowing only their multiplication X = AS, under some conditions, expressed either in terms of A and sparsity of S (identifiability conditions), or in terms of X (sparse component analysis (SCA) conditions). We present algorithms for such identification and illustrate them by examples.
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paper_title: Blind Source Separation: the Sparsity Revolution

paper_content:

Over the last few years, the development of multi-channel sensors motivated interest in methods for the coherent processing of multivariate data. Some specific issues have already been addressed as testified by the wide literature on the so-called blind source separation (BSS) problem. In this context, as clearly emphasized by previous work, it is fundamental that the sources to be retrieved present some quantitatively measurable diversity. Recently, sparsity and morphological diversity have emerged as a novel and effective source of diversity for BSS. We give here some essential insights into the use of sparsity in source separation and we outline the essential role of morphological diversity as being a source of diversity or contrast between the sources. This paper overviews a sparsity-based BSS method coined Generalized Morphological Component Analysis (GMCA) that takes advantages of both morphological diversity and sparsity, using recent sparse overcomplete or redundant signal representations. GMCA is a fast and efficient blind source separation method. In remote sensing applications, the specificity of hyperspectral data should be accounted for. We extend the proposed GMCA framework to deal with hyperspectral data. In a general framework, GMCA provides a basis for multivariate data analysis in the scope of a wide range of classical multivariate data restorate. Numerical results are given in color image denoising and inpainting. Finally, GMCA is applied to the simulated ESA/Planck data. It is shown to give effective astrophysical component separation.
---
paper_title: Morphological diversity and source separation

paper_content:

This letter describes a new method for blind source separation, adapted to the case of sources having different morphologies. We show that such morphological diversity leads to a new and very efficient separation method, even in the presence of noise. The algorithm, coined multichannel morphological component analysis (MMCA), is an extension of the morphological component analysis (MCA) method. The latter takes advantage of the sparse representation of structured data in large overcomplete dictionaries to separate features in the data based on their morphology. MCA has been shown to be an efficient technique in such problems as separating an image into texture and piecewise smooth parts or for inpainting applications. The proposed extension, MMCA, extends the above for multichannel data, achieving a better source separation in those circumstances. Furthermore, the new algorithm can efficiently achieve good separation in a noisy context where standard independent component analysis methods fail. The efficiency of the proposed scheme is confirmed in numerical experiments
---
paper_title: Sparsity and Morphological Diversity in Blind Source Separation

paper_content:

Over the last few years, the development of multichannel sensors motivated interest in methods for the coherent processing of multivariate data. Some specific issues have already been addressed as testified by the wide literature on the so-called blind source separation (BSS) problem. In this context, as clearly emphasized by previous work, it is fundamental that the sources to be retrieved present some quantitatively measurable diversity. Recently, sparsity and morphological diversity have emerged as a novel and effective source of diversity for BSS. Here, we give some new and essential insights into the use of sparsity in source separation, and we outline the essential role of morphological diversity as being a source of diversity or contrast between the sources. This paper introduces a new BSS method coined generalized morphological component analysis (GMCA) that takes advantages of both morphological diversity and sparsity, using recent sparse overcomplete or redundant signal representations. GMCA is a fast and efficient BSS method. We present arguments and a discussion supporting the convergence of the GMCA algorithm. Numerical results in multivariate image and signal processing are given illustrating the good performance of GMCA and its robustness to noise.
---
paper_title: Blind source separation by sparse decomposition

paper_content:

The blind source separation problem is to extract the underlying source signals from a set of their linear mixtures, where the mixing matrix is unknown. This situation is common, eg in acoustics, radio, and medical signal processing. We exploit the property of the sources to have a sparse representation in a corresponding signal dictionary. Such a dictionary may consist of wavelets, wavelet packets, etc., or be obtained by learning from a given family of signals. Starting from the maximum a posteriori framework, which is applicable to the case of more sources than mixtures, we derive a few other categories of objective functions, which provide faster and more robust computations, when there are an equal number of sources and mixtures. Our experiments with artificial signals and with musical sounds demonstrate significantly better separation than other known techniques.
---
paper_title: Under-determined source separation via mixed-norm regularized minimization

paper_content:

We consider the problem of extracting the source signals from an under-determined convolutive mixture assuming that the mixing filters are known. We wish to exploit the sparsity and approximate disjointness of the time-frequency representations of the sources. However, classical time-frequency masking techniques cannot be directly applied due to the convolutive nature of the mixture. To address this problem, we first formulate it as the minimization of a functional combining a classical l 2 discrepancy term between the observed mixture and the mixture reconstructed from the estimated sources and a sparse regularization term defined in terms of mixed l 2 /l 1 norms of source coefficients in a time-frequency domain. The minimum of the functional is then obtained by a thresholded Landweber iteration algorithm. Preliminary results are discussed for two synthetic audio mixtures.
---
paper_title: Sparsity and Morphological Diversity in Blind Source Separation

paper_content:

Over the last few years, the development of multichannel sensors motivated interest in methods for the coherent processing of multivariate data. Some specific issues have already been addressed as testified by the wide literature on the so-called blind source separation (BSS) problem. In this context, as clearly emphasized by previous work, it is fundamental that the sources to be retrieved present some quantitatively measurable diversity. Recently, sparsity and morphological diversity have emerged as a novel and effective source of diversity for BSS. Here, we give some new and essential insights into the use of sparsity in source separation, and we outline the essential role of morphological diversity as being a source of diversity or contrast between the sources. This paper introduces a new BSS method coined generalized morphological component analysis (GMCA) that takes advantages of both morphological diversity and sparsity, using recent sparse overcomplete or redundant signal representations. GMCA is a fast and efficient BSS method. We present arguments and a discussion supporting the convergence of the GMCA algorithm. Numerical results in multivariate image and signal processing are given illustrating the good performance of GMCA and its robustness to noise.
---
paper_title: Blind Source Separation: the Sparsity Revolution

paper_content:

Over the last few years, the development of multi-channel sensors motivated interest in methods for the coherent processing of multivariate data. Some specific issues have already been addressed as testified by the wide literature on the so-called blind source separation (BSS) problem. In this context, as clearly emphasized by previous work, it is fundamental that the sources to be retrieved present some quantitatively measurable diversity. Recently, sparsity and morphological diversity have emerged as a novel and effective source of diversity for BSS. We give here some essential insights into the use of sparsity in source separation and we outline the essential role of morphological diversity as being a source of diversity or contrast between the sources. This paper overviews a sparsity-based BSS method coined Generalized Morphological Component Analysis (GMCA) that takes advantages of both morphological diversity and sparsity, using recent sparse overcomplete or redundant signal representations. GMCA is a fast and efficient blind source separation method. In remote sensing applications, the specificity of hyperspectral data should be accounted for. We extend the proposed GMCA framework to deal with hyperspectral data. In a general framework, GMCA provides a basis for multivariate data analysis in the scope of a wide range of classical multivariate data restorate. Numerical results are given in color image denoising and inpainting. Finally, GMCA is applied to the simulated ESA/Planck data. It is shown to give effective astrophysical component separation.
---
paper_title: Sparsity and Morphological Diversity in Blind Source Separation

paper_content:

Over the last few years, the development of multichannel sensors motivated interest in methods for the coherent processing of multivariate data. Some specific issues have already been addressed as testified by the wide literature on the so-called blind source separation (BSS) problem. In this context, as clearly emphasized by previous work, it is fundamental that the sources to be retrieved present some quantitatively measurable diversity. Recently, sparsity and morphological diversity have emerged as a novel and effective source of diversity for BSS. Here, we give some new and essential insights into the use of sparsity in source separation, and we outline the essential role of morphological diversity as being a source of diversity or contrast between the sources. This paper introduces a new BSS method coined generalized morphological component analysis (GMCA) that takes advantages of both morphological diversity and sparsity, using recent sparse overcomplete or redundant signal representations. GMCA is a fast and efficient BSS method. We present arguments and a discussion supporting the convergence of the GMCA algorithm. Numerical results in multivariate image and signal processing are given illustrating the good performance of GMCA and its robustness to noise.
---
paper_title: Estimator for number of sources using minimum description length criterion for blind sparse source mixtures

paper_content:

In this paper I present a Minimum Description Length Estimator for number of sources in an anechoic mixture of sparse signals. The criterion is roughly equal to the sum of negative normalized maximum log-likelihood and the logarithm of number of sources. Numerical evidence supports this approach and compares favorabily to both the Akaike (AIC) and Bayesian (BIC) Information Criteria.
---
paper_title: Blind Source Separation: the Sparsity Revolution

paper_content:

Over the last few years, the development of multi-channel sensors motivated interest in methods for the coherent processing of multivariate data. Some specific issues have already been addressed as testified by the wide literature on the so-called blind source separation (BSS) problem. In this context, as clearly emphasized by previous work, it is fundamental that the sources to be retrieved present some quantitatively measurable diversity. Recently, sparsity and morphological diversity have emerged as a novel and effective source of diversity for BSS. We give here some essential insights into the use of sparsity in source separation and we outline the essential role of morphological diversity as being a source of diversity or contrast between the sources. This paper overviews a sparsity-based BSS method coined Generalized Morphological Component Analysis (GMCA) that takes advantages of both morphological diversity and sparsity, using recent sparse overcomplete or redundant signal representations. GMCA is a fast and efficient blind source separation method. In remote sensing applications, the specificity of hyperspectral data should be accounted for. We extend the proposed GMCA framework to deal with hyperspectral data. In a general framework, GMCA provides a basis for multivariate data analysis in the scope of a wide range of classical multivariate data restorate. Numerical results are given in color image denoising and inpainting. Finally, GMCA is applied to the simulated ESA/Planck data. It is shown to give effective astrophysical component separation.
---
paper_title: Blind Source Separation: the Sparsity Revolution

paper_content:

Over the last few years, the development of multi-channel sensors motivated interest in methods for the coherent processing of multivariate data. Some specific issues have already been addressed as testified by the wide literature on the so-called blind source separation (BSS) problem. In this context, as clearly emphasized by previous work, it is fundamental that the sources to be retrieved present some quantitatively measurable diversity. Recently, sparsity and morphological diversity have emerged as a novel and effective source of diversity for BSS. We give here some essential insights into the use of sparsity in source separation and we outline the essential role of morphological diversity as being a source of diversity or contrast between the sources. This paper overviews a sparsity-based BSS method coined Generalized Morphological Component Analysis (GMCA) that takes advantages of both morphological diversity and sparsity, using recent sparse overcomplete or redundant signal representations. GMCA is a fast and efficient blind source separation method. In remote sensing applications, the specificity of hyperspectral data should be accounted for. We extend the proposed GMCA framework to deal with hyperspectral data. In a general framework, GMCA provides a basis for multivariate data analysis in the scope of a wide range of classical multivariate data restorate. Numerical results are given in color image denoising and inpainting. Finally, GMCA is applied to the simulated ESA/Planck data. It is shown to give effective astrophysical component separation.
---
paper_title: Morphological Diversity and Sparsity for Multichannel Data Restoration

paper_content:

Over the last decade, overcomplete dictionaries and the very sparse signal representations they make possible, have raised an intense interest from signal processing theory. In a wide range of signal processing problems, sparsity has been a crucial property leading to high performance. As multichannel data are of growing interest, it seems essential to devise sparsity-based tools accounting for such specific multichannel data. Sparsity has proved its efficiency in a wide range of inverse problems. Hereafter, we address some multichannel inverse problems issues such as multichannel morphological component separation and inpainting from the perspective of sparse representation. In this paper, we introduce a new sparsity-based multichannel analysis tool coined multichannel Morphological Component Analysis (mMCA). This new framework focuses on multichannel morphological diversity to better represent multichannel data. This paper presents conditions under which the mMCA converges and recovers the sparse multichannel representation. Several experiments are presented to demonstrate the applicability of our approach on a set of multichannel inverse problems such as morphological component decomposition and inpainting.
---
paper_title: Sparsity and Morphological Diversity in Blind Source Separation

paper_content:

Over the last few years, the development of multichannel sensors motivated interest in methods for the coherent processing of multivariate data. Some specific issues have already been addressed as testified by the wide literature on the so-called blind source separation (BSS) problem. In this context, as clearly emphasized by previous work, it is fundamental that the sources to be retrieved present some quantitatively measurable diversity. Recently, sparsity and morphological diversity have emerged as a novel and effective source of diversity for BSS. Here, we give some new and essential insights into the use of sparsity in source separation, and we outline the essential role of morphological diversity as being a source of diversity or contrast between the sources. This paper introduces a new BSS method coined generalized morphological component analysis (GMCA) that takes advantages of both morphological diversity and sparsity, using recent sparse overcomplete or redundant signal representations. GMCA is a fast and efficient BSS method. We present arguments and a discussion supporting the convergence of the GMCA algorithm. Numerical results in multivariate image and signal processing are given illustrating the good performance of GMCA and its robustness to noise.
---
paper_title: Dictionary learning with spatio-spectral sparsity constraints

paper_content:

Devising efficient sparse decomposition algorithms in large redundant dictionaries has attracted much attention recently. However, choosing the right dictionary for a given data set remains an issue. An interesting approach is to learn the best dictionary from the data itself. The purpose of this contribution is to describe a new dictionary learning algorithm for multichannel data analysis purposes under specific assumptions. We assume a large number of contiguous channels as in so-called hyperspectral data. In this case it makes sense to consider a priori that the collected data exhibits sparse spectral signatures and sparse spatial morphologies in specified dictionaries of spectral and spatial waveforms. Building on GMCA, the proposed algorithm gives a practical way to enforce the additional a priori spectral sparsity constraint on the dictionary space. Numerical experiments with synthetic and real hyperspectral data illustrate the efficiency of the proposed algorithm.
---
paper_title: Sparsity and Morphological Diversity in Blind Source Separation

paper_content:

Over the last few years, the development of multichannel sensors motivated interest in methods for the coherent processing of multivariate data. Some specific issues have already been addressed as testified by the wide literature on the so-called blind source separation (BSS) problem. In this context, as clearly emphasized by previous work, it is fundamental that the sources to be retrieved present some quantitatively measurable diversity. Recently, sparsity and morphological diversity have emerged as a novel and effective source of diversity for BSS. Here, we give some new and essential insights into the use of sparsity in source separation, and we outline the essential role of morphological diversity as being a source of diversity or contrast between the sources. This paper introduces a new BSS method coined generalized morphological component analysis (GMCA) that takes advantages of both morphological diversity and sparsity, using recent sparse overcomplete or redundant signal representations. GMCA is a fast and efficient BSS method. We present arguments and a discussion supporting the convergence of the GMCA algorithm. Numerical results in multivariate image and signal processing are given illustrating the good performance of GMCA and its robustness to noise.
---
paper_title: MCALab: Reproducible Research in Signal and Image Decomposition and Inpainting

paper_content:

Morphological component analysis of signals and images has far-reaching applications in science and technology, but some consider it problematic and even intractable. Reproducible research is essential to give MCA a firm scientific foundation. Researchers developed MCALab to demonstrate key MCA concepts and make them available to interested researchers.
---

Title: Image Decomposition and Separation Using Sparse Representations: An Overview

Section 1: INTRODUCTION
Description 1: Provide an introduction to the problem of image decomposition and separation, emphasizing the significance of the billion-dollar problems in signal and image processing and the role of sparsity and morphological diversity in solving these problems.

Section 2: Organization of the Paper
Description 2: Outline the intent and structure of the paper, specifying that the first part covers monochannel sparse image decomposition and the second part focuses on blind sparse source separation, with examples and pointers to software toolboxes provided.

Section 3: Notations
Description 3: Introduce the notations and terminology used throughout the paper to facilitate understanding.

Section 4: Morphological Component Analysis
Description 4: Explain the basics of Morphological Component Analysis (MCA), its assumptions, and the process of creating atomic sparse representations.

Section 5: MCA Algorithm
Description 5: Describe the MCA algorithm, including the steps involved, thresholding techniques, and how coordinates are relaxed during iterations.

Section 6: Dictionary Choice
Description 6: Discuss the importance of choosing an appropriate dictionary for sparse decomposition and the various options available, such as wavelet, curvelet, ridgelet, and other transforms.

Section 7: Thresholding Strategy
Description 7: Detail the different thresholding strategies used in practice, with an emphasis on hard thresholding and the strategies for decreasing thresholds.

Section 8: Handling Bounded Noise
Description 8: Describe how MCA can handle data perturbed by additive noise and provide examples of its application to real images.

Section 9: Applications
Description 9: Present various applications of MCA to different types of real images, including the specifics of the dictionaries used and the results obtained.

Section 10: The Blind Source Separation Problem
Description 10: Introduce the problem of blind source separation (BSS) and the importance of devising quantitative measures of contrast and diversity.

Section 11: Generalized Morphological Component Analysis
Description 11: Explain the Generalized Morphological Component Analysis (GMCA) framework, including its assumptions and the optimization problem it aims to solve.

Section 12: GMCA Algorithm
Description 12: Summarize the GMCA algorithm, comparing it to MCA, and describe the salient-to-fine strategy used to refine source estimations progressively.

Section 13: Unknown Number of Sources
Description 13: Discuss the challenge of estimating the number of sources in BSS and present methods used to approach this problem within the GMCA framework.

Section 14: Hyperspectral Data
Description 14: Explain how GMCA can be adapted to handle hyperspectral data, including the use of spectral waveforms and the optimization problem specific to hyperspectral data.

Section 15: BSS Applications
Description 15: Provide examples of BSS applications using GMCA, including a comparison with other BSS techniques and the quantitative performance measures used.

Section 16: REPRODUCIBLE RESEARCH SOFTWARE
Description 16: Introduce the software toolboxes MCALab and GMCALab, their availability, and their purpose for demonstrating key concepts and allowing researchers to reproduce the experiments.

Section 17: CONCLUSION
Description 17: Conclude the paper by summarizing the advantages of using sparsity and morphological diversity for image decomposition and BSS, and highlight the open problems and future research directions in the field.