Morphological Diversity and Sparsity for Multichannel Data Restoration

International audienceOver 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

Morphological diversity and sparsity : new insights into multivariate data analysis

International audienceOver 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

Sparsity and morphological diversity for multivalued data analysis

International audienceThe 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 work1 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

Sparsity constraints for hyperspectral data analysis: linear mixture model and beyond

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

Sparsity and morphological diversity for hyperspectral data analysis

Recently morphological diversity and sparsity have emerged as new and effective sources of diversity for Blind Source Separation. Based on these new concepts, novelmethods such as Generalized Morphological Component Analysis have been put forward. The latter takes advantage of the very sparse representation of structured data in large overcomplete dictionaries, to separate sources based on their morphology. Building on GMCA, the purpose of this contribution is to describe a new algorithm for hyperspectral data processing. Large-scale hyperspectral data refers to collected data that exhibit sparse spectral signatures in addition to sparse spatial morphologies, in specified dictionaries of spectral and spatial waveforms. Numerical experiments are reported which demonstrate the validity of the proposed extension for solving source separation problems involving hyperspectral data

Image Decomposition and Separation Using Sparse Representations: An Overview

This paper gives essential insights into the use of sparsity and morphological diversity in image decomposition and source separation by reviewing our recent work in this field. The idea to morphologically decompose a signal into its building blocks is an important problem in signal processing and has far-reaching applications in science and technology. Starck , proposed a novel decomposition method—morphological component analysis (MCA)—based on sparse representation of signals. MCA assumes that each (monochannel) signal is the linear mixture of several layers, the so-called morphological components, that are morphologically distinct, e.g., sines and bumps. The success of this method relies on two tenets: sparsity and morphological diversity. That is, each morphological component is sparsely represented in a specific transform domain, and the latter is highly inefficient in representing the other content in the mixture. Once such transforms are identified, MCA is an iterative thresholding algorithm that is capable of decoupling the signal content. Sparsity and morphological diversity have also been used as a novel and effective source of diversity for blind source separation (BSS), hence extending the MCA to multichannel data. Building on these ingredients, we will provide an overview the generalized MCA introduced by the authors in and as a fast and efficient BSS method. We will illustrate the application of these algorithms on several real examples. We conclude our tour by briefly describing our software toolboxes made available for download on the Internet for sparse signal and image decomposition and separation

Blind Source Separation: the Sparsity Revolution

International audienceOver 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

Image decomposition and separation using sparse representations: an overview

International audienceThis paper gives essential insights into the use of sparsity and morphological diversity in image decomposition and source separation by overviewing our recent work in this field. The idea to morphologically decompose a signal into its building blocks is an important problem in signal processing and has far-reaching applications in science and technology. Starck et al. [1], [2] proposed a novel decomposition method - Morphological Component Analysis (MCA) - based on sparse representation of signals. MCA assumes that each (monochannel) signal is the linear mixture of several layers, the so-called Morphological Components, that are morphologically distinct, e.g. sines and bumps. The success of this method relies on two tenets: sparsity and morphological diversity. That is, each morphological component is sparsely represented in a specific transform domain, and the latter is highly inefficient in representing the other content in the mixture. Once such transforms are identified, MCA is an iterative thresholding algorithm that is capable of decoupling the signal content. Sparsity and morphological diversity have also been used as a novel and effective source of diversity for blind source separation (BSS), hence extending the MCA to multichannel data. Building on these ingredients, we will overview the Generalized MCA (GMCA) introduced by the authors in [3], [4] as a fast and efficient BSS method. We will illustrate the application of these algorithms on several real examples. We conclude our tour by briefly describing our software toolboxes made available for download on the Internet for sparse signal and image decomposition and separation

Learning the Morphological Diversity

International audienceThis article proposes a new method for image separation into a linear combination of morphological components. Sparsity in global dictionaries is used to extract the cartoon and oscillating content of the image. Complicated texture patterns are extracted by learning adapted local dictionaries that sparsify patches in the image. These global and local sparsity priors together with the data fidelity define a non-convex energy and the separation is obtained as a stationary point of this energy. This variational optimization is extended to solve more general inverse problems such as inpainting. A new adaptive morphological component analysis algorithm is derived to find a stationary point of the energy. 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