Generalized morphological component analysis (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. Building on GMCA, the purpose of this contribution is to describe a new algorithm for BSS applications in hyperspectral data processing. It assumes the collected data is a 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.

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.

Recently morphological diversity and sparsity have emerged as new and effective sources of diversity for blind source separation. Based on these new concepts, novel methods 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.

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-caIled 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 emergedas 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.

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 l1 norm, but MCA is much faster and gives us the possibility of handling large scale data sets.

This paper describes two extensions of the recent morphological component analysis (MCA) method to multichannel data. MCA takes advantage of the sparse representation of structured data in large overcomplete dictionaries to separate features in the data based on their morphology. It was shown to be an efficient technique in such problems as separating an image into texture and piecewise smooth parts or for inpainting applications. A first extension, MMCA, achieves a similar source separation objective based on morphological diversity. A second extension, GMMCA, takes advantage of the highly sparse representations of the sources that can be built using MCA. Indeed, sparsity is now generally recognized as a valuable property for blind source separation. The efficiency of MMCA and GMMCA is confirmed in numerical experiments

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

Spherical maps occur in a range of applications for instance in geophysics or in astrophysics with the study of the cosmic microwave background (CMB) radiation field, where observations are over the whole sky. Analyzing these images requires specific tools. This paper describes a new multiscale decomposition for data on the sphere, namely the curvelet transform on the sphere. The curvelet transform, in its first step, requires the use of an isotropic wavelet transform. Therefore, our new curvelet transform also includes a new wavelet transform on the sphere which has properties similar to those of the a trous isotropic wavelet transform.

In this paper, a real-time CAR (Continuous AR) parameter estimation method from randomly sampled observations is presented. For this purpose, a new concept is introduced: the Pseudo Correlation Vector. This vector, which is equal to the Correlation Vector in the uniform sampling case, reflects the statistical dependencies between successive values of the CAR signal. As a matter of fact, being the limit of a series recursively related to the observations, its real-time estimation becomes especially easy. An inversion of its dependence on the CAR parameters leads then to an estimate of the latter. Theoretical and simulation results regarding the proposed estimator are given.