# GMCA

## Introduction to blind source separation

Blind source separation (BSS) is a very effective mathematical method to analyze data which are modeled as the linear combination of elementary sources or components. The underlying linear model used so far in BSS is the instantaneous linear mixture model :
$\forall i,\quad x_i = \sum\limits_j a_{i j } s_j + n_j \qquad \mathbf{X} = \mathbf{A} \mathbf{S} + \mathbf{N}$
Following this model, each observation is modeled as the linear combination of the same sources plus some noise/perturbation term. The goal of BSS is to jointly estimate the mixing matrix A and the sources S. This yield an inverse problem that is clearly ill-posed. Attempting to make this problem better posed is classically done by assuming further a priori information about either the sources and/or the mixing matrix.

Since the beginning of the 90s, this problem has been well studied in statistics yielding the famous ICA framework (Independent Component Analysis). In the beginning the XXI-st century, fostered by recent advances in harmonic analysis and statistics, the sparsity of the sources has been put forward as an efficient way to disentangle between the sources.
Based on the concept of morphological diversity, the Generalized Morphological Component Analysis (GMCA) algorithm allows for the separation of sources which are sparse in any signal representation (orthonormal, redundant, … etc). It has been proved to be quite robust and performs very well to process noisy data. In a nutshell, the GMCA algorithm tackles the following optimization problem:
${A, S} =\mathrm{argmin}_{A, S} \sum\limits_j \lambda_j \parallel s_j \mathbf{W} \parallel_1 + \parallel \mathbf{X} – \mathbf{A} \mathbf{S} \parallel_{ F , \Sigma}^2$
This problem is solved using iterative thresholding algorithms with an particularly. An exhaustive description of the GMCA algorithm can be found in this review article.

## GMCA for hyperspectral data analysis

The GMCA algorithm has been further extended to process hyperspectral data. In contrast to usual multispectral data, the number of observations can be large (~100) in the hyperspectral case. Furthermore, the columns of the mixing matrix A are generally related to the electromagnetic spectra of the sought-after components; these spectra generally exhibit some regularities/structures which have sparse distributions in an adequate signal representation (e.g. wavelets).

In the hyperspectral case, the HypGMCA algorithm further enforce the sparsity of the sources as well as the sparsity of the columns of the mixing matrix. A description of the HypGMCA algorithm as well as an illustration on Mars Express data are presented here.

## Local and multiscale mixture model and L-GMCA

The GMCA algorithm assumes that all data sample follows a linear mixture that is described by the same mixing matrix. However, in some physical applications (and more precisely in astrophysics), this model should be replaced by a local mixture model. For the purpose of CMB estimation from microwave data, the GMCA algorithm has been improved to perform the separation of sources assuming a local and multiscale mixture model. This model has particularly been proved to be efficient for the estimation of the CMB map from the Planck data.

J. Bobin, J.-L. Starck, F. Sureau and S. Basak, “Sparse component separation for accurate CMB map estimation” , Astronomy and Astrophysics , 550, A73, 2013.

## Non-negativity and sparsity in BSS

Sparse non-negative matrix factorization or BSS is a very active field. However, properly enforcing sparsity together with non-negativity is non-trivial. The GMCA has recently been extended to handle both constraint in a proper way thanks to recent advances in optimization and more precisely in proximal calculus. Further details can be found at this location.

## Sparse blind source separation in the presence of outliers

While real-world data are often grossly corrupted, most techniques of blind source separation (BSS) give erroneous results in the presence of outliers. We propose two robust algorithms, coined rGMCA and rAMCA, that jointly estimate the sparse sources and outliers without requiring any prior knowledge on the outliers. More precisely, they use an alternative weighted scheme to weaken the influence of the estimated outliers. The latest rAMCA algorithm, based on AMCA, provides good performances in terms of accuracy and reliability, both in the determined and over-determined case. Further details can be found in this report

## Publications

J. Rapin, J. Bobin, A. Larue and J.-L. Starck, “Sparse and Non-negative BSS for Noisy Data”, IEEE Transactions on Signal Processing, 21, 62, pp 5620-5632, 2013.

J. Rapin, J. Bobin, A. Larue and J.-L. Starck, “NMF with Sparse Regularizations in Transformed Domains”, SIAM Journal on Imaging Sciences, 4, 6, pp 2020-2047, 2014.

J. Rapin, J. Bobin, A. Larue and J.-L. Starck, “Sparse and Non-negative BSS for Noisy Data”, IEEE Transactions on Signal Processing, 21, 62, pp 5620-5632, 2013.

Y.Moudden and J.Bobin, Hyperspectral BSS using GMCA with spatio-spectral sparsity constraintsIEEE Transactions on image processing – Vol 20. Issue 3. pages 872-879 (2011)

J.Bobin, J.-L. Starck, Y.Moudden, J. Fadili, Blind Source Separation: the Sparsity Revolution, Advances in Imaging and Electron Physics, Vol. 152, p. 221-298 – Peter W. Hawkes Ed. – 2008.

J.Bobin, J.-L. Starck, J. Fadili, Y.Moudden, Sparsity and Morphological Diversity in Blind Source Separation, IEEE Transactions on Image Processing, Vol.16, N.11, p. 2662-2674, November 2007.

J. Bobin, Y. Moudden, J.-L. Starck and M. Elad, Morphological Diversity and Source Separation, IEEE Signal Processing Letters, Vol.13, N.7, p. 409-412, July 2006.

In Astrophysics :

J. Bobin, F. Sureau, J.-L. Starck, A. Rassat and P. Paykari, “Joint Planck and WMAP CMB Map Reconstruction”, Astronomy and Astrophysics , accepted.

J. Bobin, F. Sureau, P. Paykari, A. Rassat, S. Basak and J.-L. Starck, “WMAP 9-year CMB estimation using sparsity”, Astronomy and Astrophysics , 553, L4, pp 10, 2013.

J. Bobin, J.-L. Starck, F. Sureau and S. Basak, “Sparse component separation for accurate CMB map estimation” , Astronomy and Astrophysics , 550, A73, 2013.

E. Chapman, F. Abdalla, J.Bobin, J.L. Starck, G. Harker, V. Jelic, P. Labropoulos, S. Zaroubi, M. Brentjens, A. De Bruyn and L. Koopmanx, The scale of the problem: recovering images of reionization with GMCA, MNRAS, 459, 2013.

J.Bobin, Y.Moudden, J.-L. Starck, J. Fadili, N. Aghanim, SZ and CMB reconstruction using GMCA, Statistical Methodology, 2008 – Vol. 4, p.307-317.

## The GMCALab toolbox

A lightweight Matlab/Octave version of the GMCALab toolbox can be found at this location.

We are now developping a python-based toolbox coined pyGMCALab, which is available at this location.