L'Analyse en Composantes Indépendantes : un outil puissant pour le traitement de l'information

- L'analyse en composantes indépendantes (ACI) est une approche très générale qui a été développée pour résoudre le problème de séparation aveugle de sources (SAS) indépendantes. Au-delà de ce problème, l'ACI s'applique à l'étude de la représentation parcimonieuse de données liées à des phénomènes cognitifs complexes. Cet article est constitué de deux parties. La première est une synthèse sur les principes de la séparation de sources dans laquelle sont abordés la question de la séparabilité, les critères d'indépendance, et l'utilisation d'informations a priori. La seconde partie présente les approches de la SAS fondées sur des représentations parcimonieuses des signaux et montre comment elles permettent de traiter des problèmes sous-déterminés, et comment elles sont liées au codage parcimonieux

Sparse image approximation with application to flexible image coding

Natural images are often modeled through piecewise-smooth regions. Region edges, which correspond to the contours of the objects, become, in this model, the main information of the signal. Contours have the property of being smooth functions along the direction of the edge, and irregularities on the perpendicular direction. Modeling edges with the minimum possible number of terms is of key importance for numerous applications, such as image coding, segmentation or denoising. Standard separable basis fail to provide sparse enough representation of contours, due to the fact that this kind of basis do not see the regularity of edges. In order to be able to detect this regularity, a new method based on (possibly redundant) sets of basis functions able to capture the geometry of images is needed. This thesis presents, in a first stage, a study about the features that basis functions should have in order to provide sparse representations of a piecewise-smooth image. This study emphasizes the need for edge-adapted basis functions, capable to accurately capture local orientation and anisotropic scaling of image structures. The need of different anisotropy degrees and orientations in the basis function set leads to the use of redundant dictionaries. However, redundant dictionaries have the inconvenience of giving no unique sparse image decompositions, and from all the possible decompositions of a signal in a redundant dictionary, just the sparsest is needed. There are several algorithms that allow to find sparse decompositions over redundant dictionaries, but most of these algorithms do not always guarantee that the optimal approximation has been recovered. To cope with this problem, a mathematical study about the properties of sparse approximations is performed. From this, a test to check whether a given sparse approximation is the sparsest is provided. The second part of this thesis presents a novel image approximation scheme, based on the use of a redundant dictionary. This scheme allows to have a good approximation of an image with a number of terms much smaller than the dimension of the signal. This novel approximation scheme is based on a dictionary formed by a combination of anisotropically refined and rotated wavelet-like mother functions and Gaussians. An efficient Full Search Matching Pursuit algorithm to perform the image decomposition in such a dictionary is designed. Finally, a geometric image coding scheme based on the image approximated over the anisotropic and rotated dictionary of basis functions is designed. The coding performances of this dictionary are studied. Coefficient quantization appears to be of crucial importance in the design of a Matching Pursuit based coding scheme. Thus, a quantization scheme for the MP coefficients has been designed, based on the theoretical energy upper bound of the MP algorithm and the empirical observations of the coefficient distribution and evolution. Thanks to this quantization, our image coder provides low to medium bit-rate image approximations, while it allows for on the fly resolution switching and several other affine image transformations to be performed directly in the transformed domain

Sparse Decompositions in ``incoherent'' dictionaries

The purpose of this paper is to generalize a result by Donoho, Huo, Elad and Bruckstein on sparse representations of signals/images in a union of two orthonormal bases. We consider general (redundant) dictionaries in finite dimension, and derive sufficient conditions on a signal/image for having a unique sparse representation in such a dictionary. In particular, it is proved that the result of Donoho and Huo, concerning the replacement of a combinatorial optimization problem with a linear programming problem when searching for sparse representations, has an analog for dictionaries that may he highly redundant. The special case where the dictionary is given by a union of several orthonormal bases is studied in more detail and some examples are given