L’électroencéphalographie permet de mesurer l’activité du cerveau à partir des variations du champ électrique à la surface du crâne. Cette mesure est utilisée pour le diagnostic médical, la compréhension du fonctionnement du cerveau ou dans les systèmes d’interface cerveau-machine. De nombreux travaux se sont attachés au développement de méthodes d’analyse de ces signaux en vue d’en extraire différentes composantes d’intérêt, néanmoins leur traitement pose encore de nombreux problèmes. Cette thèse s’intéresse à la mise en place de méthodes permettant l’obtention de représentations redondantes pour ces signaux. Ces représentations se sont avérées particulièrement efficaces ces dernières années pour la description de nombreuses classes de signaux grâce à leur grande flexibilité. L’obtention de telles représentations pour les mesures EEG présente certaines difficultés du fait d’un faible rapport signal à bruit des composantes recherchées. Nous proposons dans cette thèse de les surmonter en guidant les méthodes considérées vers des représentations physiologiquement plausibles des signaux EEG à l’aide de régularisations. Ces dernières sont construites à partir de connaissances a priori sur les propriétés spatiales et temporelles de ces signaux. Pour chacune d’entre elles, des algorithmes sont proposés afin de résoudre les problèmes d’optimisation associés à l’obtention de ces représentations. L’évaluation des approches proposées sur des signaux EEG souligne l’efficacité des régularisations proposées et l’intérêt des représentations obtenues. / The electroencephalography measures the brain activity by recording variations of the electric field on the surface of the skull. This measurement is usefull in various applications like medical diagnosis, analysis of brain functionning or whithin brain-computer interfaces. Numerous studies have tried to develop methods for analyzing these signals in order to extract various components of interest, however, none of them allows to extract them with sufficient reliabilty. This thesis focuses on the development of approaches considering redundant (overcomoplete) representations for these signals. During the last years, these representations have been shown particularly efficient to describe various classes of signals due to their flexibility. Obtaining such representations for EEG presents some difficuties due to the low signal-to-noise ratio of these signals. We propose in this study to overcome them by guiding the methods considered to physiologically plausible representations thanks to well-suited regularizations. These regularizations are built from prior knowledge about the spatial and temporal properties of these signals. For each regularization, an algorithm is proposed to solve the optimization problem allowing to obtain the targeted representations. The evaluation of the proposed EEG signals approaches highlights their effectiveness in representing them.

Overcomplete representations are currently one of the highly researched areas especially in the field of signal processing due to their strong potential to generate sparse representation of signals. Sparse representation implies that given signal can be represented with components that are only rarely significantly active. It has been strongly argued that the mammalian visual system is highly related towards sparse and overcomplete representations. The primary visual cortex has overcomplete responses in representing an input signal which leads to the use of sparse neuronal activity for further processing. This work investigates the sparse coding with an overcomplete basis set representation which is believed to be the strategy employed by the mammalian visual system for efficient coding of natural images. This work analyzes the Sparse Code Learning algorithm in which the given image is represented by means of linear superposition of sparse statistically independent events on a set of overcomplete basis functions. This algorithm trains and adapts the overcomplete basis functions such as to represent any given image in terms of sparse structures. The second part of the work analyzes an inhibition based sparse coding model in which the Gabor based overcomplete representations are used to represent the image. It then applies an iterative inhibition algorithm based on competition between neighboring transform coefficients to select subset of Gabor functions such as to represent the given image with sparse set of coefficients. This work applies the developed models for the image compression applications and tests the achievable levels of compression of it. The research towards these areas so far proves that sparse coding algorithms are inefficient in representing high frequency sharp image features. So this work analyzes the performance of these algorithms only on the natural images which does not have sharp features and compares the compression results with the current industrial standard coding schemes such as JPEG and JPEG 2000. It also models the characteristics of an image falling on the retina after the distortion effects of the eye and then applies the developed algorithms towards these images and tests compression results.