Robust Focalized Brain Activity Reconstruction using ElectroEncephaloGrams

This thesis is developed around the problem of brain activity reconstruction/identification using non-invasive ElectroEncephaloGrams. Multiple research lines branch from the central research line, some of them constituting distinct reconstruction methods, and others serving as aiding tools to the reconstructions. Noticeably, the application area of some of the proposed tools is broader than the original considered problem. The presented algorithms share the common preoccupation for statistical robustness and focalization, aiming to provide the users of the methods with solutions allowing as much as possible straightforward identification/discrimination of active areas from inactive areas. The work tackles first the problem of 2D surfacic cortical reconstruction, proposing an anisotropic edge-preserving vector diffusion algorithm, inspired by the relatively recent emergence of diffusion algorithms in the image processing community. Subsequently, the 3D volumetric reconstruction problem is attacked under the perspective of statistical estimation. A thorough analysis of the reconstruction bounds, based on the introduction of the sensitivity functions and on the Cramér-Rao lower bound, results in principles for optimal system design. Then an alternate statistical modeling is proposed for non-Gaussian distributions, under the form of Infinite Mixture of Gaussians, using the newly introduced Gaussian Transform of distributions. Relying on the obtained results, new generic estimation algorithms are proposed for non-Gaussian data and tested in a denoising application. Their extension to the EEG inverse problem produces a new class of EEG inversion methods, yielding robust focalized solutions under non-Gaussian priors.