Macq, BenoîtCole, LucLucCole2025-02-052025-02-052004https://dial-mem.test.bib.ucl.ac.be/handle/123456789/42453This work adopts the notion of feature extraction for three-dimensional Magnetic Resonance Images of Multiple Sclerosis lesions. Multiple sclerosis is a progressive disease that requires an evolution study through time. The evolution of the disease can be followed on a patient with a temporal series of examinations that produce three-dimensional images of the brain. The purpose is to segment the Multiple Sclerosis lesions using a sophisticated computerized algorithm based on interface evolution. Image segmentation is the process of extracting objects from a background thus partitioning the image into distinct regions. There exist three main methods for segmentation: Edge-based methods, pixel-based classification methods and region-based methods. This work gives a review of the mathematical background and the application for a specific region-based method called the level set method introduced by J.A. Sethian from University of California, Berkeley. The level let method is in fact a numerical technique, which can follow the evolution of interfaces. These interfaces can develop sharp corners, break apart, and merge together. This method is a powerful numerical technique for tracking the evolution of interfaces moving under a variety of complex motions. Moreover, the level let approach is based on computing viscosity solutions to the appropriate equations of motion, using techniques borrowed from hyperbolic conservation laws. This work considers the evolution of a front propagating along its normal vector field with curvature dependent speed. Furthermore, it defines an "energy- like" quantity of the moving front, the total variation, and it shows a general result relating the growth/decay of this energy to the speed. In addition it studies a front moving with speed dependent on the curvature, and shows that the curvature term plays a smoothing role in the solution. When the speed is constant, the solution is seen to blow up, differentiability is lost, and an entropy condition can be formulated to provide an explicit construction for a weak solution beyond blow up time. The equations of motion are numerically solved and it is also showed that the solution converges to the constructed weak solution as the curvature smoothing term vanishes. Finally, there is a discussion on the difficulties involved in numerically solving such problems and a possible remedy is described.text::thesis::master thesisthesis:49497