Fonctionnelles d'Ambrosio-Tortorelli : approximation via la convergence Γ de la fonctionnelle de Mumford-Shah

François, Corentin
(2020)

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Supervisors
Olbermann, Heiner
Faculty
Faculté des sciences
Degree label
Master [120] en sciences mathématiques
Abstract
enIn this thesis, we study the Mumford-Shah functional, introduced by the two eponymous mathematicians in image segmentation theory. More precisely, we introduce the Ambrosio-Tortorelli functionals, functionals depending on a parameter h designed to Γ-converge to the Mumford-Shah functional. This special type of convergence guarantee, with some little extra conditions, the convergence of the minimizers of the approximating functionals towards the minimizers of the limit functional. In this way, we will prove the existence of minimizers for the Mumford-Shah functional and provide a practical way to approximate them. Through this thesis, we will also briefly examine the Hausdorff measures and bounded variation functions theories.