Curve-and-Surface Evolutions for Image Processing

Abstract : The goal of this manuscript was to study several problems which appear in image processing and which involves hypersurfaces of the Euclidian space R^n. Denoising a image basically consists in smoothing its lines. This smoothing can appear either as a minimizer of a suitable functional or results from a regularizing flow on the level sets of the image. In this thesis, we study two examples of these approaches. In the first chapter, we smooth by minimization. More precisely, we work on generalizations of the procedure suggested by Rudin Osher and Fatemi, which penalizes the total variation. We prove that under different assumptions on the domain, on the way to link the image to the data and on the choice of the total variation (isotropic, anisotropic,...), the continuity of the source image is preserved by the minimizing procedure. In Chapter 2, we study Mean curvature flow and add some obstacles which constraint the evolution. We choose the level-set approach: the surface is the preimage of 0 by a function which therefore satisfies a PDE. We prove existence and uniqueness of a (viscosity) solution for this equation. and study its asymptotic in time using comparison with a discrete minimizing scheme. In Chapter 3 (with M. Novaga), we add some information to the result of Chapter 2 by focusing on the geometric formulation of the mean curvature flow with obstacles. We follow the approach by Ecker and Huisken to show that there exists a unique solution of the motion in short times. Finally, in the last chapter (with M. Novaga and P. Pozzi), we make a first step towards the understanding of crystalline motion. Restricted to the planar framework, we show (using an approximation by a smooth motion) that there exists a short time of existence for an anisotropic curvature motion of an immersed curve.
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Submitted on : Thursday, October 22, 2015 - 4:05:26 PM
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  • HAL Id : tel-01219407, version 1

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Gwenael Mercier. Curve-and-Surface Evolutions for Image Processing. Analysis of PDEs [math.AP]. École polytechnique 2015. English. ⟨tel-01219407⟩

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