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Theses

Le Modèle elliptique de l'équation d'osmose et ses applications

Abstract : This thesis deals with the elliptic osmosis equation and several of itsapplications. One application in particular is the texturation of 3D modelswith multi-date satellite images. The osmosis equation is similar to Poissonequation but with an illumination-invariant data term. It was first introduced forimage editing in a parabolic formulation for a domain with Neumann boundaryconditions.The first chapter describes the elliptic formulation of the osmosis model andseveral of its associated boundary-value problems: Dirichlet, Neumann and mixedboundary-value conditions. Theoretical results for the existence and uniquenessof solutions to these problems are proved for regular domains. These resultsare extended to manifolds with one local chart. The second chapter gives thesame results for the discrete case. For arbitrary domains, Neumann and mixedboundary conditions are difficult to implement with a classic finite differencescheme. For this reason a graph formulation of the problem is introduced thatallows much more flexibility for the manipulation of these boundary conditions.Unlike a finite difference scheme this formulation can be directlyapplied to triangular meshes.The third chapter presents applications of the elliptic osmosis model to theproblems of seamless cloning, shadow removal and image fusion. For seamlesscloning the results are compared to the ones obtained with Poisson editing. Thisshows the interest of having an illumination-invariant term when dealing withinput images whose contrasts are very different. The experiments also presenttheadvantages of solving the problem locally with Dirichlet conditions instead ofon the whole image domain with Neumann boundary conditions.The illumination-invariance of the equation encourages its use for the problemof shadow removal. This application showcases the interest of using mixed boundaryconditions as it allows the user to deal with both cast and attached shadows.This chapter also shows several methods to fuse more than two images of ascene. Several aggregator functions are proposed and the results of thedifferent fusions are compared. It illustrates the interest of PDE-based fusionover the simple fusion of the colour information.A more concrete application related to art is presented in the fourth chapter:the digital restoration of censored medieval illuminations when infraredreflectograms are provided along with the colour images. This applicationneeds the use of the methods already described for seamless cloning andshadow removal. It also showcases the importance of mixed boundary conditions.The last chapter proposes a pipeline to texture a given 3D model frommulti-date satellite images. We automatically detect the shadows,distinguishing the cast and attached shadows. The final texture is a PDE basedfusion of the satellite images weighted by the presence of shadows and theorientation of the satellite sensor.
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Submitted on : Friday, November 29, 2019 - 2:39:36 PM
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Marie de Masson d'Autume. Le Modèle elliptique de l'équation d'osmose et ses applications. Mathématiques générales [math.GM]. Université Paris Saclay (COmUE), 2019. Français. ⟨NNT : 2019SACLN028⟩. ⟨tel-02372750v2⟩

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