Upper bounds on the coarsening rate of discrete, ill-posed nonlinear diffusion equations

We prove a weak upper bound on the coarsening rate of the discrete-in-space version of an ill-posed, nonlinear diffusion equation. The continuum version of the equation violates parabolicity and lacks a complete well-posedness theory. In particular, numerical simulations indicate very sensitive dependence on initial data. Nevertheless, models based on its discrete-in-space version, which we study, are widely used in a number of applications, including population dynamics (chemotactic movement of bacteria), granular flow (formation of shear bands), and computer vision (image denoising and segmentation). Our bounds have implications for all three applications. © 2008 Wiley Periodicals, Inc.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/61227/1/20259_ftp.pd

On a coupled PDE model for image restoration

In this paper, we consider a new coupled PDE model for image restoration. Both the image and the edge variables are incorporated by coupling them into two different PDEs. It is shown that the initial-boundary value problem has global in time dissipative solutions (in a sense going back to P.-L. Lions), and several properties of these solutions are established. This is a rough draft, and the final version of the paper will contain a modelling part and numerical experiments

PDE-Based Signal Quantization

La version librement accessible est celle du working paper, intitulé "PDE model with memory term applied to signal quantization. Mathematical and numerical analysis."We presente a new method for signal restoration/quantization based on diffusion reaction model with memory term. We prove that the model is stable, with the existence and uniqueness results. We also propose a numerical approximation that we prove the convergence and present some experiments on noisy signals.ou

Information denoising and quantization by diffusion reaction model

We present a new diffusion reaction model for signal denoising and quantization. We first discuss on classical quantization methods and present the most popular denoising models. Then we construct our method as a combination of quantization and denoising terms and show its efficiency on noisy signals

Equations aux dérivées partielles appliquées à la restauration et à l'agrandissement des images

Antonin Chambolle (Directeur de thèse) Jean-Michel Morel (Président) Gilles Aubert (Raporteur) George-Henri Cottet (Raporteur) Françoise dibos (Examinateur) Frédéric Guichard (Examinateur)This thesis consists in two parts in which we study two problems of computer vision : image restoration in the first part and image zooming in the second part. In the introduction of our first part we present the main mathematical models that have been proposed for the restoration (denoising and edge enhancement) of digital images. Then we discuss edge-detection theory and the Malik and Perona model. We introduce various variants that have been proposed to stabilized the (ill-posed) Malik and Perona equation and in particular the model which we will study in the following chapters. For this model in all dimensions, we prove in small time existence and uniqueness of a classical solution. We then construct (in one and two dimensions) a numerical approximation and prove its convergence to a weak solution. As conclusion of this part, we present some experiments. We begin the second part by presenting existing image zooming methods. Using geometrical arguments, we propose a new approach based on a partial differential equation. Next we prove the well posedness of the model using the theory of the viscosity solutions. We then discuss the discretisation of the model, and finally present some experiments.Dans ce travail, nous avons étudié deux problèmes fondamentaux de la vision : la restauration d'image dans la première partie, puis dans la deuxième partie l'agrandissement d' image. L'introduction de la première partie est une présentation des méthodes mathématiques de restauration d'image. En suite, nous présentons les différents travaux théoriques établis sur le modèle de Malik et Perona et son lien avec la doctrine de détection de bord. Ceci nous conduit au choix du modèle que nous étudions dans la suite. Pour ce modèle nous prouvons en petits temps et en toutes dimensions, existence et unicité d'une solution classique. Ensuite, en dimensions un et deux, nous construisons un schéma numérique dont nous prouvons la convergence vers une solution faible. Nous clôturons cette partie par illustrer quelques exemples d'application du modéle. Dans la deuxième partie nous abordons le problème de l'agrandissement des images. Après avoir exposé les méthodes qui existent dans la littérature, nous proposons une approche basée sur l'analyse géométrique de l'image. Nous utilisons ensuite la théorie des solutions de viscosité pour prouver que le modèle proposé est bien posé. Enfin nous discutons de la résolution numérique du modèle et nous présentons quelques exemples d'applications

TIME-DELAY REGULARIZATION OF ANISOTROPIC DIFFUSION AND IMAGE PROCESSING

02-2004We study a time-delay regularization of the anisotropic diffusion model for image denoising of Malik and Perona, which has been proposed by Nitzberg and Shiota. In the two-dimensional case, we show the convergence of a numerical approximation and the existence of a weak solution. Finally, we show some experiments on images