The Cauchy problem for a nonlinear elliptic equation: Nash-game approach and application to image inpainting

Moez  Kallel; Maher  Moakher; Anis  Theljani

doi:10.3934/ipi.2015.9.853

Article Contents

2015, Volume 9, Issue 3: 853-874. Doi: 10.3934/ipi.2015.9.853

This issue Previous Article Point-wise behavior of the Geman--McClure and the Hebert--Leahy image restoration models Next Article A reweighted $l^2$ method for image restoration with Poisson and mixed Poisson-Gaussian noise

The Cauchy problem for a nonlinear elliptic equation: Nash-game approach and application to image inpainting

1.
Laboratoire LAMSIN, ENIT, University of Tunis El Manar, B.P. 37, 1002 Tunis-Belvédère

Received: May 31, 2014

Revised: February 28, 2015

Published: July 2015

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Abstract

Image inpainting or disocclusion, which refers to the process of restoring a damaged image with missing information, has many applications in different fields. Different techniques can be applied to solve this problem. In particular, many variational models have appeared in the literature. These models give rise to partial differential equations for which Dirichlet boundary conditions are usually used. The basic idea of the algorithms that have been proposed in the literature is to fill-in damaged regions with available information from their surroundings. The aim of this work is to treat the case where this information is not available in a part of the boundary of the damaged region. We formulate the image inpainting problem as a nonlinear Cauchy problem. Then, we give a Nash-game formulation of this Cauchy problem and we present different numerical experiments using the finite-element method for solving the image inpainting problem.

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Mathematics Subject Classification: Primary: 65M32, 68U10; Secondary: 35G25, 91A10.

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