In this section, we present an application of the topological asymptotic analysis to the inpainting problem. The goal of inpainting is to fill a hidden part of an image. In other words, if we denote by 
the original image and 
the hidden part of the image, the goal is to recover the hidden part
from the known part of the image
. There are many applications, for instance removing some spots on a badly preserved movie or image, or deleting encrusted logos and images on television programs, ...
This problem has been widely studied. Several methods have been considered: learning approaches (neural networks, radial basis functions, support vector machine, ...), in which the learning data is taken in
, and then the approximate function is evaluated in
[116,117]; minimization of an energy cost function in
based on a total variation norm [46,47]; morphological analysis for the reconstruction of both cartoon and texture [58]; ...
In order to study the inpainting problem, we first consider a crack localization method. Crack detection allows us to identify the edges of the hidden part of the image, and the inpainting problem can then be easily solved. We will consider the classical thermal diffusion technique [89,45,113,114,95] and improve it by modeling the edges by cracks. These cracks are supposed to be highly insulating and to allow the temperature to jump across edges. As both the Dirichlet and Neumann conditions are known on the boundary of the hidden subset, we can define a criterion measuring the discrepancy between the solutions of a Dirichlet and a Neumann problem respectively [75]. This problem is similar to the inverse conductivity problem, also known as the Calderón problem [44], which consists of identifying the coefficients of a partial differential equation from the knowledge of the Dirichlet to Neumann operator. Only two measurements are needed to recover several simple cracks [3,4,31]. From the numerical point of view, several methods [10,32,33,43,63,97,96] have been proposed, but the topological gradient approach seems to be the most efficient method for crack localization. The minimization of the criterion allows us to identify the main edges inside the hidden part of the image. The image is finally filled between the edges thanks to the Laplace operator.
This section summarizes the work introduced in [26,27]. We also refer to these references for the results of many numerical experiments.