Introduction

The idea of topological asymptotic analysis is to measure the impact of a perturbation of the domain on a cost function. We only consider here the approach that has been introduced for topological optimization purpose, in which the goal is to identify an optimal shape and its complementary in a given domain [83,64,68].

Topological shape optimization seems particularly well adapted to solve image processing problems (like classification, segmentation, enhancement, inpainting, ...), as they mainly consist of identifying a particular subdomain of the image: its edges.

At first sight, the main issue of topological shape analysis is the non-differentiability of the problem. To find the optimal domain is indeed equivalent to identify its characteristic function. Several classical approaches have been developed to make this problem differentiable. We can cite here the relaxation technique, which allows the characteristic function to take all possible values in the interval , and the level set approach where the characteristic function is replaced by a regular level set function which is positive inside the optimal domain and negative outside [83,6,5,7,34,100].

The idea of topological asymptotic analysis is to switch the characteristic function from one to zero (or from zero to one) in a (infinitely) small area. Thus, the variation of the cost function is small when we switch a very small part from the subdomain to its complementary. The topological asymptotic expansion provides this variation, and allows one to derive a topological gradient of the cost function [83,64,101,100].

In this chapter, we first present the basic tools of topological asymptotic analysis, and we then study several applications to image processing problems: inpainting (where the goal is to fill a hidden part of an image), restoration and enhancement, classification, and segmentation. Then, we present a very efficient way to speed up all the algorithms introduced in this chapter, based on discrete cosine transforms and an appropriate preconditioning. Finally, we present a coupled approach combining the topological gradient and the minimal path technique in order to improve the edge detection, and to avoid non-connex contours.