Topological derivative: A tool for image processing

The increasing complexity in several fields of science and technology has motivated the use of techniques originally conceived in other areas of applications. An illustrative example of this is given by the topological derivative which quantifies the sensitivity of a problem when the domain under consideration is perturbed by changing its topology. This concept, initially conceived to deal with topology optimization problems, has also been successfully applied to inverse problems and material properties characterization. Our aim in this paper is to present an other field of application for the topological derivative: image processing. An appropriate functional and a variational problem are associated to the cost endowed to an specific image processing application. Thus, the corresponding topological derivative can be used as an indicator function that leads, through a minimization process, to the processed image. We focus our attention on two image processing application. In the first, the topological derivative is used in image restoration, i.e. to restore an image that was somehow degraded (acquisition process, transmission, storage, etc.). Moreover, a novel fully discrete algorithm based on the topological derivative concept is presented. In the second application, we use the topological derivative to derive a “continuous” and a fully discrete novel image segmentation algorithms, i.e. for objects identification in an image. Finally and in order to show the performance of these algorithms, several numerical examples are also presented in this work.