J Math Imaging Vis (2009) 33: 122–134 DOI 10.1007/s10851-008-0121-2 Image Processing by Topological Asymptotic Expansion

Abstract The aim of this article is to recall the applications of the topological asymptotic expansion to major image processing problems. We briefly review the topological as-ymptotic analysis, and then present its historical application to the crack localization problem from boundary measure-ments. A very natural application of this technique in image processing is the inpainting problem, which can be solved by identifying the optimal localization of the missing edges. A second natural application is then the image restoration or enhancement. The identification of the main edges of the image allows us to preserve them, and smooth the image outside the edges. If the conductivity outside edges goes to infinity, the regularized image is piecewise constant and pro-vides a natural solution to the segmentation problem. The numerical results presented for each application are very promising. Finally, we must mention that all these problems are solved with a O(n. log(n)) complexity