Image processing by topological asymptotic analysis

The aim of this article is to recall the applications of the topological asymptotic expansion to many image processing problems. We briefly review the topological asymptotic analysis. 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 problem. The identification of the main edges of the image allows us to preserve them, and to smooth the image outside the edges. We present then an application to the regularized and unsupervised classification problem. If the conductivity outside the edges goes to infinity, the regularized image is piecewise constant and provides a natural solution to the segmentation problem. We also mention that all these problems are solved with a $\mathcal{O}(n.\log(n))$ complexity. Finally, we present a hybrid scheme for contour detection and completion based on topological gradient and fast marching algorithms. We briefly illustrate all these applications with numerical results