Edge Detection Using Topological Gradients: A Scale-Space Approach

We provide in this paper a link between two methods of edge detection: edge detection using scale-space analysis, and edge detection using topological asymptotic analysis. More precisely, we show that the topological gradient associated with an image u is given by a combination of the gradients of two smoothed versions of the image u at two different scales, namely phi star u and (phi star phi) star u, where phi is the fundamental solution of the elliptic restoration equation. In the same setting, we propose a new edge detector based on the maximization of the variance of the image. Then we extend our approach to Gaussian kernels through a topological asymptotic analysis of the parabolic heat equation. A numerical comparison of these detectors together with the Canny edge detector is presented.