Area-preserving curve-shortening flows: from phase separation to image processing

Abstract

Some known models in phase separation theory (Hele-Shaw, nonlocal mean curvature motion) and their approximations by means of Cahn-Hilliard and nonlocal Allen-Cahn equations are proposed as a tool to generate planar curve-shortening flows without shrinking. This procedure can be seen as a level set approach to area-preserving geometric flows in the spirit of Sapiro and Tannenbaum [38], with application to shape recovery. We discuss the theoretical validation of this method and its implementation to problems of shape recovery in Computer Vision. The results of some numerical experiments on image processing are presented.