Variational approach for edge-preserving regularization using coupled PDEs

This paper deals with edge-preserving regularization for inverse problems in image processing. We first present a synthesis of the main results we have obtained in edge-preserving regularization by using a variational approach. We recall the model involving regularizing functions ʹ and we analyze the geometry-driven diffusion process of this model in the threedimensional (3-D) case. Then half-quadratic theorem is used to give a very simple reconstruction algorithm. After a critical analysis of this model, we propose another functional to minimize for the edge-preserving reconstruction purpose. It results in solving two coupled partial differential equations (PDE’s): one processes the intensity, the other the edges. We study the relationship with similar PDE systems in particular with the functional proposed by Ambrosio–Tortorelli [1], [2] in order to approach the Mumford–Shah functional [3] developed in the segmentation application. Experimental results on synthetic and real images are presented.