A region-based alternative for edge-preserving smoothing

We examine the problem of edge-preserving smoothing from the perspective of explicit regions rather than the explicit edge approach of Shah (1996), Ambrosio and Tortorelli (1992), and others. We reformulate Mumford and Shah´s (1985) original edge-preserving smoothing functional in terms of regions rather than their boundaries. We then introduce an energy functional that is similar to Ambrosio and Tortorelli´s continuous approximation of Mumford and Shah´s functional, but uses a region label function rather than an edge-strength function. We also show that the spatially discretized version of our energy functional results from a particular continuation method as applied to Mumford and Shah´s energy functional. We base this continuation method on Hummel and Zucker´s (1983) relaxation labeling approach to solving discrete Markov random field labeling problems. We demonstrate this approach for a test case