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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.