A variant of the level set method and applications to image segmentation

In this paper we propose a variant of the level set formulation for identifying curves separating regions into different phases. In classical level set approaches, the sign of $n$ level set functions are utilized to identify up to $2^n$ phases. The novelty in our approach is to introduce a piecewise constant level set function and use each constant value to represent a unique phase. If $2^n$ phases should be identified, the level set function must approach $2^n$ predetermined constants. We just need one level set function to represent $2^n$ unique phases, and this gains in storage capacity. Further, the reinitializing procedure requested in classical level set methods is superfluous using our approach. The minimization functional for our approach is locally convex and differentiable and thus avoids some of the problems with the nondifferentiability of the Delta and Heaviside functions. Numerical examples are given, and we also compare our method with related approaches.