Bregman-Proximal Augmented Lagrangian Approach to Multiphase Image Segmentation

Abstract

This work studies the optimization problem of assigning multiple labels with the minimum perimeter, namely Potts model, in the spatially continuous setting. It was extensively studied within recent years and used to many different applications of image processing and computer vision, especially image segmentation. The existing convex relaxation approaches use total-variation functionals directly encoding perimeter costs, which result in pixelwise simplex constrained optimization problems and can be efficiently solved under a primal-dual perspective in numerics. Among most efficient approaches, such challenging simplex constraints are tackled either by extra projection steps to the simplex set at each pixel, which requires intensive simplex-projection computations, or by introducing extra dual variables resulting in the dual optimization-based continuous max-flow formulation to the studied convex relaxed Potts model. However, dealing with such extra dual flow variables needs additional loads in both computation and memory; particularly for the cases with many labels. To this end, we propose a novel optimization approach upon the Bregman-Proximal Augmented Lagrangian Method (BPALM), for which the Bregman distance function, instead of the classical quadratic Euclidean distance function, is integrated in the algorithmic framework of Augmented Lagrangian Methods. The new optimization method has significant numerical advantages; it naturally avoids extra computational and memory burden in enforcing the simplex constraints and allows parallel computations over different labels. Numerical experiments show competitive performance in terms of quality and significantly reduced memory load compared to the state-of-the-art convex optimization methods for the convex relaxed Potts model.