Convex Non-Convex Segmentation over Surfaces

Abstract

The paper addresses the segmentation of real-valued functions having values on a complete, connected, 2-manifold embedded in ${{\mathbb {R}}}^3$. We present a three-stage segmentation algorithm that first computes a piecewise smooth multi-phase partition function, then applies clusterization on its values, and finally tracks the boundary curves to obtain the segmentation on the manifold. The proposed formulation is based on the minimization of a Convex Non-Convex functional where an ad-hoc non-convex regularization term improves the treatment of the boundary lengths handled by the $\ell _1$ norm in [2]. An appropriate numerical scheme based on the Alternating Directions Methods of Multipliers procedure is proposed to efficiently solve the nonlinear optimization problem. Experimental results show the effectiveness of this three-stage procedure.