Convex non-convex segmentation of scalar fields over arbitrary triangulated surfaces

An extension of the Mumford–Shah model for image segmentation is introduced to segment real-valued functions having values on a complete, connected, 2-manifold embedded in R3. The proposed approach consists of three stages: first, a multi-phase piecewise smooth partition function is computed, then its values are clustered and, finally, the curve tracking is computed on the segmented boundaries. The first stage, which constitutes the key novelty behind our proposal, relies on a Convex Non-Convex variational model where an ad-hoc non-convex regularization term coupled with a space-variant regularization parameter allows to effectively deal with both the boundaries and the inner parts of the segments. The cost functional is minimized by means of an efficient numerical scheme based on the Alternating Directions Methods of Multipliers. Experimental results are presented which demonstrate the effectiveness of the proposed three-stage segmentation approach.