Shape Segmentation and Denoising Based on Total-Variation Functionals

We study variational methods for non-smooth discrete shape processing problems involving total variation regularization.

First, we investigate the proximal map of the weighted mean absolute error functional by presenting an efficient algorithm for its vectorized evaluation and showing 2 large-scale applications of the algorithm.

Further, we address the problem of shape segmentation for triangular meshes. We propose a variational formulation employing total-variation-based regularization and compare two different regularizers: one acting directly on the assignment function and one defined in the label space. The resulting optimization problems are solved using tailored split Bregman (ADMM) schemes. Our experiments show that the label space regularizer, while computationally more expensive, yields superior segmentation quality and improved robustness in regions of constant curvature. In order to mitigate the computational cost, we implement a specialized Riemannian Newton method for the most expensive step of the ADMM iteration, where the optimization domain is a product manifold of spheres.

Finally, we introduce a novel variational framework for geometry denoising that incorporates prior knowledge about the shape’s normal vectors in the form of preferred label vectors. In this setting, segmentation arises naturally as an integral component of the denoising process. Regularization is achieved through a total-variation term, and the optimization problem is solved using a split Bregman (ADMM) approach. We show that the model yields effective noise removal while preserving geometric features consistent with the prescribed normal vectors.