Spectral Total-Variation Processing of Shapes -- Theory and Applications

In this work we present a comprehensive analysis of total variation (TV) on non Euclidean domains and its eigenfunctions. We specifically address parameterized surfaces, a natural representation of the shapes used in 3D graphics. Our work sheds new light on the celebrated Beltrami and Anisotropic TV flows, and explains experimental findings from recent years on shape spectral TV [Fumero et al. 2020] and adaptive anisotropic spectral TV [Biton and Gilboa 2022]. A new notion of convexity on manifolds is derived, by characterizing structures that are stable throughout the TV flow, performed on manifolds. We further propose a time efficient nonlinear and non Euclidean spectral framework for shape processing that is based on zero homogeneous flows, and propose three different such methods. Each method satisfies distinct characteristics, demonstrated through smoothing, enhancing and exaggerating filters.