Affiliation:
1. Technion - Israel Institute of Technology, Israel
2. Helmholtz Imaging, Deutsches Elektronen-Synchrotron DESY, Germany and Fachbereich Mathematik, Universität Hamburg, Germany
Abstract
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 surfaces is derived by characterizing structures that are stable throughout the TV flow, performed on surfaces. We establish and numerically demonstrate quantitative relationships between TV, area, eigenvalue, and eigenfunctions of the TV operator on surfaces. Moreover, we expand the shape spectral TV toolkit to include zero-homogeneous flows, leading to efficient and versatile shape processing methods. These methods are exemplified through applications in smoothing, enhancement, and exaggeration filters. We introduce a novel method which, for the first time, addresses the shape deformation task using TV. This deformation technique is characterized by the concentration of deformation along geometrical bottlenecks, shown to coincide with the discontinuities of eigenfunctions. Overall, our findings elucidate recent experimental observations in spectral TV, provide a diverse framework for shape filtering, and present the first TV-based approach to shape deformation.
Publisher
Association for Computing Machinery (ACM)
Subject
Computer Graphics and Computer-Aided Design
Reference76 articles.
1. Scale Invariant Geometry for Nonrigid Shapes
2. Minimizing total variation flow;Andreu F.;Differential and Integral Equations,2001
3. Fuensanta Andreu Coloma Ballester Vicent Caselles and José M Mazón. 2001. Minimizing total variation flow. (2001).
4. Jean-François Aujol, Guy Gilboa, Tony Chan, and Stanley Osher. 2006. Structure-texture image decomposition—modeling, algorithms, and parameter selection. International journal of computer vision 67 (2006), 111–136.
5. The Total Variation Flow in RN