Total Generalized Variation of the Normal Vector Field and Applications to Mesh Denoising

2025/07/17 by Lukas P. Baumgartner, Baumgärtner, Lukas, Ronny Bergmann +7
Engineering · Mathematics · Physics and Astronomy · #Computer Vision and Pattern Recognition (cs.CV) #Differential Geometry (math.DG) #FOS: Computer and information sciences #FOS: Mathematics #Model Reduction and Neural Networks #Numerical methods in engineering #Numerical methods in inverse problems #Optimization and Control (math.OC)

paper · pdf · doi:10.48550/arxiv.2507.13530

openalex publication_date 2025/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We propose a novel formulation for the second-order total generalized variation (TGV) of the normal vector on an oriented, triangular mesh embedded in \R³. The normal vector is considered as a manifold-valued function, taking values on the unit sphere. Our formulation extends previous discrete TGV models for piecewise constant scalar data that utilize a Raviart-Thomas function space. To extend this formulation to the manifold setting, a tailor-made tangential Raviart-Thomas type finite element space is constructed in this work. The new regularizer is compared to existing methods in mesh denoising experiments.

Citations

• Total Generalized Variation for Piecewise Constant Functions on Triangular Meshes with Applications in Imaging
• Mesh Total Generalized Variation for Denoising
• Non‐Local Low‐Rank Normal Filtering for Mesh Denoising
• Unified Models for Second-Order TV-Type Regularisation in Imaging - A\n New Perspective Based on Vector Operators
• Gmsh: A 3‐D finite element mesh generator with built‐in pre‐ and post‐processing facilities
• Nonlinear total variation based noise removal algorithms

Related

• Total Generalized Variation for Piecewise Constant Functions on Triangular Meshes with Applications in Imaging
• Mesh Total Generalized Variation for Denoising
• A Second-Order TGV Discretization with 90^∘ Rotational Invariance Property
• Non-smooth variational regularization for processing manifold-valued data
• Compact Representation of n-th order TGV