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

2025-07-17

AI Summary

This work addresses normal vector field denoising on 3D triangular meshes. We propose the first second-order Total Generalized Variation (TGV²) regularization model for spherical manifold-valued functions. Our method constructs a tangential Raviart–Thomas-type finite element space to extend classical TGV to non-Euclidean geometry while rigorously preserving the unit-norm constraint on normals. By integrating discrete variational modeling with manifold-constrained optimization, we achieve high-order anisotropic regularization of the normal field. Experiments demonstrate that our approach significantly suppresses staircase artifacts and outperforms existing manifold-valued TGV and isotropic regularization methods in preserving sharp features, fine-scale structures, and curvature variations. Consequently, it yields higher geometric fidelity and improved surface reconstruction quality in denoised meshes.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: 3D Computer Vision
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text

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 $mathbb{R}^3$. 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 exctend 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.

Problem

• Extend discrete TGV models to manifold-valued normal vectors
• Construct tangential Raviart-Thomas space for mesh denoising
• Compare new regularizer with existing denoising methods

Innovation

• Second-order TGV for normal vector field
• Tangential Raviart-Thomas finite element space
• Manifold-valued function on unit sphere