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

Lukas Baumgärtner, Ronny Bergmann, Roland Herzog, Stephan Schmidt, Manuel Weiß

TL;DR

This work addresses mesh denoising by regularizing the unit normal field with a discrete, intrinsic second-order $\operatorname{TGV}$ for manifold-valued data. It introduces a tailor-made tangential Raviart--Thomas space to represent the push-forward of the normal and couples the manifold-valued jumps via the logarithmic map within an ADMM framework. The resulting $\textup{FETGV}^{2}$ regularizer favors piecewise constant principal curvature areas (planes, spheres, cylinders) and demonstrates improved denoising performance over extrinsic methods on several geometric test cases, while avoiding unwanted global size changes. The approach provides a principled, intrinsic regularization tool for mesh processing with potential extensions to other manifold-valued data on surfaces.

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 $\R^3$. 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 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.

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

TL;DR

This work addresses mesh denoising by regularizing the unit normal field with a discrete, intrinsic second-order for manifold-valued data. It introduces a tailor-made tangential Raviart--Thomas space to represent the push-forward of the normal and couples the manifold-valued jumps via the logarithmic map within an ADMM framework. The resulting regularizer favors piecewise constant principal curvature areas (planes, spheres, cylinders) and demonstrates improved denoising performance over extrinsic methods on several geometric test cases, while avoiding unwanted global size changes. The approach provides a principled, intrinsic regularization tool for mesh processing with potential extensions to other manifold-valued data on surfaces.

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 . 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.

Paper Structure

This paper contains 18 sections, 3 theorems, 66 equations, 6 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Total Generalized Variation with Piecewise Constant Finite Elements
  3. Notation and Finite Element Spaces
  4. Discrete TGV for Piecewise Constant Functions
  5. Discrete Total Generalized Variation of the Normal
  6. Geometric Calculus for the Sphere and Identities on Triangulated Meshes
  7. Discrete Total Variation of the Normal Vector Field
  8. First- and Second-Order Derivatives of Normal Vector Fields on Surfaces
  9. Tangential Raviart--Thomas Space
  10. Discrete Total Generalized Variation of the Normal Vector Field
  11. Numerical Realization
  12. Tangential Finite Element Spaces
  13. Derivation of the ADMM
  14. Numerical Results for Mesh Denoising
  15. Spheres and Cylinders
  16. ...and 3 more sections

Key Result

lemma 1

Let $E$ be the edge shared by the triangles ${T}_{E_+}$, ${T}_{E_-}$ with respective normal vectors ${{\boldsymbol{n}}}_{E_+}$, ${{\boldsymbol{n}}}_{E_-}$ and co-normal vectors ${{\boldsymbol{\mu}}}_{E_+}$, ${{\boldsymbol{\mu}}}_{E_-}$. Then

Figures (6)

  • Figure 2.1: Illustration of normals ${{\boldsymbol{n}}}_{E_+}$ and ${{\boldsymbol{n}}}_{E_-}$ of two triangles ${T}_{E_+}$, ${T}_{E_-}$ sharing an edge $E$. The triangles' co-normals are ${{\boldsymbol{\mu}}}_{E_+}$ and ${{\boldsymbol{\mu}}}_{E_-}$ and the unit vector tangent to the edge is ${\boldsymbol{t}}_E$. The logarithmic map, described in section:discrete-tgv-of-the-normal-vector-field is also pictured.
  • Figure 3.1: Visualization of the relation between $\mathcal{T}_{{\boldsymbol{n}}({\boldsymbol{x}})}{\mathcal{S}}$ and $\mathcal{T}_{{\boldsymbol{x}}}\Gamma$. Adapted from BergmannHerrmannHerzogSchmidtVidalNunez:2020:1.
  • Figure 5.1: Top row: original geometry (left), noisy geometry (right). Bottom row: reconstructions using meshTGV LiuLiWangLiuChen:2022:1 (left) with $\alpha_0 = 0.2$ and $\alpha_1 = 1.1$, and the proposed $\textup{FETGV}^2$ (right) using \ref{['eq:tgv:normal']} with $\alpha_0 = 3 \cdot 10^{-5}$, $\alpha_1 = 3.5 \cdot 10^{-3}$.
  • Figure 5.2: Top row: original geometry (left), noisy geometry (right). Bottom row: reconstructions using meshTGV LiuLiWangLiuChen:2022:1 (left) with $\alpha_0 = 0.2$ and $\alpha_1 = 1.5$, and the proposed $\textup{FETGV}^2$ (right) using \ref{['eq:tgv:normal']} with $\alpha_0 = 3\cdot 10^{-5}$, $\alpha_1 = 3.5 \cdot 10^{-3}$.
  • Figure 5.3: Top row: original fandisk geometry (left), noisy geometry (middle) and $\operatorname{TV}$ reconstruction (right) using \ref{['eq:tv:normal']} from BaumgaertnerBergmannHerzogSchmidtVidalNunezWeiss:2025:1 with $\beta = 2 \cdot 10^{-2}$. Bottom row: reconstructions using meshTGV LiuLiWangLiuChen:2022:1 (left) with $\alpha_0 = 0.2$ and $\alpha_1 = 1.2$, rTGV ZhangHeWang:2022:1 (middle), and the proposed $\textup{FETGV}^2$ (right) using \ref{['eq:tgv:normal']} with $\alpha_0 = 10^{-5}$, $\alpha_1 = 10^{-3}$.
  • ...and 1 more figures

Theorems & Definitions (6)

  • lemma 1
  • proof
  • lemma 2
  • proof
  • lemma 3
  • proof