Mesh Denoising and Inpainting using the Total Variation of the Normal and a Shape Newton Approach

TL;DR

This paper addresses denoising and inpainting of triangular meshes by regularizing with the discrete total variation of the unit normal field. It introduces a novel, equivalent TV formulation that avoids parallel transport and couples a split Bregman (ADMM) framework with a second-order shape Newton method for the geometry updates, delivering substantial speed-ups over prior approaches. The authors derive first- and second-order shape derivatives via shape calculus, implement a globalized inexact Newton scheme with a truncated CG solver and adaptive penalty parameter updates, and demonstrate strong qualitative and quantitative results on complex geometries including the fandisk and Stanford bunny. The approach yields faithful feature preservation with efficient convergence, enabling practical 3D denoising and inpainting applications with robust performance.

Abstract

We present a novel approach to denoising and inpainting problems for surface meshes. The purpose of these problems is to remove noise or fill in missing parts while preserving important features such as sharp edges. A discrete variant of the total variation of the unit normal vector field serves as a regularizing functional to achieve these goals. In order to solve the resulting problem, we use a version of the split Bregman (ADMM) iteration adapted to the problem. A new formulation of the total variation regularizer, as well as the use of an inexact Newton method for the shape optimization step, bring significant speed-up compared to earlier methods. Numerical examples are included, demonstrating the performance of our algorithm with some complex 3D geometries.

Figures (7)

• Figure 3.1: Illustration of the geodesic distance (angle) between normals ${\boldsymbol{n}}_+$ and ${\boldsymbol{n}}_-$ and the logarithmic map $\mathop{\mathrm{log}}\nolimits_{{\boldsymbol{n}}_+}\IfNoValueF{-NoValue-}{ -NoValue- } {{\boldsymbol{n}}_-}$ of two triangles $T_+$, $T_-$, which share the edge $E$. The triangles' co-normals are ${\boldsymbol{\mu}}_+$ and ${\boldsymbol{\mu}}_-$. Note that $\mathop{\mathrm{log}}\nolimits_{{\boldsymbol{n}}_+}\IfNoValueF{-NoValue-}{ -NoValue- } {{\boldsymbol{n}}_-}$ is parallel to ${\boldsymbol{\mu}}_+$.
• Figure 5.1: Mesh denoising result using algorithm:split_Bregman applied to problem \ref{['eq:mesh_denoising_problem']} with $\beta = 2\cdot 10^{-2}$, $\tau = 10^{-8}$ and initial penalty parameter $\rho = 10^{-2}$. Original geometry (left), noisy geometry (middle) and reconstruction (right).
• Figure 5.2: Visualization of the number of iterations used for the globalized Newton scheme (algorithm:Newton_solve) in each iteration of algorithm:split_Bregman (left) applied to the fandisk denoising problem described in section:Mesh_Denoising_Problem. Evolution of the penalty parameter~$\rho$ over the iterations (right).
• Figure 5.3: Combined residual norms \ref{['eq:residuals_alg']}, \ref{['eq:residuals_RM']} for the fandisk denoising problem over iterations (left) as well as combined residual norms over CPU time on an AMD Ryzen 5 3600 desktop CPU (right). For all three methods, parameters $\beta = 2\cdot 10^{-2}$ and $\tau = 10^{-8}$ are chosen. For algorithm:split_Bregman, the adaptive penalty parameter selection is used starting with $\rho = 10^{-2}$. The Riemannian ADMM method uses the constant penalty parameter $\rho = 1$ for all iterations. Every $20$th iteration is plotted.
• Figure 5.4: Mesh denoising results using algorithm:split_Bregman on problem \ref{['eq:mesh_denoising_problem']} with $\beta = 5 \cdot 10^{-3}$, $\tau = 10^{-8}$ and initial penalty parameter $\rho = 10^{-3}$. Original geometry (left), noisy geometry (middle) and reconstruction (right).

Theorems & Definitions (5)

• remark 1
• lemma 1
• proof
• lemma 2
• proof