Semi-sparsity Generalization for Variational Mesh Denoising

TL;DR

The paper addresses the challenge of denoising 3D mesh surfaces while preserving sharp features and smooth regions on irregular triangulated meshes. It introduces a variational framework that employs semi-sparsity priors, discretizes higher-order operators on meshes, and uses a two-stage approach: first refine face normals via a semi-sparsity regularization, then update vertex positions, with a multi-block ADMM solver that yields closed-form subproblems. Key contributions include designing a semi-sparsity normal estimation model with dynamic edge/line weights, an efficient three-block ADMM algorithm with soft/hard thresholding for subproblems, and a shape-preserving vertex update; extensive experiments demonstrate superior or competitive performance against state-of-the-art methods on CAD, non-CAD, and scanned data. The method achieves robust feature preservation and smooth surface recovery, offering a practical denoising pipeline for geometry processing and potential extensions to other geometry tasks.

Abstract

In this paper, we propose a new variational framework for 3D surface denoising over triangulated meshes, which is inspired by the success of semi-sparse regularization in image processing. Differing from the uniformly sampled image data, mesh surfaces are typically represented by irregular, non-uniform structures, which thus complicate the direct application of the standard formulation and pose challenges in both model design and numerical implementation. To bridge this gap, we first introduce the discrete approximations of higher-order differential operators over triangle meshes and then develop a semi-sparsity regularized minimization model for mesh denoising. This new model is efficiently solved by using a multi-block alternating direction method of multipliers (ADMM) and achieves high-quality simultaneous fitting performance -- preserving sharp features while promoting piecewise-polynomial smoothing surfaces. To verify its effectiveness, we also present a series of experimental results on both synthetic and real scanning data, showcasing the competitive and superior results compared to state-of-the-art methods, both visually and quantitatively.

Figures (9)

• Figure 1: The illustration of $D_{1}\left(\tau_{i}\right), B_{1}\left(\tau_{i}\right), B_{2}\left(\tau_{i}\right), M_{1}\left(v_i\right)$, and $N_{1}\left(v_i\right)$(see also in liu2019triangulated). The elements contained in $D_{1}\left(\tau_{i}\right)$,$B_{1}\left(\tau_{i}\right)$, $B_{2}\left(\tau_{i}\right)$, $M_{1}\left(v_i\right)$, and $N_{1}\left(v_i\right)$ are plotted in green, cyan, blue, gray, and orange, respectively. (a) $D_{1}\left(\tau_{i}\right)$ is the 1-ring of the triangle $\tau_{i}$, which refers to three triangles; (b) $B_{1}\left(\tau_{i}\right)$ is the set of lines connecting the barycenter and vertices of $\tau_{i}$, which refers to three lines, and $B_{2}\left(\tau_{i}\right)$ is the set of lines connecting vertices of $\tau_{i}$ and barycenter of triangles contained in $D_{1}\left(\tau_{i}\right)$, which refers to six lines; (c) $M_{1}\left(v_i\right)$ is the 1-disk of the vertex $v_i$, which refers to seven triangles; (d) $N_{1}\left(v_i\right)$ is the 1-neighborhood of $v_i$, which refers to seven vertices.
• Figure 2: The illustration of (a) $[\mathrm{v}]_l$ (or $[[\mathrm{u}]]_l$) over the line $l$, and the associated $B_1(e)$ containing all the line $l_1, l_2$ and $l_3$ and $l_4$ in two adjacent oriented triangles sharing an edge $e$.
• Figure 3: Visual denoising comparison of varying $\alpha$ with the fixed $\beta$ and $\lambda$. The noisy mesh is corrupted by Gaussian noise ($\sigma=0.1\,\bar{l}_e$).
• Figure 4: Visual comparison of varying $\beta$ with the fixed $\alpha$ and $\lambda$. The noisy mesh is corrupted by Gaussian noise ($\sigma=0.15\,\bar{l}_e$). Zoom in for better view.
• Figure 5: Visual comparison of different denoising methods under noise level $\sigma = 0.35\, \bar{l}_e$. The second row shows the corresponding error maps of the angular difference between the face normals of the denoised meshes and the ground truth. (Zoom in for a better view.)