Mesh Denoising

TL;DR

The paper addresses denoising of 3D meshes and compares four approaches, including linear filtering, heat diffusion, Sobolev regularization, and a Sinkhorn-based barycenter method. It builds on optimal transport theory and the Sinkhorn algorithm, as well as graph-based diffusion and Sobolev regularization, to denoise meshes of very different sizes. Empirical results on a large elephant mesh and a small Nefertiti bust show Sobolev regularization as the fastest method with strong performance, while filtering and heat diffusion offer competitive accuracy depending on parameters; the Sinkhorn barycenter approach provides a theoretical alternative but requires more adaptation for geometric data. The findings highlight a regime where large-scale meshes benefit more from diffusion-based and filtering approaches, and they outline limitations and directions for geometry-aware Sinkhorn methods. The study provides practical guidance for selecting denoising strategies and underscores the potential of entropic optimal transport in mesh processing.

Abstract

In this paper, we study four mesh denoising methods: linear filtering, a heat diffusion method, Sobolev regularization, and, to a lesser extent, a barycentric approach based on the Sinkhorn algorithm. We illustrate that, for a simple image denoising task, a naive choice of a Gibbs kernel can lead to unsatisfactory results. We demonstrate that while Sobolev regularization is the fastest method in our implementation, it produces slightly less faithful denoised meshes than the best results obtained with iterative filtering or heat diffusion. We empirically show that, for the large mesh considered, the heat diffusion method is slower and not more effective than filtering, whereas on a small mesh an appropriate choice of diffusion parameters can improve the quality. Finally, we observe that all three mesh-based methods perform markedly better on the large mesh than on the small one.

Figures (6)

• Figure 1: The various noisy meshes studied in Table \ref{['datamaillage']}. From left to right: $\rho = 0.015$, $\rho = 0.2$, $\rho = 1$ (bottom row only).
• Figure 2: From left to right: a photo of a cat; the same photo corrupted with white noise of amplitude $\rho = 0.2$; the kernel of the noisy image; the barycenter $b$ sought after one iteration of the algorithm.
• Figure 3: The original meshes, an elephant NTmeshd and the bust of Nefertiti NTmeshp.
• Figure 4: Top: Processing time versus the number of filtering iterations (left, in seconds) and denoising level versus the number of iterations (right), for the elephant mesh with noise $\rho = 0.2$. Bottom: Processing time versus the choice of parameter $\tau$ to achieve the best possible result (left, in seconds) and denoising level versus the number of iterations for the most effective and fastest algorithm, with $\tau = 1.01$, maximum achieved at $T = 37.4$, with an execution time of 70 ms (right), for the elephant mesh with noise $\rho = 0.2$. Note that to achieve a denoising level similar to Sobolev regularization (25 dB), it takes 20 ms for filtering, as well as for the heat equation method. Since the execution time of the Sobolev algorithm does not depend on the parameter $\mu$, we have all the necessary information for comparison.
• Figure 5: Best noise reduction after processing by heat diffusion on the Nefertiti busts. From left to right: $\rho = 0.015$, $\rho = 0.2$, $\rho = 1$.

Theorems & Definitions (16)

• Definition 1
• Remark 1
• Remark 2
• Definition 2
• Definition 3
• Definition 4
• Proposition 1
• proof
• Remark 3
• Definition 5