Point Cloud Denoising With Fine-Granularity Dynamic Graph Convolutional Networks

TL;DR

This paper introduces fine-granularity dynamic graph convolutional networks called GD-GCN, a novel approach to denoising in 3-D point clouds that approximates the Riemannian metric and incorporates robust graph spectral filters based on the Bernstein polynomial approximation, providing theoretical guarantees for BIBO stability.

Abstract

Due to limitations in acquisition equipment, noise perturbations often corrupt 3-D point clouds, hindering down-stream tasks such as surface reconstruction, rendering, and further processing. Existing 3-D point cloud denoising methods typically fail to reliably fit the underlying continuous surface, resulting in a degradation of reconstruction performance. This paper introduces fine-granularity dynamic graph convolutional networks called GD-GCN, a novel approach to denoising in 3-D point clouds. The GD-GCN employs micro-step temporal graph convolution (MST-GConv) to perform feature learning in a gradual manner. Compared with the conventional GCN, which commonly uses discrete integer-step graph convolution, this modification introduces a more adaptable and nuanced approach to feature learning within graph convolution networks. It more accurately depicts the process of fitting the point cloud with noise to the underlying surface by and the learning process for MST-GConv acts like a changing system and is managed through a type of neural network known as neural Partial Differential Equations (PDEs). This means it can adapt and improve over time. GD-GCN approximates the Riemannian metric, calculating distances between points along a low-dimensional manifold. This capability allows it to understand the local geometric structure and effectively capture diverse relationships between points from different geometric regions through geometric graph construction based on Riemannian distances. Additionally, GD-GCN incorporates robust graph spectral filters based on the Bernstein polynomial approximation, which modulate eigenvalues for complex and arbitrary spectral responses, providing theoretical guarantees for BIBO stability. Symmetric channel mixing matrices further enhance filter flexibility by enabling channel-level scaling and shifting in the spectral domain.

Figures (8)

• Figure 1: (a) Evolving trajectory of the noise points relative to the underlying surface by coarse-grained dynamics GCN. (b) Evolving trajectory of the noise points relative to the underlying surface by fine-granularity dynamic GCN.
• Figure 2: The progression of point cloud denoising. (a) is the noisy input, (b) is the result after integer-step graph convolution, (c) shows the early result of micro-step graph convolution, and (d) shows the final refined output after several micro-steps.
• Figure 3: (a) Overall architecture of proposed fine-grained dynamics graph convolution networks (GD-GCN). (b) The details of Riemannian metric learning. (c) The details of micro-step temporal graph convolution at time $t$.
• Figure 4: Denoising results for airplane and car from the ShapeNet repository. From top left to bottom right: Groud Truth, GPDNet pistilli2020learning, TD hermosilla2019total, DMR luo2020differentiable, ScoreNet luo2021score, Deeprs chen2022deep, and PathNet wei2024pathnet and the proposed method. CD values in $10^{-6}$ are reported.
• Figure 5: Denoising results for the bunny and dragon from the Stanford repository using both supervised (top) and unsupervised (bottom) modes. From top left to bottom right: TD hermosilla2019total, DMR luo2020differentiable, ScoreNet luo2021score, ours and ground truth. CD values in $10^{-6}$ are reported.

Theorems & Definitions (2)

• Proposition 1
• Remark 1