Feature Graph Learning for 3D Point Cloud Denoising

Abstract

Identifying an appropriate underlying graph kernel that reflects pairwise similarities is critical in many recent graph spectral signal restoration schemes, including image denoising, dequantization, and contrast enhancement. Existing graph learning algorithms compute the most likely entries of a properly defined graph Laplacian matrix $\mathbf{L}$, but require a large number of signal observations $\mathbf{z}$'s for a stable estimate. In this work, we assume instead the availability of a relevant feature vector ${\mathbf{f}}_i$ per node $i$, from which we compute an optimal feature graph via optimization of a feature metric. Specifically, we alternately optimize the diagonal and off-diagonal entries of a Mahalanobis distance matrix $\mathbf{M}$ by minimizing the graph Laplacian regularizer (GLR) ${\mathbf{z}}^{\top }\mathbf{Lz}$, where edge weight is $w_{i,j}=\exp \{-({\mathbf{f}}_i-{\mathbf{f}}_j{)}^{\top }\mathbf{M}({\mathbf{f}}_i-{\mathbf{f}}_j)\}$, given a single observation $\mathbf{z}$. We optimize diagonal entries via proximal gradient (PG), where we constrain $\mathbf{M}$ to be positive definite (PD) via linear inequalities derived from the Gershgorin circle theorem. To optimize off-diagonal entries, we design a block descent algorithm that iteratively optimizes one row and column of $\mathbf{M}$. To keep $\mathbf{M}$ PD, we constrain the Schur complement of sub-matrix ${\mathbf{M}}_{2,2}$ of $\mathbf{M}$ to be PD when optimizing via PG. Our algorithm mitigates full eigen-decomposition of $\mathbf{M}$, thus ensuring fast computation speed even when feature vector ${\mathbf{f}}_i$ has high dimension. To validate its usefulness, we apply our feature graph learning algorithm to the problem of 3D point cloud denoising, resulting in state-of-the-art performance compared to competing schemes in extensive experiments.

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Cited by in corpus (17)

• Score-Based Point Cloud Denoising
• Differentiable Manifold Reconstruction for Point Cloud Denoising
• Hypergraph Spectral Analysis and Processing in 3D Point Cloud
• Dynamic Point Cloud Denoising via Manifold-to-Manifold Distance
• Graph Sampling for Matrix Completion Using Recurrent Gershgorin Disc Shift
• MODNet: Multi-offset Point Cloud Denoising Network Customized for Multi-scale Patches
• Graph-Based Depth Denoising & Dequantization for Point Cloud Enhancement
• Graph Signal Processing for Geometric Data and Beyond: Theory and Applications
• Mask-GVAE: Blind Denoising Graphs via Partition
• 3D Dynamic Point Cloud Denoising via Spatial-Temporal Graph Learning
• Rethinking Point Cloud Filtering: A Non-Local Position Based Approach
• Graph Metric Learning via Gershgorin Disc Alignment
• Incomplete Gamma Kernels: Generalizing Locally Optimal Projection Operators
• Signal Processing in the Retina: Interpretable Graph Classifier to Predict Ganglion Cell Responses
• 3D Point Cloud Enhancement using Graph-Modelled Multiview Depth Measurements
• Deep graph learning for semi-supervised classification
• Signed Graph Metric Learning via Gershgorin Disc Perfect Alignment