Topological Signal Processing for 3D Point Cloud Data

TL;DR

The proposed approach provides a topology-based representation to characterize the geometry and attributes of PCs, and introduces higher-order Laplacian operators that enable the processing of signals over triangular meshes.

Abstract

Our goal in this paper is to apply the topological signal processing (TSP) framework to the analysis of 3D Point Clouds (PCs) represented on simplicial complexes. Building on Discrete Exterior Calculus (DEC) theory for vector fields, we introduce higher-order Laplacian operators that enable the processing of signals over triangular meshes. Unlike traditional approaches, the proposed approach allows us to characterize both color attributes, modeled as 3D vectors on nodes, and geometry, modeled as 3D vectors on the barycenter of each triangle. Then, we show as TSP tools may efficiently be used to sample, recover and filter PCs attributes treating them as edge signals. Numerical results on synthetic PCs demonstrate accurate color reconstruction with robustness to sparse data and geometry refinement in the case of noisy PC coordinates. The proposed approach provides a topology-based representation to characterize the geometry and attributes of PCs.

Figures (5)

• Figure 1: Example of geometric simplicial complex.
• Figure 2: (a) Toroidal PC and (b) associated discrete color vector field.
• Figure 3: MSE versus observed samples.
• Figure 4: MSE of the denoised normals with respect to the noiseless case, as a function of the SNR.
• Figure 5: MSE of the denoised normals as a function of the SNR for Red and Black PC varying $\gamma$ with a fixed $\lambda=0.1$.