Topological Signal Processing for 3D Point Cloud Data

TL;DR

This paper introduces a topological signal processing framework for 3D point clouds using discrete exterior calculus, enabling efficient analysis and filtering of geometric and attribute data with robustness to noise and sparsity.

Contribution

It extends topological signal processing to 3D point clouds by developing higher-order Laplacians based on DEC, allowing topology-aware analysis of geometry and attributes.

Findings

Accurate color reconstruction on synthetic point clouds.

Robustness to sparse data and noisy coordinates.

Topology-based characterization of 3D point cloud attributes.

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.