Fractional harmonic transform on point cloud manifolds

TL;DR

This paper introduces a fractional harmonic transform for point cloud manifolds, enabling more flexible spectral analysis and improved geometric feature processing compared to traditional methods.

Contribution

It proposes a novel fractional harmonic transform that generalizes existing spectral methods by introducing adjustable fractional orders for enhanced geometric analysis.

Findings

Enables richer spectral representations of point clouds.

Improves filtering and feature enhancement tasks.

Provides a new theoretical framework for manifold spectral analysis.

Abstract

Three-dimensional point clouds can be viewed as discrete samples of smooth manifolds, allowing spectral analysis using the Laplace-Beltrami operator (LBO). However, the traditional point cloud manifold harmonic transform (PMHT) is limited by its fixed basis functions and single spectral representation, which restricts its ability to capture complex geometric features. This paper proposes a point cloud manifold fractional harmonic transform (PMFHT), which generalizes PMHT by introducing fractional-order parameters and constructs a continuously adjustable intermediate fractional-order spectral domain between the spatial domain and the frequency domain. This fractional-order framework supports more flexible transformation and filtering operations. Experiments show that choosing different transformation orders can enrich the spectral representation of point clouds and achieve excellent results in tasks such as filtering and feature enhancement. Therefore, PMFHT not only expands the theoretical framework of point cloud spectral analysis, but also provides a powerful new tool for manifold geometry processing.