Fractional harmonic transform on point cloud manifolds

TL;DR

The paper addresses the limitation of PMHT's fixed harmonic basis by introducing a fractional-order PMFHT that creates a continuum between the spatial and frequency representations via the parameter $\alpha$. PMFHT is realized by applying fractional powers to the PMHT spectral matrix, with forward transform $\hat{f}^{(a)} = F_M^{(a)} f$ and $F_M^{(a)} = P J^a P^{-1}$, recovering PMHT at $a=1$ and reducing to the identity at $a=0$. Key properties include $F_0 = I$, $F_1 = F_M$, and index additivity $F_\alpha F_\beta = F_{\alpha+\beta}$. The method yields richer spectral representations and improves tasks such as denoising, feature enhancement, and shape analysis on point clouds. Experiments on Stanford horse and bunny show flexible spectral representation and effective high-/low-pass filtering across fractional orders.

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.

Figures (5)

• Figure 1: Mathematical principles flowchart
• Figure 2: Point cloud manifold harmonic orthogonal bases $H_1-H_6$.
• Figure 3: PMFHT spectra of horse point cloud
• Figure 4: PMFHT spectra of bunny point cloud
• Figure 5: Point cloud manifold fractional harmonic filtering results.