Manifold Fractional Harmonic Transform for 3D Point Clouds

TL;DR

This paper introduces a novel fractional harmonic transform for 3D point clouds, enabling spectral analysis in the fractional domain, and develops algorithms for encryption and target detection validated by experiments.

Contribution

It extends manifold spectral analysis to the fractional domain, providing a theoretical foundation and practical algorithms for encryption and maritime target detection.

Findings

The PMFHT framework is theoretically rigorous and stable.

The encryption scheme has a large key space and high sensitivity.

The target detection method effectively suppresses sea clutter.

Abstract

Point clouds can be regarded as discrete samples of smooth manifolds and are typically analyzed via the eigenfunctions of the Laplace-Beltrami operator. This paper extends manifold spectral analysis to the fractional domain, enabling continuous interpolation between the spatial and spectral domains for point cloud data. First, a point cloud manifold fractional harmonic transform (PMFHT) is proposed, with its fundamental properties rigorously derived, along with the associated convolution, correlation, and sampling theorems. These theoretical results establish a solid foundation for stable fractional-order spectral representation on manifolds. Second, within the PMFHT framework, two representative algorithms are developed. On the one hand, by integrating multi-order PMFHT with chaotic phase modulation, a point cloud encryption scheme is constructed, characterized by a large key space and high sensitivity to key perturbations. On the other hand, an optimal filter is designed in the fractional manifold spectral domain, leading to a maritime target detection method specifically tailored for point cloud data, which effectively suppresses sea clutter while preserving weak target energy under low signal-to-clutter ratio conditions. Finally, experiments on measured data validate the effectiveness of the proposed algorithms.