Title: Fractional harmonic transform on point cloud manifolds Show Chinese title

Title: 点云流形上的分数调和变换

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. Show Chinese abstract

Abstract: 三维点云可以看作是平滑流形的离散采样，允许使用拉普拉斯-贝尔特拉米算子（LBO）进行谱分析。然而，传统的点云流形谐波变换（PMHT）由于其固定的基函数和单一的谱表示而受到限制，这限制了其捕捉复杂几何特征的能力。本文提出了一种点云流形分数阶谐波变换（PMFHT），通过引入分数阶参数对PMHT进行了推广，并在空间域和频率域之间构建了一个可连续调整的中间分数阶谱域。这种分数阶框架支持更灵活的变换和滤波操作。实验表明，选择不同的变换阶数可以丰富点云的谱表示，并在滤波和特征增强等任务中取得优异的结果。因此，PMFHT不仅扩展了点云谱分析的理论框架，还为流形几何处理提供了一个强大的新工具。