用信息几何方法稳定比较3D点云形状，提升精度与可靠性。
Enhanced 3D Shape Analysis via Information Geometry
- 将点云建模为高斯混合模型，构建统计流形上的形状分析框架。
- 提出有界MSKL散度，在两个数据集上均实现稳定且单调变化的相似性度量。
- 适合需要精确、鲁棒形状比对的计算机视觉与机器人应用。
三维点云能提供物体的高度精确数字表示，广泛应用于计算机图形学、摄影测量、计算机视觉和机器人领域。然而，由于其非结构化特性和表面几何复杂性，点云比较面临重大挑战。传统几何度量如豪斯多夫距离和昌弗距离难以捕捉全局统计结构，且对异常值敏感；而现有高斯混合模型（GMM）的相对熵近似可能导致无界或数值不稳定。本文提出一种基于信息几何的3D点云形状分析框架，将点云表示为统计流形上的高斯混合模型（GMM）。我们证明了GMM空间构成统计流形，并提出改进的对称相对熵（MSKL）度量，理论上保证上下界，确保所有GMM比较的数值稳定性。在人体姿态辨别（MPI-FAUST数据集）和动物形状对比（G-PCD数据集）的综合实验中，MSKL展现出稳定且单调变化的取值，直接反映几何差异，优于传统距离和现有KL近似。
原文摘要 · Abstract (English)
Three-dimensional point clouds provide highly accurate digital representations of objects, essential for applications in computer graphics, photogrammetry, computer vision, and robotics. However, comparing point clouds faces significant challenges due to their unstructured nature and the complex geometry of the surfaces they represent. Traditional geometric metrics such as Hausdorff and Chamfer distances often fail to capture global statistical structure and exhibit sensitivity to outliers, while existing Kullback-Leibler (KL) divergence approximations for Gaussian Mixture Models can produce unbounded or numerically unstable values. This paper introduces an information geometric framework for 3D point cloud shape analysis by representing point clouds as Gaussian Mixture Models (GMMs) on a statistical manifold. We prove that the space of GMMs forms a statistical manifold and propose the Modified Symmetric Kullback-Leibler (MSKL) divergence with theoretically guaranteed upper and lower bounds, ensuring numerical stability for all GMM comparisons. Through comprehensive experiments on human pose discrimination (MPI-FAUST dataset) and animal shape comparison (G-PCD dataset), we demonstrate that MSKL provides stable and monotonically varying values that directly reflect geometric variation, outperforming traditional distances and existing KL approximations.
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