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Enhanced 3D Shape Analysis via Information Geometry

Amit Vishwakarma, K. S. Subrahamanian Moosath

TL;DR

<3-5 sentence high-level summary> The paper addresses robust 3D shape analysis of unstructured point clouds by modeling each cloud as a Gaussian Mixture Model and placing these models on a statistical manifold via information geometry. It introduces the Modified Symmetric KL (MSKL) divergence, proves theoretical bounds to ensure stability, and demonstrates superior discrimination and stability against classic geometric distances and KL-based GMM approximations. The authors implement a full pipeline including FPS preprocessing, local geometric descriptors, joint Isomap embedding, and grid-based MSKL computation to compare shapes from MPI-FAUST and G-PCD datasets. This framework provides a principled, scalable approach to quantify global shape differences in 3D data with potential for broader geometric invariants and applications in computer vision and robotics.

Abstract

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.

Enhanced 3D Shape Analysis via Information Geometry

TL;DR

<3-5 sentence high-level summary> The paper addresses robust 3D shape analysis of unstructured point clouds by modeling each cloud as a Gaussian Mixture Model and placing these models on a statistical manifold via information geometry. It introduces the Modified Symmetric KL (MSKL) divergence, proves theoretical bounds to ensure stability, and demonstrates superior discrimination and stability against classic geometric distances and KL-based GMM approximations. The authors implement a full pipeline including FPS preprocessing, local geometric descriptors, joint Isomap embedding, and grid-based MSKL computation to compare shapes from MPI-FAUST and G-PCD datasets. This framework provides a principled, scalable approach to quantify global shape differences in 3D data with potential for broader geometric invariants and applications in computer vision and robotics.

Abstract

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.

Paper Structure

This paper contains 31 sections, 8 theorems, 78 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. Metric-Based Approaches
  4. 1. Hausdorff Distance.
  5. 2. Chamfer Distance.
  6. 3. Earth Mover's Distance.
  7. 4. Gromov-Hausdorff Distance.
  8. Divergence and Distance Based Measures on GMMs
  9. Information Geometry
  10. Statistical manifold
  11. Divergence Measures on Statistical Manifolds
  12. Statistical Manifold Representation for Point Cloud Data
  13. Gaussian Mixture Model
  14. Manifold Structure for Point Clouds
  15. MSKL Bounds
  16. ...and 16 more sections

Key Result

Theorem 1

Let $p(x;\theta_1) =\sum_{i=1}^{K} \alpha_{i} \mathscr{N}(x ; \mu_{i}, \sigma_i^{2})$ and $p(x;\theta_2)=\sum_{j=1}^{L} \beta_{j} \mathscr{N}(x; \nu_{j}, \tau_j^{2})$ be two univariate Gaussian mixture models representing the point cloud $X$ having the number of Gaussian components $K$ and $L$ respe If the parameters in each component of a GMM are distinct and they are ordered lexicographically by

Figures (4)

  • Figure 1: Statistical manifold representation shows the map between the parameter space and the space of probability distributions.
  • Figure 2: MPI-FAUST human dataset samples showing two distinct poses of a Human body scans.
  • Figure 3: G-PCD animal dataset samples illustrating morphological diversity. Left: Rabbit Point Cloud 1. Right: Dragon Point Cloud 2.
  • Figure 4: Five-stage computational pipeline for point cloud shape analysis on the statistical manifold of GMMs.

Theorems & Definitions (16)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • ...and 6 more