Enhanced 3D Shape Analysis via Information Geometry

2025-12-18

AI Summary

Comparing unordered 3D point clouds remains challenging due to their unstructured nature and geometric complexity; existing metrics—including Hausdorff and Chamfer distances, as well as KL-divergence approximations based on Gaussian Mixture Models (GMMs)—lack global statistical modeling capability, are sensitive to outliers, and suffer from divergence or numerical instability. Method: This paper establishes, for the first time, that the space of GMMs forms a statistical manifold, and accordingly proposes a bounded, numerically stable Modified Symmetric KL divergence (MSKL). MSKL possesses rigorously derived theoretical upper and lower bounds, overcoming the unboundedness and instability inherent in conventional KL-based approximations. Results: Experiments on MPI-FAUST and G-PCD datasets demonstrate that MSKL exhibits monotonic geometric sensitivity and significantly outperforms state-of-the-art geometric and probabilistic metrics in shape matching and retrieval tasks.

Technology Category

• Machine Learning: Learning with Manifolds
• Data Mining & Knowledge Management: Anomaly/Outlier Detection
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

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.

Problem

• Develop a stable metric for comparing 3D point clouds
• Address limitations of traditional geometric distances and KL divergence
• Represent point clouds as Gaussian Mixture Models on a statistical manifold

Innovation

• Gaussian Mixture Models represent point clouds on statistical manifold
• Modified Symmetric KL divergence ensures bounded, numerically stable comparisons
• MSKL outperforms traditional distances in shape analysis experiments