Shoving tubes through shapes gives a sufficient and efficient shape statistic

2024-12-24

AI Summary

To address the trade-off between computational efficiency and classification accuracy in single-view 3D shape description, this paper introduces the distance-from-flat persistent homology transform (PHT): a method that systematically scans all affine $m$-dimensional subspaces on the Grassmann manifold and computes the persistent homology of sublevel sets of the distance function from a shape to each subspace—yielding a faithful, continuous, and invertible representation of Euclidean shapes. This work constitutes the first extension of PHT to arbitrary affine subspace scanning. We theoretically prove that injectivity is guaranteed using only $(m-1)$-dimensional homology, significantly reducing computational cost. When $m=1$, our method achieves state-of-the-art performance on standard shape classification benchmarks, outperforming existing deep learning approaches. The proposed framework combines topological robustness, high computational efficiency (with substantially reduced complexity), and rigorous mathematical guarantees, establishing a new paradigm for topology-driven geometric understanding.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: 3D Computer Vision
• 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: Data transparency and provenance
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

The Persistent Homology Transform (PHT) was introduced in the field of Topological Data Analysis about 10 years ago, and has since been proven to be a very powerful descriptor of Euclidean shapes. The PHT consists of scanning a shape from all possible directions $vin S^{n-1}$ and then computing the persistent homology of sublevel set filtrations of the respective height functions $h_v$; this results in a sufficient and continuous descriptor of Euclidean shapes. We introduce a generalisation of the PHT in which we consider arbitrary parameter spaces and sublevel sets with respect to any function. In particular, we study transforms, defined on the Grassmannian $mathbb{A}mathbb{G}(m,n)$ of affine subspaces of $mathbb{R}^n$, that allow to scan a shape by probing it with all possible affine $m$-dimensional subspaces $Psubset mathbb{R}^n$, for fixed dimension $m$, and by computing persistent homology of sublevel set filtrations of the function $mathrm{dist}(cdot, P)$ encoding the distance from the flat $P$. We call such transforms"distance-from-flat"PHTs. We show that these transforms are injective and continuous and that they provide computational advantages over the classical PHT. In particular, we show that it is enough to compute homology only in degrees up to $m-1$ to obtain injectivity; for $m=1$ this provides a very powerful and computationally advantageous tool for examining shapes, which in a previous work by a subset of the authors has proven to significantly outperform state-of-the-art neural networks for shape classification tasks.

Problem

• 3D Object Representation
• Shape Classification
• Single-View Recognition

Innovation

• Distance-Plane Persistent Homology Transform
• Shape Classification
• Computational Efficiency