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Shoving tubes through shapes gives a sufficient and efficient shape statistic

Adam Onus, Nina Otter, Renata Turkes

TL;DR

This work generalises the persistent homology transform (PHT) by introducing the distance-from-flat PHT, $ ext{PHT}_{ ext{AG}(m,n), ext{d}}$, which probes Euclidean shapes with respect to all affine $m$-dimensional flats. By establishing definability and using Radon-transform ideas, the authors prove that this transform is continuous (with respect to suitable diagram metrics) and that truncating the homology degrees to $ ext{0,1,...,m-1}$ yields an injective descriptor of shapes, offering computational advantages, especially for tubular-type filtrations when $m=1$ and $k=0$. They connect the generalized PHT to the classical one, show relationships to Euler calculus and Schapira’s inversion, and discuss stability properties, instabilities, and potential extensions to non-Euclidean spaces and broader parameter spaces. The results provide a theoretically solid, computationally attractive framework for injective shape statistics with practical implications for classification and geometric analysis. Overall, the paper advances topological data analysis by broadening the domain and filtration choices for PHTs while ensuring injectivity and continuity under well-defined geometric conditions.

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 $v\in S^{n-1}$ 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 $P\subset \mathbb{R}^n$, 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.

Shoving tubes through shapes gives a sufficient and efficient shape statistic

TL;DR

This work generalises the persistent homology transform (PHT) by introducing the distance-from-flat PHT, , which probes Euclidean shapes with respect to all affine -dimensional flats. By establishing definability and using Radon-transform ideas, the authors prove that this transform is continuous (with respect to suitable diagram metrics) and that truncating the homology degrees to yields an injective descriptor of shapes, offering computational advantages, especially for tubular-type filtrations when and . They connect the generalized PHT to the classical one, show relationships to Euler calculus and Schapira’s inversion, and discuss stability properties, instabilities, and potential extensions to non-Euclidean spaces and broader parameter spaces. The results provide a theoretically solid, computationally attractive framework for injective shape statistics with practical implications for classification and geometric analysis. Overall, the paper advances topological data analysis by broadening the domain and filtration choices for PHTs while ensuring injectivity and continuity under well-defined geometric conditions.

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 and then computing the persistent homology of sublevel set filtrations of the respective height functions ; 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 of affine subspaces of , that allow to scan a shape by probing it with all possible affine -dimensional subspaces , for fixed dimension , and by computing persistent homology of sublevel set filtrations of the function encoding the distance from the flat . 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 to obtain injectivity; for 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.

Paper Structure

This paper contains 33 sections, 19 theorems, 76 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Grassmannian of affine subspaces
  4. A plurality of perspectives
  5. Distances and metrics on affine Grassmannians
  6. Distances on Grassmannians
  7. Distances on affine Grassmannians
  8. Schubert calculus
  9. The Grassmannian of all m-flats containing a specific k-flat
  10. Euler characteristic of Grassmannians
  11. Persistent homology
  12. A tameness framework for topology: o-minimal structures
  13. Euler characteristic and calculus
  14. Radon integral transform
  15. (Distance-from-flat) persistent homology transforms
  16. ...and 18 more sections

Key Result

Lemma 2.3

The deaffine map $p$ and embedding $j$ from Definition D:maps aff gr are a surjection and inclusion of sets, respectively.

Figures (7)

  • Figure 1: For a shape $X$, the classical $\mathrm{PHT}(X)$ consists of $\mathrm{PD}_k(X, h_v)$ with respect to the height filtration $h_v$ for any direction $v$, and for any homological degree $k$. For the example image $X$, the figure illustrates $\mathrm{PHT}$ on the height filtration from two directions (in blue).
  • Figure 2: Illustration of an added value of distance-from-flat $\mathrm{PHT}_{\mathbb{AG}(m,n),\mathrm{d}}$ on $\mathbb{AG}(m', n)$ over $\mathbb{AG}(m, n)$ for $m' < m$, in particular of tubular $\mathrm{PHT}_{\mathbb{AG}(1,n),\mathrm{d}}$ over height $\mathrm{PHT}_{\mathbb{AG}(n-1,n),\mathrm{d}}$. $\mathrm{PD}_0(X)$ with respect to the adjusted height, i.e., distance from a plane (top rows) cannot discriminate between a ball- and sphere-like shape: there is always the one connected components in the filtration for both shapes; higher homological dimensions are needed. However, $\mathrm{PD}_0(X)$ with respect to the tubular filtration function, i.e., distance from a line (bottom rows) is sufficient to differentiate the two shapes: the sphere sees a second connected component in the filtration.
  • Figure 3: The height $\mathrm{PHT}_{\mathbb{AG}(n-1,n),\mathrm{d}}$ is more refined than the classical $\mathrm{PHT}_{\mathbb{S}^{n-1}, h}.$ Indeed, for any direction $v,$ the sublevel set (highlighted in green) of the height function $h_v$ can be realised as the sublevel set of the distance function $d_P$ for the appropriate hyperplane $P,$$X_r^-(h_v) = X_{r+M}^-(d_P)$ (a). However, the sublevel sets of the distance from hyperplanes $P$ that pass through the shape (b) cannot be realised as sublevel sets of the height function $h_v$ for any direction $v$: the former can be seen as the region of the shape squished between two hyperplanes, whereas the latter is the intersection of the shape with a halfspace.
  • Figure 4: Illustration of the added value of $\mathrm{PHT}_{\mathbb{P},f}$ over $\mathrm{PH}$ in every homological dimension: $\mathrm{PH}$ describes some topological and geometric features (such as the number, size or position of cycles) and therefore loses some information about the shape --- it is not injective, but considering multiple filtrations or "directions" with $\mathrm{PHT}_{\mathbb{P},f}$ can help to completely recover the shape. For the example circle and square, their $\mathrm{PD}$s are the same in any homological degree for some filtration (top rows), but considering different filtration can help to discriminate between the two shapes (bottom rows).
  • Figure 5: Distance-from-flat $\mathrm{PHT}_{\mathbb{AG}(m,n),\mathrm{d}}$ is not stable, since a small shape perturbation can yield a large change in $\mathrm{PHT}$. For the example shapes $X$ and $Y$ which are at small distance $d(X, Y),$ and any flat $P \in \mathbb{AG}(m, 2),$ we have that $d(\mathrm{PD}_1(X, d_P), \mathrm{PD}_1(Y, d_P)) = \infty$, since even a single-pixel hole results in an essential $1$-dimensional cycle.
  • ...and 2 more figures

Theorems & Definitions (70)

  • Definition 2.1: Grassmannian and affine Grassmannian spaces $\mathbb{G}(m, n)$ and $\mathbb{AG}(m, n)$
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Remark 2.5
  • Remark 2.6
  • Proposition 2.7
  • Definition 2.8: Filtration
  • Definition 2.9
  • Definition 2.10
  • ...and 60 more