Shoving tubes through shapes gives a sufficient and efficient shape statistic

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.