Persistent Intrinsic Volumes

TL;DR

This paper introduces a new method combining geometric measure theory and persistent homology to estimate intrinsic volumes of a shape from noisy data, with proven stability and linear convergence rates.

Contribution

It presents a novel estimator for intrinsic volumes that leverages persistent homology, extending noise filtering from topology to geometry with stability guarantees.

Findings

Estimator achieves linear convergence rate.

Method extends persistent homology to geometric measure estimation.

Provides stability results for intrinsic volume estimation.

Abstract

We develop a new method to estimate the area, and more generally the intrinsic volumes, of a compact subset $X$ of $\mathbb{R}^d$ from a set $Y$ that is close in the Hausdorff distance. This estimator enjoys a linear rate of convergence as a function of the Hausdorff distance under mild regularity conditions on $X$. Our approach combines tools from both geometric measure theory and persistent homology, extending the noise filtering properties of persistent homology from the realm of topology to geometry. Along the way, we obtain a stability result for intrinsic volumes.