Euler Integration of Gaussian Random Fields and Persistent Homology

TL;DR

This paper extends Euler characteristic to persistent homology, relates Euler integrals to persistent homology, and computes the expected Euler integral for Gaussian random fields, providing a new quantitative descriptor.

Contribution

It introduces a novel extension of Euler characteristic to persistent homology and derives a closed-form expression for the expected Euler integral of Gaussian fields.

Findings

Derived a closed-form expression for the expected Euler integral of Gaussian fields

Established a relationship between Euler integrals and persistent homology

Provided the first explicit mean calculation for a persistent homology descriptor

Abstract

In this paper we extend the notion of the Euler characteristic to persistent homology and give the relationship between the Euler integral of a function and the Euler characteristic of the function's persistent homology. We then proceed to compute the expected Euler integral of a Gaussian random field using the Gaussian kinematic formula and obtain a simple closed form expression. This results in the first explicitly computable mean of a quantitative descriptor for the persistent homology of a Gaussian random field.