Euler integration of Gaussian random fields and persistent homology (Q2885380)

The Euler characteristic is a finitely additive operator on compact sets. However for certain classes of functions it is possible to consider it as a measure and integrate with respect to it. Specifically this can be done for tame functions: a function \(f: X \rightarrow {\mathbb R}\) is tame if \(X\) has finite Euler characteristic and if the homotopy types of \(f^{-1}(-\infty, u]\) and \(f^{-1}[u, \infty)\) change only a finite number of times as \(u\) varies over \(\mathbb R\). The persistent homology of a real valued function \(f\) tracks changes in the homology of the sublevel sets \(f^{-1}(-\infty, u]\), and so one would expect a relationship between the Euler integrals and the persistent homology. Such a relationship is encapsulated by the concept of a barcode and the paper contains results on this. The main result of this paper is the following one.NEWLINENEWLINELet \(M\) be a compact \(d\)-dimensional stratified space, and let \(f :M \rightarrow {\mathbb R}^{k}\) be a \(k\)-dimensional Gaussian random field satisfying the Gaussian kinematic formula conditions. For a tame piecewise-\(C^{2}\) function \(G : {\mathbb R}^{k} \rightarrow {\mathbb R}\), let \(g = G \circ f\). If we set \(D_{u} = G^{-1}(-\infty, u]\), then NEWLINE\[NEWLINE \operatorname{E}\{\int_{M} g \lceil d\chi \rceil \} = \chi(M)\operatorname{E}\{g\} - \sum_{j=1}^d (2\pi)^{-j/2} {\mathcal L}_{j} (M) \int_{\mathbb R} {\mathcal M}_{j}(D_{u})du, NEWLINE\]NEWLINE where \(\operatorname{E}\{g\} := \operatorname{E}\{g(t)\}\) and the \({\mathcal L}_{i}\), and \({\mathcal M}_{j}\) are the Lipschitz-Killing curvatures and Gaussian Minkowski functionals, respectively.NEWLINENEWLINEThe authors give specific examples of computations of the \({\mathcal M}\) term and an application to target counting in a sensor field.