The invariant joint distribution of a stationary random field and its derivatives: Euler characteristic and critical point counts in 2 and 3D

TL;DR

This paper derives explicit formulas for the joint distribution of an isotropic random field, its derivatives, and invariants in 2D and 3D, enabling calculation of topological and extremal features based on the field's spectral properties.

Contribution

It provides a comprehensive moments expansion of the joint distribution, facilitating explicit computation of Euler characteristic and extrema counts in multiple dimensions.

Findings

Explicit formulas for Euler characteristic in ND

Calculation of extrema counts as functions of threshold

Application to model examples demonstrating the approach

Abstract

The full moments expansion of the joint probability distribution of an isotropic random field, its gradient and invariants of the Hessian is presented in 2 and 3D. It allows for explicit expression for the Euler characteristic in ND and computation of extrema counts as functions of the excursion set threshold and the spectral parameter, as illustrated on model examples.