Weakly non-Gaussian formula for the Minkowski functionals in general dimensions

TL;DR

This paper derives a general analytic formula for the expectation values of Minkowski functionals in any dimension, accounting for weak non-Gaussianity, useful for analyzing the morphology of random fields in various scientific contexts.

Contribution

It extends previous limited formulas to a comprehensive expression applicable to any dimension and weak non-Gaussianity, broadening the analytical tools for morphological analysis.

Findings

Derived a second-order correction formula for Minkowski functionals

Applicable to any dimension and weak non-Gaussianity

Useful for analyzing cosmic and other random fields

Abstract

The Minkowski functionals are useful statistics to quantify the morphology of various random fields. They have been applied to numerous analyses of geometrical patterns, including various types of cosmic fields, morphological image processing, etc. In some cases, including cosmological applications, small deviations from the Gaussianity of the distribution are of fundamental importance. Analytic formulas for the expectation values of Minkowski functionals with small non-Gaussianity have been derived in limited cases to date. We generalize these previous works to derive an analytic expression for expectation values of Minkowski functionals up to second-order corrections of non-Gaussianity in a space of general dimensions. The derived formula has sufficient generality to be applied to any random fields with weak non-Gaussianity in a statistically homogeneous and isotropic space of any dimensions.