Integral Geometry of Linearly Combined Gaussian and Student-t, and Skew Student’s t Random Fields

Abstract

The integral geometry of random fields has been investigated since the 1980s, and the analytic formulae of the Minkowski functionals (also called Lipschitz-Killing curvatures, shortly denoted LKCs) of their excursion sets on a compact subset S in the n-dimensional Euclidean space have been reported in the specialized literature for Gaussian and student-t random fields. Very recently, explicit analytical formulae of the Minkowski functionals of their excursion sets in the bi-dimensional case (n = 2) have been defined on more sophisticated random fields, namely: the Linearly Combined Gaussian and Student-t, and the Skew Student-t random fields. This paper presents the theoretical background, and gives the explicit analytic formulae of the three Minkowski functionals. Simulation results are also presented both for illustration and validation, together with a real application example on an worn engineered surface.