The integral geometry of random sets

In various scientific fields from astro- and high energy physics to neuroimaging, researchers observe entire images or functions rather than single observations. The integral geometric properties, notably the Euler characteristic of the level/excursion sets of these functions, typically modelled as Gaussian random fields have found some interesting applications in these domains.

In this talk, I will describe some statistical applications of the (average) integral geometric properties of these random sets. What makes all calculations possible is the use of the random functions themselves as Morse functions and the Rice-Kac formula for counting the average number of zeros of a function. Most of what I describe is joint work with Robert Adler and Keith Worsley.