The mean Euler characteristic and excursion probability of Gaussian random fields with stationary increments

Representations of Gaussian processes with stationary increments

Boundary Corrections for the Expected Euler Characteristic of Excursion Sets of Random Fields, with an Application to Astrophysics

Certain images arising in astrophysics and medicine are modelled as smooth random fields inside a fixed region, and experimenters are interested in the number of ‘peaks’, or more generally, the topological structure of ‘hot-spots’ present in such an image. This paper studies the Euler characteristic of the excursion set of a random field; the excursion set is the set of points where the image e...

Interpolation of nonstationary fields with stationary increments

The problem of the linear interpolationof nonstationarymultidimensional processes with stationary increments is studied. The expressions of the interpolation filters and of the estimation error are derived, which generalize the results of the interpolation theory for stationary processes. Both finite and infinite extent interpolation are considered. An application to the interpolation of an und...

Limit theorems for excursion sets of stationary random fields

We give an overview of the recent asymptotic results on the geometry of excursion sets of stationary random fields. Namely, we cover a number of limit theorems of central type for the volume of excursions of stationary (quasi–, positively or negatively) associated random fields with stochastically continuous realizations for a fixed excursion level. This class includes in particular Gaussian, P...

Local maxima and the expected Euler characteristic of excursion sets of χ, F and t fields

The maximum of a Gaussian random field was used by Worsley et al. (1992) to test for activation at an unknown point in positron emission tomography images of blood flow in the human brain. The Euler characteristic of excursion sets was used as an estimator of the number of regions of activation. The expected Euler characteristic of excursion sets of stationary Gaussian random fields has been de...