Statistics for Gaussian random fields with unknown location and scale using Lipschitz-Killing curvatures

In the present article we study the average of Lipschitz-Killing (LK) curvatures of the excursion set of a stationary isotropic Gaussian field X on Double-struck capital R2. The novelty is that the field can be nonstandard, that is, with unknown mean and variance, which is more realistic from an applied viewpoint. To cope with the unknown location and scale parameters of X, we introduce novel fundamental quantities called effective level and effective spectral moment. We propose unbiased and asymptotically normal estimators of these parameters. From these asymptotic results, we build a test to determine if two images of excursion sets can be compared. This test is applied on both synthesized and real mammograms. Meanwhile, we establish the consistency of the empirical variance estimators of the third LK curvature under a weak condition on the correlation function of X.