Statistics for Gaussian Random Fields with Unknown Location and Scale using Lipschitz-Killing Curvatures

Elena Di Bernardino (1, 2) , Céline Duval (3)

1 CEDRIC - Centre d'études et de recherche en informatique et communications
2 LJAD - Laboratoire Jean Alexandre Dieudonné
3 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

In this paper we study some statistics linked to the average of Lipschitz-Killing (LK) curvatures of the excursion set of a stationary non-standard isotropic Gaussian field X on R 2. Under this hypothesis of unknown location and scale parameters of X, we introduce novel fundamental quantities, that we call effective level and effective spectral moment, and we derive unbiased and asymptotically normal estimators of these parameters. Furthermore, empirical variance estimators of the asymptotic variance of the third LK curvature of the excursion set (i.e., the area) and of the effective level are proposed. Their consistency is established under a weak condition on the correlation function of X. Finally, using the previous asymptotic results, we built a test to determine if two images of excursion sets can be compared. This test is applied on both synthesized and real mammograms.

Mots clés: Gaussian random fields; Excursion sets; Euler characteristic; Test of Gaussianity; Image analysis

Domaines: Théorie [stat.TH]

Elena Di Bernardino, Céline Duval. Statistics for Gaussian Random Fields with Unknown Location and Scale using Lipschitz-Killing Curvatures. Scandinavian Journal of Statistics, 2022, 49, pp.143-184. ⟨10.1111/sjos.12500⟩. ⟨hal-02317747⟩