Redundancy in Gaussian random fields

Valentin de Bortoli (1) , Agnès Desolneux (2, 3, 1) , Bruno Galerne (4) , Arthur Leclaire (5)

1 CB - CB - Centre Borelli - UMR 9010
2 CNRS - Centre National de la Recherche Scientifique
3 ENS Paris Saclay - Ecole Normale Supérieure Paris-Saclay
4 IDP - Institut Denis Poisson
5 IMB - Institut de Mathématiques de Bordeaux

In this paper, we introduce a notion of spatial redundancy in Gaussian random fields. This study is motivated by applications of the a contrario method in image processing. We define similarity functions on local windows in random fields over discrete or continuous domains. We derive explicit Gaussian asymptotics for the distribution of similarity functions when computed on Gaussian random fields. Moreover, for the special case of the squared L2 norm, we give non-asymptotic expressions in both discrete and continuous periodic settings. Finally, we present fast and accurate approximations of these non-asymptotic expressions using moment methods and matrix projections.

Mots clés: Random fields; spatial redundancy; central limit theorem; law of large numbers; eigenvalues approximation; moment methods

Domaines: Mathématiques [math]

Valentin de Bortoli, Agnès Desolneux, Bruno Galerne, Arthur Leclaire. Redundancy in Gaussian random fields. ESAIM: Probability and Statistics, 2020, 24, pp.627-660. ⟨10.1051/ps/2020010⟩. ⟨hal-01931737v1⟩