The effect of discretization on the mean geometry of a 2D random field

Biermé, Hermine; Desolneux, Agnès

Annales Henri Lebesgue, Tome 4 (2021), pp. 1295-1345.

L’étude de la géométrie des ensembles d’excursion des champs aléatoires 2D est une question importante tant d’un point de vue théorique qu’appliqué. Dans cet article nous nous intéressons à la relation qu’il existe entre le périmètre (resp. la courbure totale, liée à la caractéristique d’Euler par le théorème de Gauss–Bonnet) des ensembles d’excursion d’une fonction et de sa discrétisée. Nous utilisons une formulation faible de cette quantité vue comme une fonction qui à un niveau lui associe le périmètre (resp. la courbure totale) de l’excursion correspondante. Nous nous intéressons également à un cadre stochastique où les fonctions sont remplacées par des champs aléatoires. Nous montrons en particulier que, sous des hypothèses de stationarité et d’isotropie sur le champ aléatoire, en moyenne, le périmètre est toujours biaisé (avec un facteur $4/\pi$) contrairement à la courbure totale. Nous illustrons nos résultats sur différents exemples de champs aléatoires.

The study of the geometry of excursion sets of 2D random fields is a question of interest from both the theoretical and the applied viewpoints. In this paper we are interested in the relationship between the perimeter (resp. the total curvature, related to the Euler characteristic by Gauss–Bonnet Theorem) of the excursion sets of a function and the ones of its discretization. Our approach is a weak framework in which we consider the functions that map the level of the excursion set to the perimeter (resp. the total curvature) of the excursion set. We will be also interested in a stochastic framework in which the sets are the excursion sets of 2D random fields. We show in particular that, under some stationarity and isotropy conditions on the random field, in expectation, the perimeter is always biased (with a $4/\pi$ factor), whereas the total curvature is not. We illustrate all our results on different examples of random fields.

Classification : 26B15, 28A75, 60G60, 60D05, 62M40, 60G10, 68R01, 60G22
Mots clés : Perimeter, Total curvature, Euler Characteristic, excursion sets, discrete geometry, stationary random field, image analysis, Gaussian random field

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