Abstract
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, in expectation, under some stationarity and isotropy conditions on the random field, the perimeter is always biased (with a 4/π factor), whereas the total curvature is not. We illustrate all our results on different examples of random fields.
Highlights
Understanding the geometry of excursion sets of random fields is a question that receives much attention from both the theoretical and the applied point of view
The considered quantities are the surface area, the perimeter and the Euler characteristic, i.e. the number of connected components minus the number of holes, of a black-and-white image obtained by thresholding a gray-level image at some fixed level, corresponding to an excursion set
In our previous paper [BD20], we have introduced functionals that allow us to give formulas for the perimeter and for the total curvature of the excursion sets of a function defined on an open set of R2
Summary
Understanding the geometry of excursion sets of random fields is a question that receives much attention from both the theoretical and the applied point of view (see [Adl00] for instance). In our previous paper [BD20], we have introduced functionals that allow us to give (weak) formulas for the perimeter and for the total curvature (related to the Euler Characteristic, by Gauss–Bonnet Theorem) of the excursion sets of a function defined on an open set of R2. Another way to obtain discrete functions is to discretize a smooth function (or random field) This is what we do, and we give the limits as the tile size goes to 0, showing that the level curvature integral behaves well, whereas the level perimeter integral has a bias that we quantify. In the Appendix, we have postponed some technical proofs and we propose an unbiased way to compute the level perimeter integral
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