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In the present paper, we study three geometric characteristics for the excursion sets of a two dimensional stationary random field. First we show that these characteristics can be estimated without bias if the considered field satisfies a kinematic formula, this is for instance the case for fields that are given by a function of smooth Gaussian fields or for some shot noise fields. By using the proposed estimators of these geometric characteristics, we describe some inference procedures for the estimation of the parameters of the considered field. An extensive simulation study illustrates the performances of each estimator. Then we use one of the previous estimators to build a test to determine whether a given field is Gaussian or not when compared to various alternatives. The test is based on a sparse information i.e. the excursion sets for two different levels of the considered field. Finally this test is adapted and applied in a real-data case: synthesized 2D digital mammograms.

Lipschitz-Killing curvatures of excursion sets for two-dimensional random fields