Gaussian random field's anisotropy using excursion sets

This paper addresses the problem of detecting and estimating the anisotropy of a stationary real-valued random field from a single realization of one of its excursion sets. This setting is challenging as it relies on observing a binary image without prior knowledge of the field's mean, variance, or the specific threshold value. Our first contribution is to propose a generalization of Caba\~na's contour method to arbitrary dimensions by analyzing the Palm distribution of normal vectors along the excursion set boundaries. We demonstrate that the anisotropy parameters can be recovered by solving a smooth and strongly convex optimization problem involving the eigenvalues of the empirical covariance matrix of these normal vectors. Our second main contribution is a new, model-agnostic statistical test for isotropy in dimension two. We introduce a statistic based on the contour method which is asymptotically distributed as a chi-squared variable with two degrees of freedom under the null hypothesis of quasi-isotropy. Unlike existing methods based on Lipschitz-Killing curvatures, this procedure does not require knowledge of the random field's covariance structure. Extensive numerical experiments show that our test is well-calibrated and more powerful than model-based alternatives as well as that the estimation of the anisotropy parameters, including the directions, is robust and efficient. Finally, we apply this framework to test the quasi-isotropy of the Cosmic Microwave Background (CMB) using the Planck data release 3 mission.