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 Cabaña'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.