Identification and isotropy characterization of deformed random fields through excursion sets

A deterministic application $\theta\,:\,\mathbb{R}^2\rightarrow\mathbb{R}^2$ deforms bijectively and regularly the plane and allows to build a deformed random field $X\circ\theta\,:\,\mathbb{R}^2\rightarrow\mathbb{R}$ from a regular, stationary and isotropic random field $X\,:\,\mathbb{R}^2\rightarrow\mathbb{R}$. The deformed field $X\circ\theta$ is in general not isotropic, however we give an explicit characterization of the deformations $\theta$ that preserve the isotropy. Further assuming that $X$ is Gaussian, we introduce a weak form of isotropy of the field $X\circ\theta$, defined by an invariance property of the mean Euler characteristic of some of its excursion sets. Deformed fields satisfying this property are proved to be strictly isotropic. Besides, assuming that the mean Euler characteristic of excursions sets of $X\circ\theta$ over some basic domains is known, we are able to identify $\theta$.