No abstract

A deterministic application θ : R 2 → R 2 deforms bijectively and regularly the plane and allows to build a deformed random field X • θ : R 2 → R from a regular, stationary and isotropic random field X : R 2 → R. The deformed field X • θ is in general not isotropic, however we give an explicit characterization of the deformations θ that preserve the isotropy. Further assuming that X is Gaussian, we introduce a weak form of isotropy of the field X • θ, 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 • θ over some basic domains is known, we are able to identify θ.Deformed fields are a class of non-stationary and non-isotropic fields obtained by deforming a fixed stationary and isotropic random field thanks to a deterministic function that transforms bijectively the index set. Deformed fields respond to the need to model spatial and physical phenomena that are in numerous cases not stationary nor isotropic. To give but one example, they are currently widely used in cosmology to model the cosmic microwave background (CMB) deformed anisotropically by the gravitational lensing effect, with mass reconstruction as an objective [14].Our framework is two-dimensional: we set X : R 2 → R the underlying stationary and isotropic field, θ : R 2 → R 2 a bijective deterministic function and X θ = X •θ the deformed field. In fact, most studies on the deformed field model deal with dimension two. The reason for this is that it is the simplest case of multi-dimensionality, the results can be illustrated easily thanks to simulations and it still covers a lot of possible applications, particularly in image analysis. For instance, deformed fields are involved in the "shape from texture" issue, that is, the problem of recovering a 3-dimensional textured surface thanks to a 2-dimensional projection [8].The model of deformed fields was introduced in 1992 in a spatial statistics framework by Sampson and Guttorp in [18], with only a stationarity assumption on X. It is also studied through the covariance function in [16] and in [17].