  Combining information both within and between sample realizations, we propose
a simple estimator for the local regularity of surfaces in the functional data
framework. The independently generated surfaces are measured with errors at
possibly random discrete times. Non-asymptotic exponential bounds for the
concentration of the regularity estimators are derived. An indicator for
anisotropy is proposed and an exponential bound of its risk is derived. Two
applications are proposed. We first consider the class of multi-fractional,
bi-dimensional, Brownian sheets with domain deformation, and study the
nonparametric estimation of the deformation. As a second application, we build
minimax optimal, bivariate kernel estimators for the reconstruction of the
surfaces.
