Nonlinear estimation in anisotropic multi-index denoising

In the framework of denoising a function depending of a multidimensional variable (for instance an image), we provide a nonparametric procedure which constructs a pointwise kernel estimation with a local selection of the multidimensional bandwidth parameter. Our method is a generalization of the Lepski's method of adaptation, and roughly consists in choosing the “coarsest” bandwidth such that the estimated bias is negligible. However, this notion becomes more delicate in a multidimensional setting. We will particularly focus on functions with inhomogeneous smoothness properties and especially providing a possible disparity of the inhomogeneous aspect in the different directions. We show, in particular that our method is able to exactly attain the minimax rate or to adapt to unknown degree of anisotropic smoothness up to a logarithmic factor, for a large scale of anisotropic Besov spaces.