Asymptotical Minimax Recovery of Sets with Smooth Boundaries

This is the authors' abstract. We don't add key points for this paper.

In this paper optimal rates of convergence are derived for estimates of sets in N-dimensional "black and white" pictures under smoothness conditions. It is assumed that the boundaries of the "black" regions have a smooth parameterisation, that is, that the boundaries are given by smooth functions from the sphere SN-1 into RN. Furthermore, classes of convex regions are considered. Two models are studied: edge estimation models motivated by image segmentation problems and density support estimation.