Asymptotical Minimax Recovery of Sets with Smooth Boundaries

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 $S^{N-1}$ into $\mathbb{R}^N$. Furthermore, classes of convex regions are considered. Two models are studied: edge estimation models motivated by image segmentation problems and density support estimation.