Granulometric Smoothing on Manifolds

Given a random sample from a density function supported on a manifold M, a new method for the estimating highest density regions of the underlying population is introduced. The new proposal is based on the empirical version of the opening operator from mathematical morphology combined with a preliminary estimator of density function. This results in an estimator that is easy-to-compute since it simply consists of a list of carefully selected centers and a radius. The new estimator is shown to be consistent, and its convergence rate in terms of the Hausdorff distance are provided. All consistency results are established uniformly on the level of the set and for any Riemannian manifold M satisfying mild assumptions. The applicability of the procedure is shown by means of some illustrative examples.