Granulometric Smoothing on Manifolds

TL;DR

This paper introduces a novel, computationally simple method for estimating high-density regions on manifolds using morphological opening and density estimation, with proven consistency and convergence properties.

Contribution

It proposes a new estimator combining morphological opening with density estimation for manifolds, providing theoretical guarantees and practical applicability.

Findings

Estimator is consistent and converges in Hausdorff distance.

Method is easy to compute, involving centers and radii from data.

Applicability demonstrated through illustrative examples.

Abstract

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 the density function. This results in an estimator that is easy-to-compute since it simply consists of a list of centers and a radius $r$ that are adequately selected from the data. The new estimator is shown to be consistent and its convergence rates 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 some illustrative examples.