The Geometry of Nonparametric Filament Estimation

TL;DR

This paper introduces nonparametric methods for estimating filamentary structures in planar point data, linking geometric properties with statistical estimation and providing convergence rates.

Contribution

It develops a novel approach that transforms boundary estimators into filament estimators, connecting computational geometry with statistical support estimation.

Findings

Filaments can be represented as the medial axis of the data support.

The proposed methods effectively estimate filaments from planar point data.

Convergence rates for the estimators are established.

Abstract

We consider the problem of estimating filamentary structure from planar point process data. We make some connections with computational geometry and we develop nonparametric methods for estimating the filaments. We show that, under weak conditions, the filaments have a simple geometric representation as the medial axis of the data distribution's support. Our methods convert an estimator of the support's boundary into an estimator of the filaments. We also find the rates of convergence of our estimators.