On the path density of a gradient field

TL;DR

This paper introduces a statistically rigorous method for detecting multiple filaments in noisy point cloud data by analyzing the concentration of steepest ascent paths derived from kernel density estimates.

Contribution

It proposes a novel approach that constructs a path density estimator to reliably identify filaments with statistical guarantees, addressing limitations of existing methods.

Findings

Method effectively detects filaments in noisy data

Provides a consistent estimator of path density

Handles multiple filaments of various sizes and shapes

Abstract

We consider the problem of reliably finding filaments in point clouds. Realistic data sets often have numerous filaments of various sizes and shapes. Statistical techniques exist for finding one (or a few) filaments but these methods do not handle noisy data sets with many filaments. Other methods can be found in the astronomy literature but they do not have rigorous statistical guarantees. We propose the following method. Starting at each data point we construct the steepest ascent path along a kernel density estimator. We locate filaments by finding regions where these paths are highly concentrated. Formally, we define the density of these paths and we construct a consistent estimator of this path density.