Robust Trajectory-based Density Estimation for Geometric Structure Recovery: Theory and Applications

TL;DR

This paper introduces a novel, efficient density estimation method using modified edge quadtrees for geometric structure recovery in spatio-temporal data, outperforming traditional KDE in speed and accuracy.

Contribution

The paper presents a new quadtree-based density estimation technique with mathematical stability guarantees, improving computational efficiency and geometric accuracy over existing methods.

Findings

Significantly faster than KDE in shape extraction tasks.

Achieves comparable or better accuracy in geometry recovery.

Provides stable density estimates under local perturbations.

Abstract

With the rise of the Internet of Things, strategies for effectively processing big data are essential for discovering meaningul insights. The time series datasets produced by groups of interconnected devices contain valuable underlying patterns. Recent works have extracted patterns from spatio-temporal datasets to aid in road network generation, activity recognition, and others. The speed and accuracy of the underlying geometry reconstruction are important in these applications. Existing methods such as kernel density estimation (KDE) have been used but are often computationally expensive. We propose modifying edge quadtrees to utilize their effective heirarchical structure. Our modification estimates density using a novel trajectory count function which provides mathematical guarantees on the stability of the count by enforcing an invariance to local perturbations. We evaluate our method's effectiveness at extracting the underlying geometry and representative subsample points. For verification, we compare against a KDE variant at extracting the underlying shape of noisy synthetic trajectories travelling alonng the shape. We compare map extraction from GPS traces against current methods. Our method significantly improves runtime while extracting the geometry better or at least comparably. We also compare against maxmin subsampling on an activity recognition data set and find a significant runtime improvement with comparable performance.