Choosing thresholds for density-based map construction algorithms

Abstract

Due to the ubiquitous use of various positioning technologies in smart phones and other devices, geospatial tracking data has become a routine data source. One of its uses that has gained recent popularity is the construction of street maps from vehicular tracking data. Due to the inherent noise in the data, many map construction algorithms are based on thresholding a density function. While kernel density estimation provides a firm theoretical foundation for computing the density from the measurements, the thresholds are generally picked in a heuristic, and often brute-force way, which results in slow algorithms with no guarantees on the map construction quality. In this paper, we formalize the selection of thresholds in a density-based street map construction algorithm. We propose a new thresholding technique that uses persistent homology combined with statistical analysis to determine a small set of thresholds that captures all relevant topological features. We formally prove that when the samples are drawn uniformly from the street map, a constant number of thresholds suffices to recover the street map. We also provide algorithms to compute the thresholds for different sampling assumptions. Finally, we show the effectiveness of our algorithms in several experiments on artificially generated data and on real GPS trajectory data.