Progressive Simplification of Polygonal Curves

TL;DR

This paper introduces an efficient algorithm for progressive polygonal curve simplification across multiple scales, compatible with various distance measures, and demonstrates practical speedups through shortcut graph techniques.

Contribution

It presents a novel $O(n^3m)$-time algorithm for multi-scale curve simplification that minimizes cumulative complexity and is compatible with multiple distance metrics.

Findings

Algorithm achieves consistent simplifications across scales.

Shortcut graph techniques significantly improve speed and memory efficiency.

Experimental results validate practical effectiveness on trajectory data.

Abstract

Simplifying polygonal curves at different levels of detail is an important problem with many applications. Existing geometric optimization algorithms are only capable of minimizing the complexity of a simplified curve for a single level of detail. We present an $O(n^3m)$-time algorithm that takes a polygonal curve of n vertices and produces a set of consistent simplifications for m scales while minimizing the cumulative simplification complexity. This algorithm is compatible with distance measures such as the Hausdorff, the Fr\'echet and area-based distances, and enables simplification for continuous scaling in $O(n^5)$ time. To speed up this algorithm in practice, we present new techniques for constructing and representing so-called shortcut graphs. Experimental evaluation of these techniques on trajectory data reveals a significant improvement of using shortcut graphs for progressive and non-progressive curve simplification, both in terms of running time and memory usage.