Similarity of Polygonal Curves in the Presence of Outliers

TL;DR

This paper introduces a robust variant of the Fréchet distance for polygonal curves that accounts for outliers by minimizing ignored subcurve lengths or maximizing preserved subcurve lengths, providing approximation algorithms with provable bounds.

Contribution

It formulates the MinEx and MaxIn problems as robust measures of curve similarity, proves their algebraic complexity, and offers an efficient approximation algorithm with explicit error bounds.

Findings

Exact solutions are algebraically intractable.

Proposed algorithm approximates solutions within additive error.

Algorithm runs in near-cubic time relative to input size.

Abstract

The Fr\'{e}chet distance is a well studied and commonly used measure to capture the similarity of polygonal curves. Unfortunately, it exhibits a high sensitivity to the presence of outliers. Since the presence of outliers is a frequently occurring phenomenon in practice, a robust variant of Fr\'{e}chet distance is required which absorbs outliers. We study such a variant here. In this modified variant, our objective is to minimize the length of subcurves of two polygonal curves that need to be ignored (MinEx problem), or alternately, maximize the length of subcurves that are preserved (MaxIn problem), to achieve a given Fr\'{e}chet distance. An exact solution to one problem would imply an exact solution to the other problem. However, we show that these problems are not solvable by radicals over $\mathbb{Q}$ and that the degree of the polynomial equations involved is unbounded in general. This motivates the search for approximate solutions. We present an algorithm, which approximates, for a given input parameter $\delta$, optimal solutions for the \MinEx\ and \MaxIn\ problems up to an additive approximation error $\delta$ times the length of the input curves. The resulting running time is upper bounded by $\mathcal{O} \left(\frac{n^3}{\delta} \log \left(\frac{n}{\delta} \right)\right)$, where $n$ is the complexity of the input polygonal curves.