Similarity of polygonal curves in the presence of outliers

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Abstract

The Fréchet 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 Fréchet 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 Fréchet 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 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 δ, optimal solutions for the MinEx and MaxIn problems up to an additive approximation error δ times the length of the input curves. The resulting running time is O(n3δlog(nδ)), where n is the complexity of the input polygonal curves.

Keywords

Fréchet distance
Similarity of polygonal curves
Approximation
Weighted shortest path

Cited by (0)

1

Research supported by Fonds de Recherche du Québec – Nature et Technologies.

2

Research supported by High Performance Computing Virtual Laboratory and SUN Microsystems of Canada.

3

Research supported by Natural Sciences and Engineering Research Council of Canada.

4

A part of this research was carried out while the author was visiting Carleton University.