Finding a Curve in a Point Set

TL;DR

This paper addresses the problem of matching a polygonal curve to a point set under various transformations, providing exact and approximation algorithms for different problem variants.

Contribution

It introduces algorithms for the Curve/Point Set Matching problem with transformations, advancing solutions for exact and approximate matching.

Findings

Algorithms for matching with translation, rotation, and affine transforms

Exact solutions for specific problem variants

Approximation algorithms with performance guarantees

Abstract

Let $P$ be a polygonal curve in $\mathbb{R}^D$ of length $n$, and $S$ be a point set of size $k$. The Curve/Point Set Matching problem consists of finding a polygonal curve $Q$ on $S$ such that its Fr\'echet distance from $P$ is less than a given $\varepsilon$. In this paper, we consider this problem with the added freedom to transform the input curve $P$ by translating it, rotating it, or applying an arbitrary affine transform. We present exact and approximation algorithms for several variations of this problem.