Polygon Matching and Indexing Under Affine Transformations

TL;DR

This paper presents efficient algorithms for polygon matching and indexing under affine and similarity transformations, enabling rapid retrieval of matching polygons in large collections even with unknown transformations.

Contribution

It introduces algorithms that perform polygon matching under affine transformations in time independent of collection size, including special cases like triangles and noisy conditions.

Findings

Matching polygons under affine transformations can be done in logarithmic time for known transformations.

Algorithms can find all matching polygons with unknown affine transformations in linear time.

Special bounds are provided for triangle matching in noisy conditions.

Abstract

Given a collection $\{Z_1,Z_2,\ldots,Z_m\}$ of $n$-sided polygons in the plane and a query polygon $W$ we give algorithms to find all $Z_\ell$ such that $W=f(Z_\ell)$ with $f$ an unknown similarity transformation in time independent of the size of the collection. If $f$ is a known affine transformation, we show how to find all $Z_\ell$ such that $W=f(Z_\ell)$ in $O(n+\log(m))$ time. For a pair $W,W^\prime$ of polygons we can find all the pairs $Z_\ell,Z_{\ell^\prime}$ such that $W=f(Z_\ell)$ and $W^\prime=f(Z_{\ell^\prime})$ for an unknown affine transformation $f$ in $O(m+n)$ time. For the case of triangles we also give bounds for the problem of matching triangles with variable vertices, which is equivalent to affine matching triangles in noisy conditions.