Geometric pattern matching for point sets in the plane under similarity transformations

Dror Aiger, Klara Kedem

Research output: Contribution to journalArticlepeer-review

11 Scopus citations

Abstract

We consider the following geometric pattern matching problem: Given two sets of points in the plane, P and Q, and some (arbitrary) δ > 0, find a similarity transformation T (translation, rotation and scale) such that h (T (P), Q) < δ, where h (ṡ, ṡ) is the directional Hausdorff distance with L as the underlying metric; or report that none exists. We are only interested in the decision problem, not in minimizing the Hausdorff distance, since in the real world, where our applications come from, δ is determined by the practical uncertainty in the position of the points (pixels). Similarity transformations have not been dealt with in the context of the Hausdorff distance and we fill the gap here. We present efficient algorithms for this problem imposing a reasonable separation restriction on the points in the set Q. If the L distance between every pair of points in Q is at least 8δ, then the problem can be solved in O (m n2 log n) time, where m and n are the numbers of points in P and Q respectively. If the L distance between every pair of points in Q is at least cδ, for some c, 0 < c < 1, we present a randomized approximate solution with expected runtime O (n2 c- 4 ε- 8 log4 m n), where ε > 0 controls the approximation. Our approximation is on the size of the subset, B ⊆ P, such that h (T (B), Q) < δ and | B | > (1 - ε) | P | with high probability.

Original languageEnglish
Pages (from-to)935-940
Number of pages6
JournalInformation Processing Letters
Volume109
Issue number16
DOIs
StatePublished - 31 Jul 2009

Keywords

  • Approximation algorithms
  • Computational geometry
  • Geometric pattern matching
  • Hausdorff distance
  • Randomized algorithms

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Signal Processing
  • Information Systems
  • Computer Science Applications

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