Minimum Partial-Matching and Hausdorff RMS-Distance under Translation: Combinatorics and Algorithms

  • Rinat Ben-Avraham
  • Matthias Henze
  • Rafel Jaume
  • Balázs Keszegh
  • Orit E. Raz
  • Micha Sharir
  • Igor Tubis
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8737)

Abstract

We consider the RMS-distance (sum of squared distances between pairs of points) under translation between two point sets in the plane. In the Hausdorff setup, each point is paired to its nearest neighbor in the other set. We develop algorithms for finding a local minimum in near-linear time on the line, and in nearly quadratic time in the plane. These improve substantially the worst-case behavior of the popular ICP heuristics for solving this problem. In the partial-matching setup, each point in the smaller set is matched to a distinct point in the bigger set. Although the problem is not known to be polynomial, we establish several structural properties of the underlying subdivision of the plane and derive improved bounds on its complexity. In addition, we show how to compute a local minimum of the partial-matching RMS-distance under translation, in polynomial time.

Keywords

partial matching Hausdorff RMS-distance polyhedral subdivision local minimum 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Rinat Ben-Avraham
    • 1
  • Matthias Henze
    • 2
  • Rafel Jaume
    • 2
  • Balázs Keszegh
    • 3
  • Orit E. Raz
    • 1
  • Micha Sharir
    • 1
  • Igor Tubis
    • 1
  1. 1.Blavatnik School of Computer ScienceTel Aviv UniversityTel AvivIsrael
  2. 2.Institut für InformatikFreie Universität BerlinBerlinGermany
  3. 3.Alfréd Rényi Institute of Mathematics, Hungarian Academy of SciencesBudapestHungary

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