Minimizing the Weighted Directed Hausdorff Distance between Colored Point Sets under Translations and Rigid Motions

  • Christian Knauer
  • Klaus Kriegel
  • Fabian Stehn
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5598)

Abstract

Matching geometric objects with respect to their Hausdorff distance is a well investigated problem in Computational Geometry with various application areas. The variant investigated in this paper is motivated by the problem of determining a matching (in this context also called registration) for neurosurgical operations. The task is, given a sequence \(\mathcal{P}\) of weighted point sets (anatomic landmarks measured from a patient), a second sequence \(\mathcal{Q}\) of corresponding point sets (defined in a 3D model of the patient) and a transformation class \(\mathcal{T}\), compute the transformations \(t\in\mathcal{T}\) that minimize the weighted directed Hausdorff distance of \(t(\mathcal{P})\) to \(\mathcal{Q}\). The weighted Hausdorff distance, as introduced in this paper, takes the weights of the point sets into account. For this application, a weight reflects the precision with which a landmark can be measured.

We present an exact solution for translations in the plane, a simple 2-approximation as well as a FPTAS for translations in arbitrary dimension and a constant factor approximation for rigid motions in the plane or in ℝ3.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Christian Knauer
    • 1
  • Klaus Kriegel
    • 1
  • Fabian Stehn
    • 1
  1. 1.Institut für InformatikFreie Universität BerlinGermany

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