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.
Supported by the German Research Foundation (DFG), grant KN 591/2-2.
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Knauer, C., Kriegel, K., Stehn, F. (2009). Minimizing the Weighted Directed Hausdorff Distance between Colored Point Sets under Translations and Rigid Motions. In: Deng, X., Hopcroft, J.E., Xue, J. (eds) Frontiers in Algorithmics. FAW 2009. Lecture Notes in Computer Science, vol 5598. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02270-8_13
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DOI: https://doi.org/10.1007/978-3-642-02270-8_13
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