Finding the upper envelope of n line segments in O(n log n) time
Information Processing Letters
The upper envelope of Voronoi surfaces and its applications
SCG '91 Proceedings of the seventh annual symposium on Computational geometry
A Method for Registration of 3-D Shapes
IEEE Transactions on Pattern Analysis and Machine Intelligence - Special issue on interpretation of 3-D scenes—part II
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
Approximation algorithms for a point-to-surface registration problem in medical navigation
FAW'07 Proceedings of the 1st annual international conference on Frontiers in algorithmics
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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 ***3.