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 P of weighted point sets (anatomic landmarks measured from a patient), a second sequence Q of corresponding point sets (defined in a 3D model of the patient) and a transformation class T, compute the transformations t@?T that minimize the weighted directed Hausdorff distance of t(P) to 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 R^3.