Minimizing the weighted directed Hausdorff distance between colored point sets under translations and rigid motions

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 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 .