Obstacle-avoiding similarity metrics and shortest path problems

Similarity metrics are functions that measure the similarity of geometric objects. The motivation for studying similarity metrics is that these functions are essential building blocks for areas such as computer vision, robotics, medical imaging, and drug design. Although similarity metrics are traditionally computed in environments without obstacles, we use shortest paths to compute similarity metrics in simple polygons, in polygons with polygonal holes, and on polyhedral surfaces. We measure the length of a path either by Euclidean distance or by the number of turns on the path.

We also compute shortest paths that steer a medical needle through a sequence of treatment points in the plane. This technique could be used in biopsy procedures to take multiple tissue samples with a single puncture of the skin. Such an algorithm could also be applied to brachytherapy procedures that implant radioactive pellets at many cancerous locations. Computing shortest paths for medical needles is a challenging problem because medical needles cut through tissue along circular arcs and have a limited ability to turn. Although optimal substructure can fail, we compute globally optimal paths with a wavefront propagation technique.