Obstacle-avoiding Similarity Metrics and Shortest Path Problems

DEDICATION This dissertation is dedicated to my father and mother and to the best advisor in the galaxy. Words cannot express my gratitude for the encouragement, humor, and inspiration you provided. ACKNOWLEDGMENTS I am eternally grateful to my advisor Carola Wenk for countless meetings and proofreading sessions. Our meetings always left me excited about research, and I would do well to emulate your cheerful and ever-optimistic demeanor. Thank you for inspiration. Special thanks also go to Tom Bylander, José Iovino, Jianhua Ruan, and Weining Zhang for their insights regarding this disser-tation. submitted to the 9th Latin American Theoretical Informatics Symposium. " This Doctoral Dissertation was produced in accordance with guidelines which permit the inclusion as part of the Doctoral Dissertation the text of an original paper, or papers, submitted for publication. The Doctoral Dissertation must still conform to all other requirements explained in the " Guide for the Preparation of a Master's Thesis/Recital Document or Doctoral Dissertation at The University of Texas at San Antonio. " It must include a comprehensive abstract, a full introduction and literature review, and a final overall conclusion. Additional material (procedural and design data as well as descriptions of equipment) must be provided in sufficient detail to allow a clear and precise judgment to be made of the importance and originality of the research reported. It is acceptable for this Doctoral Dissertation to include as chapters authentic copies of papers already published, provided these meet type size, margin, and legibility requirements. In such cases, connecting texts, which provide logical bridges between different manuscripts, are mandatory. Where the student is not the sole author of a manuscript, the student is required to make an explicit statement in the introductory material to that manuscript describing the student's contribution to the work and acknowledging the contribution of the other author(s). The signatures of the Supervising Committee which precede all other material in the Doctoral Dissertation attest to the accuracy of this statement. 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 …