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Algorithms for Three-Dimensional Free-Form Object Matching by Kwang Hee Ko B.S. in Naval Architecture and Ocean Engineering (1995) Seoul National University, Republic of Korea M.S. in Naval Architecture and Marine Engineering (2001) M.S. in Mechanical Engineering (2001) Massachusetts Institute of Technology Submitted to the Department of Ocean Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy MASSACHUSETTS INSTITUTE OF TECHNOLOGY at the AUG 2 5 2003 MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2003 LIBRARIES @ Massachusetts Institute of Technology 2003. All rights reserved. Author........ ............................... V Department of Ocean Engineering March 13, 2003 C ertified b y ................. ............................................ .......... Nicholas M. Patrikalakis, Kawasaki Professor of Engineering Thesis Co-Supervisor Certified by ... Takasdi Maekawa, Principal Research Scientist Thesis Co-Supervisor Accepted by.. ................ Michael S. Triantafyllou, Professor of Ocean Engineering Chairman, Departmental Committee on Graduate Students Department of Ocean Engineering Algorithms for Three-Dimensional Free-Form Object Matching by Kwang Hee Ko Submitted to the Department of Ocean Engineering on March 13, 2003, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract This thesis addresses problems of free-form object matching for the point vs. NURBS surface and the NURBS surface vs. NURBS surface cases, and its application to copy- right protection. Two new methods are developed to solve a global and partial match- ing problem with no a priori information on correspondence or initial transformation and no scaling effects, namely the KH and the umbilic method. The KH method es- tablishes a correspondence between two objects by utilizing the Gaussian and mean curvatures. The umbilic method uses the qualitative properties of umbilical points to find correspondence information between two objects. These two methods are ex- tended to deal with uniform scaling effects. The umbilic method is enhanced with an algorithm for scaling factor estimation using the quantitative properties of umbilical points. The KH method is used as a building block of an optimization scheme based on the golden section search which recovers iteratively an optimum scaling factor. Since the golden section search only requires an initial interval for the scaling factor, the solution process is simplified compared to iterative optimization algorithms, which require good initial estimates of the scaling factor and the rigid body transformation. The matching algorithms are applied to problems of copyright protection. A suspect model is aligned to an original model through matching methods so that similarity between two geometric models can be assessed to determine if the suspect model contains part(s) of the original model. Three types of tests, the weak, intermediate and strong tests, are proposed for similarity assessment between two objects. The weak and intermediate tests are performed at node points obtained through shape intrinsic wireframing. The strong test relies on isolated umbilical points which can be used as fingerprints of an object for supporting an ownership claim to the original model. The three tests are organized in two decision algorithms so that they produce systematic and statistical measures for a similarity decision between two objects in a hierarchical manner. Based on the systematic statistical evaluation of similarity, a decision can be reached whether the suspect model is a copy of the original model. Thesis Co-Supervisor: Nicholas M. Patrikalakis, Kawasaki Professor of Engineering Thesis Co-Supervisor: Takashi Maekawa, Principal Research Scientist Acknowledgments First of all, I want to thank my wife, Suyeon, for her love and emotional support, and my family for their love and understanding during my study at MIT. I would like to thank my thesis supervisors, Professor Nicholas M. Patrikalakis and Dr. Takashi Maekawa, for their expert advice on my research work and instructive guidance on my academic studies, and Professors D. C. Gossard, H. Masuda, S. Sarma and F.-E. Wolter for their helpful advice as members of my doctoral thesis committee. Thanks also go to Professor Takis Sakkalis for his comments, Dr. Constantinos Evangelinos for helpful discussions and efforts for stable hardware environment for my thesis work, Mr. Fred Baker for efforts in the laboratory management, Design Laboratory fellows Dr. Wonjoon Cho, Ms. Hongye Liu, Mr. Da Guo and Mr. Harish Mukundan for making a good laboratory environment, and Dr. Yonghwan Kim, Mr. Jaehyeok Auh, Mr. Sungjoon Kim, Mr. Youngwoong Lee for their help and good advice. Funding for this research was obtained from the National Science Foundation (NSF), under grant number DMI-0010127. Contents Abstract 3 Acknowledgments 4 Contents 5 List of Figures 8 List of Tables 10 List of Symbols 11 1 Introduction 12 1.1 Background and Motivation ............. .......... 12 1.2 Research Objectives. ........... ............. ... 14 1.3 Thesis Organization. ....... .................... 14 2 Theoretical Background 16 2.1 Review of Differential Geometry ... ........ ......... 16 2.1.1 Basic Theory of Surfaces .... ............ ..... 16 2.1.2 Lines of Curvature ....... ............ ..... 18 2.1.3 G eodesics ................ ............. 18 2.1.4 U m bilics .... ........... ............ ... 19 2.2 Review of NURBS Curves and Surfaces ................. 22 3 Mathematical and Computational Prerequisites 25 3.1 Literature Review .............. ............... 25 3.1.1 U m bilics ................. ............. 25 3.1.2 Principal Patches .. ....................... 26 3.2 Rotation and Translation .................... ..... 27 3.3 Lines of Curvature ... ......................... 28 3.4 G eodesics ........ ......................... 29 3.5 Orthogonal Projection of Points and Curves ...... ...... .. 31 3.5.1 Introduction ......... ........... ........ 31 3.5.2 P oints ............ ........... ........ 31 3.5.3 C urves ........ ....................... 32 5 3.5.4 Lines of Curvatures .... ......... .... ..... 33 3.5.5 Geodesics ...................... ...... 34 3.5.6 Calculation of Initial Values ......... ... ... .. 35 3.5.7 Examples ......... ......... .... .. .. 35 3.6 Extraction of Umbilical Points ......... ...... .. .. 36 3.7 Shape Intrinsic Wireframing ................ .. .. 41 3.7.1 Overall Structure .................. .. .. 41 3.7.2 Algorithms for Constructing Quadrilateral Meshes . .. .. 44 3.7.3 Implementation .. .............. .. .. 46 3.7.4 Analysis of the Algorithm .......... .. .. 48 3.8 Interval Projected Polyhedron Algorithm ...... .. .. 48 3.8.1 Robustness in Numerical Computation . .. .. .. 48 3.8.2 Brief Review of Interval Projected Polyhedron Algorithm . .. 49 3.9 Conclusions .............. ........ .. .. .. 49 4 Object Matching 50 4.1 Literature Review .............. ............... 52 4.1.1 Moment Theory .......................... 52 4.1.2 Principal Component Analysis ..... ............. 53 4.1.3 Contour and Silhouette Matching ................ 54 4.1.4 New Representation Scheme .... ............ ... 54 4.1.5 Matching Through Localization/Registration ....... ... 56 4.1.6 Miscellaneous Approaches .................... 57 4.2 Problem Statement .. ............. ............ 59 4.2.1 Matching Objects ......................... 59 4.2.2 Distance Metric ........... ............ ... 59 4.2.3 Distance between a Point and a Parametric Surface ...... 59 4.2.4 Distance Metric Function ........... ......... 60 4.3 Surface Fitting .............................. 60 4.4 Matching Criteria ............................. 60 4.4.1 E-Offset Test .... ............ ........... 61 4.4.2 Principal Curvature and Direction ................ 61 4.4.3 Umbilic Test ............................ 61 4.4.4 Assessment of Matching ......... ........... 61 4.5 Moment Method ............................. 62 4.6 Correspondence Search .......................... 62 4.6.1 Algorithm using Umbilical Points ................ 63 4.6.2 Algorithm using Curvatures ................... 63 4.7 Algorithms with Uniform Scaling Effects ................ 67 4.7.1 Use of Umbilical Points ....... ............ ... 68 4.7.2 Optimization . ........... ............ ... 69 4.7.3 Complexity Analysis ....................... 71 4.7.4 Accuracy Analysis ........... ............. 72 4.7.5 Convergence of the Optimization Method ............ 73 4.8 Performance Considerations ....................... 74 4.9 Conclusions ...................... ......... 76 5 Shape Intrinsic Fingerprints 77 5.1 Introduction ....... ......... ...... 77 5.2 Algorithms ..... ........... ....... 79 5.2.1 Algorithm I .................. 79 5.2.2 Algorithm 2 .. .......... ...... 80 5.3 Conclusions .......... ............ 80 6 Examples and Applications 83 6.1 Object Matching ................... 83 6.1.1 Moment Method . ............... 83 6.1.2 Matching using Umbilics with Scaling Effects 84 6.1.3 Matching using Curvatures ........... 87 6.2 Copyright Protection ..... ......... .... 93 7 Conclusions and Recommendations 109 7.1 Conclusions .... ..... .... 109 7.2 Recommendations for Future Work . 110 A Classification of Umbilical Points 112 A.1 Cubic Form .. ... ... ... .. ... ... 112 A.2 Characteristic Lines vs. Cubic Form ... ... 113 0 ) . A.2.1 F: 0 -+ j(2e' +e-iO . ..... 113 A .2.2 Jw J = 1 .. ............... 113 0 0 A.2.3 F 2 : 0 -+ (2ei + e-2 ) ..... .... 113 A.3 Inverse Transformation .. .... .... ..
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