In this thesis we present a general framework of geometric partial differential equations from the viewpoint of geometric energy functional. The proposed geometric functional involves the Gaussian curvature, the mean curvature and the squared norms of their gradients. The geometric partial differential equations are given as the Euler-Lagrangian Equations of the geometric energy functionals by using the calculus of variation method. As a special example, we focus on Gaussian curvature related geometric energy functionals and the corresponding partial differential equations. We present three numerical methods to solve the resulting geometric partial differential equations: the direct discretization method, the finite element method and the level set method. We test these numerical schemes with a large class of geometric models. Potential applications of our proposed geometric partial differential equations include mesh optimization, surface smoothing, surface blending, surface restoration and physical simulation. Finally, we point out some possible directions of future work including singular analysis of the derived geometric partial differential equations and numerical error estimates of our numerical schemes. / Yan, Yinhui. / "September 2008." / Adviser: Kwong Chung Piney. / Source: Dissertation Abstracts International, Volume: 73-01, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 121-134). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [201-] System requirements: Adobe Acrobat Reader. Available via World Wide Web.

Wang, Yingze. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 42-47). / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Related Work --- p.5 / Chapter 2.1 --- Line labeling and realization problem --- p.5 / Chapter 2.2 --- 3D reconstruction from multiple views --- p.6 / Chapter 2.3 --- 3D reconstruction from single line drawings --- p.7 / Chapter 2.3.1 --- Face identification from the line drawings --- p.7 / Chapter 2.3.2 --- 3D geometry reconstruction --- p.9 / Chapter 2.4 --- Our research topic and contributions --- p.13 / Chapter 3 --- Reconstruction of Curved Manifold Objects --- p.14 / Chapter 3.1 --- Assumptions and terminology --- p.14 / Chapter 3.2 --- Reconstruction of curved manifold objects --- p.17 / Chapter 3.2.1 --- Distinguishing between curved and planar faces --- p.17 / Chapter 3.2.2 --- Transformation of Line Drawings --- p.20 / Chapter 3.2.3 --- Regularities --- p.23 / Chapter 3.2.4 --- 3D Wireframe Reconstruction --- p.26 / Chapter 3.2.5 --- Generating Curved Faces --- p.28 / Chapter 3.2.6 --- The Complete 3D Reconstruction Algorithm --- p.33 / Chapter 4 --- Experiments --- p.35 / Chapter 5 --- Conclusions and Future Work --- p.40 / Chapter 5.1 --- Conclusions --- p.40 / Chapter 5.2 --- Future work --- p.40 / Bibliography --- p.42

Xue, Tianfan. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (p. 52-55). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Previous Approaches on Face Identification --- p.3 / Chapter 1.1.1 --- Face Identification --- p.3 / Chapter 1.1.2 --- General Objects --- p.4 / Chapter 1.1.3 --- Manifold Objects --- p.7 / Chapter 1.2 --- Previous Approaches on 3D Reconstruction --- p.9 / Chapter 1.3 --- Our approach for Face Identification --- p.11 / Chapter 1.4 --- Our approach for 3D Reconstruction --- p.13 / Chapter 2 --- Face Detection --- p.14 / Chapter 2.1 --- GAFI and its Face Identification Results --- p.15 / Chapter 2.2 --- Our Face Identification Approach --- p.17 / Chapter 2.2.1 --- Real Face Detection --- p.18 / Chapter 2.2.2 --- The Weak Face Adjacency Theorem --- p.20 / Chapter 2.2.3 --- Searching for Type 1 Lost Faces --- p.22 / Chapter 2.2.4 --- Searching for Type 2 Lost Faces --- p.23 / Chapter 2.3 --- Experimental Results --- p.25 / Chapter 3 3 --- D Reconstruction --- p.30 / Chapter 3.1 --- Assumption and Terminology --- p.30 / Chapter 3.2 --- Finding Cuts from a Line Drawing --- p.34 / Chapter 3.2.1 --- Propositions for Finding Cuts --- p.34 / Chapter 3.2.2 --- Searching for Good Cuts --- p.35 / Chapter 3.3 --- Separation of a Line Drawing from Cuts --- p.38 / Chapter 3.4 3 --- D Reconstruction from a Line Drawing --- p.45 / Chapter 3.5 --- Experiments --- p.45 / Chapter 4 --- Conclusion --- p.50

published_or_final_version / abstract / Computer Science / Master / Master of Philosophy

Du, Hao. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (p. 61). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- 3D Reconstruction from 2D Line Drawings and its Applications --- p.1 / Chapter 1.2 --- Algorithmic Development of 3D Reconstruction from 2D Line Drawings --- p.3 / Chapter 1.2.1 --- Line Labeling and Realization Problem --- p.4 / Chapter 1.2.2 --- 3D Reconstruction from Multiple Line Drawings --- p.5 / Chapter 1.2.3 --- 3D Reconstruction from a Single Line Drawing --- p.6 / Chapter 1.3 --- Research Problems and Our Contributions --- p.12 / Chapter 2 --- Adaptive Parameter Setting --- p.15 / Chapter 2.1 --- Regularities in Optimization-Based 3D Reconstruction --- p.15 / Chapter 2.1.1 --- Face Planarity --- p.18 / Chapter 2.1.2 --- Line Parallelism --- p.19 / Chapter 2.1.3 --- Line Verticality --- p.19 / Chapter 2.1.4 --- Isometry --- p.19 / Chapter 2.1.5 --- Corner Orthogonality --- p.20 / Chapter 2.1.6 --- Skewed Facial Orthogonality --- p.21 / Chapter 2.1.7 --- Skewed Facial Symmetry --- p.22 / Chapter 2.1.8 --- Line Orthogonality --- p.24 / Chapter 2.1.9 --- Minimum Standard Deviation of Angles --- p.24 / Chapter 2.1.10 --- Face Perpendicularity --- p.24 / Chapter 2.1.11 --- Line Collinearity --- p.25 / Chapter 2.1.12 --- Whole Symmetry --- p.25 / Chapter 2.2 --- Adaptive Parameter Setting in the Objective Function --- p.26 / Chapter 2.2.1 --- Hill-Climbing Optimization Technique --- p.28 / Chapter 2.2.2 --- Adaptive Weight Setting and its Explanations --- p.29 / Chapter 3 --- Parameter Learning --- p.33 / Chapter 3.1 --- Construction of A Large 3D Object Database --- p.33 / Chapter 3.2 --- Training Dataset Generation --- p.34 / Chapter 3.3 --- Parameter Learning Framework --- p.37 / Chapter 3.3.1 --- Evolutionary Algorithms --- p.38 / Chapter 3.3.2 --- Reconstruction Error Calculation --- p.39 / Chapter 3.3.3 --- Parameter Learning Algorithm --- p.41 / Chapter 4 --- Experimental Results --- p.45 / Chapter 4.1 --- Adaptive Parameter Setting --- p.45 / Chapter 4.1.1 --- Use Manually-Set Weights --- p.45 / Chapter 4.1.2 --- Learn the Best Weights with Different Strategies --- p.48 / Chapter 4.2 --- Evolutionary-Algorithm-Based Parameter Learning --- p.49 / Chapter 5 --- Conclusions and Future Work --- p.53 / Bibliography --- p.55