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Conformal Geometry Computational Algorithms and Engineering Appl....pdf

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    Preface
    Contents
    1 Introduction
    1.1 Riemann Mapping Theorem
    1.2 Holomorphic Differential
    1.3 Uniformization Theorem
    1.4 Shape Space
    1.5 Geometric Structures
    Part I Computational Algorithms
    2 Topological Algorithms
    2.1 Halfedge Data Structure
    2.2 Cut Graph and Fundamental Domain
    2.3 Homotopy and Homology Group Basis
    2.4 Canonical Homotopy Group Generators
    3 Harmonic Map
    3.1 Discrete Harmonic Energy
    3.2 Topological Disk Surfaces
    3.3 Topological Sphere Surfaces
    3.4 Riemann Mapping
    4 Harmonic and Holomorphic Forms
    4.1 Characteristic Forms
    4.2 Cohomology Basis
    4.3 Harmonic 1-Form
    4.4 Hodge Star Operator
    4.5 Holomorphic 1-Form
    5 Discrete Ricci Flow
    5.1 Discrete Background
    5.2 Discrete Surface Ricci Flow
    5.3 Isometric Planar Embedding
    5.4 Hyperbolic Ricci Flow
    5.5 Isometric Hyperbolic Embedding
    Part II Engineering Applications
    6 Computer Graphics
    6.1 Introduction
    6.2 Global Surface Parametrization
    6.2.1 Optimal Global Conformal Parametrization Using Discrete One-Form
    6.2.2 Global Conformal Parameterization Using Discrete Euclidean Ricci Flow
    6.3 Uniform Remeshing
    6.3.1 Voronoi Diagram in Different Spaces
    6.3.2 Centroid Voronoi Tessellation in Different Spaces
    6.3.3 CVT Energy in Different Spaces
    6.3.4 Computing Centroidal Voronoi Tessellations
    6.3.5 Uniform Remeshing and Partition
    6.4 Metric-Driven RoSy Fields Design
    6.4.1 Theory of Compatibility
    6.4.2 Computing N-RoSy Fields
    6.4.3 Remeshing
    6.4.4 Celtic Knot on Surfaces
    6.4.5 Pen-and-Ink Sketching of Surfaces
    6.5 Computing Shortest Homotopic Cycles on Polyhedral Surfaces
    6.5.1 Geodesic Uniqueness
    6.5.2 Fuchsian Group Method
    6.5.3 Birkhoff Curve Shortening Method
    6.5.4 Extremal Quasiconformal Mapping
    6.5.5 Homotopy Detection
    6.6 Summary and Further Reading
    7 Computer Vision
    7.1 Introduction
    7.2 Teichmuller Shape Space
    7.2.1 Luo Coordinates
    7.2.2 Fenchel-Nielsen Coordinates
    7.2.3 Robustness of Teichmuller Space Coordinates
    7.2.4 Surface Indexing and Classification
    7.3 3D Facial Shape Index
    7.3.1 Generalized Discrete Ricci Flow
    7.3.2 Doubly Connected Domain
    7.3.3 Multiply Connected Domain
    7.4 Shape Signature
    7.4.1 Beltrami Equation
    7.4.2 Conformal Module
    7.4.3 Holomorphic Differentials
    7.4.4 Conformal Welding
    7.4.5 Computing Shape Signatures of Planar Domains
    7.4.6 Shape Clustering
    7.5 Summary and Further Reading
    8 Geometric Modeling
    8.1 Introduction
    8.2 Triangular B-Splines
    8.2.1 Definition
    8.2.2 Properties
    8.2.3 Surface Reconstruction
    8.3 Theoretical Foundation of Manifold Splines
    8.3.1 Definitions and Concepts
    8.3.2 Equivalence to Affine Atlas
    8.3.3 Existence
    8.4 Constructing Affine Atlas
    8.4.1 Computing Affine Structure Using Holomorphic 1-form
    8.4.2 Computing Affine Structure Using Discrete Ricci Flow
    8.4.3 Computing Affine Structure Using Polycube Maps
    8.5 Extending Planar Splines to Manifold Domains
    8.5.1 Manifold Triangular B-spline
    8.5.2 Manifold Powell–Sabin Spline
    8.5.3 Manifold T-spline
    8.6 Handling Extraordinary Points
    8.7 Discussions
    8.7.1 Domain Manifold
    8.7.2 Manifold Splines Versus Planar Splines
    8.7.3 Manifold Splines Versus Spherical Triangular B-Splines
    8.7.4 Manifold Splines Versus Subdivision Surfaces
    8.7.5 General Manifold Spline Program
    8.8 Summary
    9 Medical Imaging
    9.1 Introduction
    9.2 Brain Surface Conformal Parameterization
    9.2.1 Spherical Brain Conformal Parameterization
    9.2.2 Planar Brain Conformal Parameterization with Holomorphic Functions
    9.2.3 Hyperbolic Brain Conformal Parameterization with Ricci Flow Method
    9.3 Brain Surface Registration
    9.3.1 Optimization of Brain Conformal Mapping with Landmarks
    9.3.2 Constrained Harmonic Map
    9.3.3 Hyperbolic Harmonic Map
    9.3.4 Surface Fluid Registration
    9.4 Global Transformation-Invariant Shape Descriptors
    9.4.1 Spherical Harmonic Analysis Based Rotation Invariant Shape Descriptor
    9.4.2 Conformal Welding Based Teichmüller Shape Descriptor
    9.4.3 Teichmüller Space Coordinates for Landmark Curve-Based Brain Morphometry Analysis
    9.4.4 Riemannian Optimal Mass Transport Map and Riemannian Wasserstein Distance
    9.5 Point-to-Point Local Surface Deformation Measurements
    9.5.1 Genetic Influence of APOE4 Genotype on Hippocampal Morphometry
    9.5.2 Ventricular Abnormalities in Mild Cognitive Impairment
    9.5.3 Computer-Assisted Diagnosis with Computational Conformal Geometry
    9.6 Summary
    10 Wireless Sensor Networks
    10.1 Introduction
    10.2 Localization
    10.2.1 Planar Sensor Network Localization
    10.2.2 Surface Sensor Network Localization
    10.3 Greedy Routing
    10.3.1 3D Wireless Sensor Networks Without Hole
    10.3.2 3D Wireless Sensor Networks with Internal Holes
    10.4 Deployment
    10.4.1 Optimal Surface Network Deployment
    10.4.2 Optimal Solution
    10.4.3 Decreased Sensing Unreliability
    10.4.4 Unreliability Function
    10.4.5 Special Scenarios
    10.5 In-Network Data-Centric Processing
    10.5.1 Double-Ruling in 3D Network
    10.5.2 Data Replication
    10.5.3 Data Retrieval
    10.5.4 Delivery of Data and Query
    10.5.5 Storage
    10.6 Marching of Autonomous Networked Robots
    10.6.1 Optimal Marching Problem
    10.6.2 An Approximated Solution
    10.6.3 Modified Harmonic Map
    10.6.4 Minor Local Adjustment
    10.6.5 Global Connectivity
    10.6.6 Moving Distance
    10.6.7 Holes
    10.7 Summary and Further Reading
    References
    Index
    Miao Jin Xianfeng Gu Ying He Yalin Wang Conformal Geometry Computational Algorithms and Engineering Applications
    Conformal Geometry
    Miao Jin Xianfeng Gu Ying He Yalin Wang Conformal Geometry Computational Algorithms and Engineering Applications 123
    Miao Jin The Centre for Advanced Computer Studies University of Louisiana Lafayette, LA USA Xianfeng Gu State University of New York Stony Brook, NY USA Ying He School of Computer Science and Engineering Nanyang Technological University Singapore Singapore Yalin Wang School of Computing, Informatics and Decision Systems Engineering Arizona State University Tempe, AZ USA ISBN 978-3-319-75330-0 https://doi.org/10.1007/978-3-319-75332-4 ISBN 978-3-319-75332-4 (eBook) Library of Congress Control Number: 2018934929 © Springer International Publishing AG, part of Springer Nature 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. trademarks, service marks, etc. Printed on acid-free paper This Springer imprint is published by the registered company Springer International Publishing AG part of Springer Nature The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
    For those who love geometry.
    Preface Conformal means angle preserving in mathematics. Conformal geometry studies conformal structure of general surfaces. Conformal structure is a natural geometric structure and a special atlas on surfaces such that angles among tangent vectors can be coherently defined on different local coordinate systems. Conformal structure governs many physics phenomena including heat diffusion and electric–magnetic fields. Computational conformal geometry focuses on algorithmic study of conformal geometry and offers powerful tools to handle a broad range of geometric problems in engineering fields. It links modern geometry theories to real engineering applications. The power of computational conformal geometry in engineering fields stems from the following fundamental reasons. Conformal geometry studies surface conformal structure. All surfaces in daily life have a natural conformal structure. Therefore, geometric algorithms based on conformal geometry benefit general surfaces. Conformal structure of a general surface is more flexible than Riemannian metric structure and more rigid than topological structure. It can handle large deformations, which Riemannian geometry cannot efficiently handle; it pre- serves a lot of geometric information during the deformation, whereas topo- logical methods lose too much information. Conformal maps are easy to control. For example, the conformal maps between two simply-connected closed surfaces form a six-dimensional space; therefore by fixing three points, the mapping is uniquely determined. This fact makes con- formal geometric method very valuable for surface matching and comparison. Conformal maps preserve local shapes; therefore, it is convenient for visual- ization purposes. All surfaces can be classified according to their conformal structures, and all the conformal equivalent classes form a finite-dimensional manifold. This manifold has rich geometric structures and can be analyzed and studied. In comparison, vii
    viii Preface the isometric classes of surfaces form an infinite-dimensional space, and it is really difficult to deal with. Computational conformal geometric algorithms are based on solving elliptic partial differential equations, which are easy to solve, and the solving process is stable; namely, the solution is insensitive to the noise of the input surfaces. Therefore, computational conformal geometry method is very practical for real engineering applications. In conformal geometry, all surfaces in daily life can be deformed to three canonical spaces, the sphere, the plane, or the disk (the hyperbolic space). In other words, any surface admits one of the three canonical geometries, spherical geometry, Euclidean geometry, or the hyperbolic geometry. Most digital geo- metric processing tasks in three-dimensional space can be converted to the task in these two-dimensional canonical spaces. The book provides an overview of computational conformal geometry applied in engineering fields. We first briefly introduce the major concepts and theorems of conformal geometry in an intuitive way with a large number of illustrative images rendered by graphics tools. In part I of the book, we detail the major computational algorithms in conformal geometry in an accessible way for computer scientists and engineers. We provide less abstract mathematical reasoning, but more intuitive explanations and imple- mentation issues from the engineering point of view. In part II of the book, we dedicate each chapter to a specific application field of computational conformal geometry including computer graphics, computer vision, geometric modeling, medical imaging, and wireless sensor networks. We discuss the fundamental problems, and how computational conformal geometric methods tackle them in a theoretically elegant and computationally efficient way in each field. Computational conformal geometry is an emerging field. There are still a lot of challenging and open problems both in theory and in practice. Applying compu- tational conformal geometric methods to broader applications and adapting them to real systems are still developing. The book will be of interest to senior under- graduates, graduates, and researchers in computer science, applied mathematics, and wide branches of engineering. Lafayette, LA, USA Stony Brook, NY, USA Singapore Tempe, AZ, USA Miao Jin Xianfeng Gu Ying He Yalin Wang
    Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Riemann Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 1.2 Holomorphic Differential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Uniformization Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Shape Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 1.5 Geometric Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part I Computational Algorithms 2 Topological Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Halfedge Data Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 2.2 Cut Graph and Fundamental Domain . . . . . . . . . . . . . . . . . . . . Homotopy and Homology Group Basis . . . . . . . . . . . . . . . . . . 2.3 2.4 Canonical Homotopy Group Generators . . . . . . . . . . . . . . . . . . 3 Harmonic Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discrete Harmonic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . Topological Disk Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . Topological Sphere Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . Riemann Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 3.2 3.3 3.4 4 Harmonic and Holomorphic Forms . . . . . . . . . . . . . . . . . . . . . . . . Characteristic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cohomology Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Harmonic 1-Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hodge Star Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Holomorphic 1-Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 4.2 4.3 4.4 4.5 5 Discrete Ricci Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Discrete Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discrete Surface Ricci Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 1 1 2 3 4 5 11 11 12 13 16 19 19 21 23 25 29 29 31 32 33 34 39 39 43 ix
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