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Digital and Discrete Geometry Li M Digital and Discrete Geometry Li M. Chen Digital and Discrete Geometry Theory and Algorithms 2123 Li M. Chen University of the District of Columbia Washington District of Columbia USA ISBN 978-3-319-12098-0 ISBN 978-3-319-12099-7 (eBook) DOI 10.1007/978-3-319-12099-7 Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number: 2014958741 © Springer International Publishing Switzerland 2014 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, trademarks, service marks, etc. 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. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) To the researchers and their supporters in Digital Geometry and Topology. Preface Discrete geometry is the study of the geometric properties of discrete objects includ- ing lines, triangles, rectangles, circles, cubes, and spheres. These shapes are usually subsets of Euclidean space. On the other hand, digital geometry has two meanings: (1) The objects are formed by digital or integer points, more narrow digital geometry; and (2) The objects are computerized formations of geometric data. Sometimes we can view digital geometry as a subcategory of discrete geometry. While discrete geometry has a long history, it has recently garnered much attention due to its large role in fulfilling computer vision and image processing needs. Such a need is the motivation behind the creation of digital geometry. The subject provides tremendous new research areas within discrete geometry. In the past, geometric tiling and counting were the primary research topics in discrete geometry. Digital geometry mainly comes from two areas: image processing and computer graphics. A digital image in 2D is in the form of digital grid points; it is a natural treatment of using geometry in image processing including segmentation, recogni- tion, and reconstruction. On the other hand, computer graphics use geometric design, object dynamics, and modification. Computerized geometry must deal with efficient algorithms for many applications including classifications of digital objects, which also uses topological properties and geometry processing. It can be applied to a vast number of areas including biomathematics, medical imaging, the film industry, etc. Digital geometry is also highly related to algorithmic geometry (computational geometry), which is more focused on algorithm design for discrete objects in Eu- clidean space. However, digital geometry has its own set of problems and challenges including those involving distance measure and the formatting of digital objects, which are different than that of discrete objects. Digital geometry also has some advantages since sampling the data can usually be directly applied in its digital form. There is no need to do a conversion from discrete forms. This book provides detailed methods and algorithms in discrete geometry, espe- cially digital geometry. We also provide the necessary knowledge in its connections to other types of geometry such as differential geometry and algebraic topology. In addition, there is much discussion on the recent development of applications in variety of methods of image processing, computer vision, and computer graphics. vii viii Preface This book is intended to offer comprehensive coverage of the modern methods for geometric problems in the computing sciences. We also discuss concurrent topics in BigData and data science as well. This book is written to be suitable to different groups of readers. Chapters 1– 6 are for junior and senior college students in computer science and mathematics; Chaps. 7–12 are for graduate students. Chapters 13–15 are written for researchers or students with advanced knowledge in geometry and topology. This book can also be categorized into three parts: (a) Chaps. 1–9 are introduc- tions to digital and discrete geometry, (b) Chaps. 10–12 mainly deal with geometric processing for readers interested in applications, and (c) Chaps. 13–15 present topics in high level mathematics that are related to discrete geometry. The sections marked with “*” may require some advanced knowledge. The book is also self-contained. Acknowledgments: Many thanks to my daughter Boxi Cera Chen who helped me correct my grammar for the whole book. Many thanks to my wife Lan Zhang and my son Kyle Chen for their patience and support while I was working on this book. I never thought that I had to put increasingly more effort into completing this book. Many thanks to my colleagues Professors Feng Luo, Petra Wiederhold, and Sherali Zeadally for their continued support and encouragement. Many thanks to my colleagues in digital topology for their help and support, Professors Reinhard Klette, Reneta Barneva, Jacques-Olivier Coeurjolly, Tae Yung Kong, and Konrad Polthier, just name a few. Thanks also to UDC for giving me one semester of sabbatical to work on this project. My goal was set to write a complete introductory and comprehensive book to digital and discrete geometry. As I was reviewing my writing today, I found that it is still too far from reaching this initial vision. I hope that this book has laid a good foundation for learning digital and discrete geometry, as well as linking to various topics as a stepping stone to future research in this relatively new discipline of computer science and mathematics. Aug., 2014 Li M. Chen Washington, DC Contents Part I Basic Geometry 1 Introduction .................................................. 3 1.1 What is Geometry ......................................... 3 1.2 Contemporary Geometries in Modern Computer Times .......... 5 1.2.1 Discrete Geometry .................................. 5 1.2.2 Digital Geometry ................................... 6 1.2.3 Computational Geometry and Numerical Geometry ...... 6 1.3 Geometry and Topology in Image Processing and Computer Graphics ................................................. 7 1.4 Problems and Concepts of Digital and Discrete Geometry ........ 9 1.4.1 Some Developments in Digital Geometry ............... 12 References .................................................... 13 2 Discrete Spaces: Graphs, Lattices, and Digital Spaces ............. 17 2.1 Objects in Discrete Spaces .................................. 17 2.2 Graphs and Simple Graphs .................................. 18 2.2.1 Basic Concepts of Graphs ............................ 18 2.2.2 Special Graphs ..................................... 20 2.3 Basic Topics and Results in Graph Theory ..................... 20 2.3.1 Graph Representation, Searching Graph, and Graph Coloring ......................................... 21 2.3.2 The Minimum Spanning Tree ......................... 23 2.3.3 The Shortest Path* .................................. 23 2.3.4 Graph Homomorphism and Graph Isomorphism* ........ 25 2.4 Lattice Graphs, Triangulated Space, and Grid Space............. 25 2.5 Basic Concepts of Digital Spaces ............................ 26 2.5.1 2D and 3D Digital Spaces ............................ 26 2.5.2 mD Digital Spaces* ................................. 28 2.5.3 Points, Line-cells, and Surface-cells in Digital Space ..... 28 2.5.4 Points in Digital Space and Data in Real World .......... 29 ix x Contents 2.6 Characteristics of General Discrete Spaces .................... 30 2.7 Historical Remarks on Digital Space ......................... 31 References .................................................... 33 3 Euclidean Space and Continuous Space .......................... 35 3.1 Euclidean Space and Properties .............................. 35 3.1.1 Euclidean Spaces ................................... 36 3.1.2 Definition of Metrics ................................ 38 3.1.3 Spheres and Distance on Spheres ...................... 38 3.1.4 Two Inequalities of Euclidean Space* .................. 39 3.2 Functions on Euclidean Space ............................... 40 3.2.1 Geometric Transformation, Linear Transformation, and Matrix Algebra ................................. 41 3.3 Topological Spaces and Manifolds ........................... 43 3.4 Decomposition: From Continuous Space to Discrete Space ....... 44 3.5 Remark .................................................. 46 References .................................................... 46 Part II Digital Curves, Surfaces, and Manifolds 4 Digital Planar Geometry: Curves and Connected Regions ......... 49 4.1 General Continuous Curves and Discrete Curves ............... 49 4.2 Curves in Discrete Forms ................................... 50 4.3 Digital Curves in Σ2 ...................................... 52
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