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Modern Mathematics and Applications in Computer Graphics and Vis....pdf

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    Modern Mathematics and Applications in Computer Graphics and Vision 8703_9789814449328_tp.indd 1 12/3/14 2:32 pm
    May 2, 2013 14:6 BC: 8831 - Probability and Statistical Theory PST˙ws This page intentionally left blank This page intentionally left blank
    Modern Mathematics and Applications in Computer Graphics and Vision Hongyu Guo University of Houston — Victoria, USA N E W J E R S E Y • L O N D O N • S I N G A P O R E • B E I J I N G • S H A N G H A I • H O N G K O N G • TA I P E I • C H E N N A I World Scientific 8703_9789814449328_tp.indd 2 12/3/14 2:32 pm
    Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. MODERN MATHEMATICS AND APPLICATIONS IN COMPUTER GRAPHICS AND VISION Copyright © 2014 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 978-981-4449-32-8 Printed in Singapore
    March 5, 2014 13:38 WSPC 9x6–Modern Mathematics and Appl.-8703 guo master page v To Yanping and Alicia v
    May 2, 2013 14:6 BC: 8831 - Probability and Statistical Theory PST˙ws This page intentionally left blank This page intentionally left blank
    March 5, 2014 13:38 WSPC 9x6–Modern Mathematics and Appl.-8703 guo master page vii Preface While I am teaching computer graphics, computer vision and game pro- gramming courses, too often I encounter students in classes, who have a great interest and ambition in these fields but only to feel limited by the lacking of proper mathematics preparation. This is because, unlike other areas of computer science, where discrete mathematics is mostly applied, computer graphics and vision, as well as game programming utilize many areas of continuous mathematics. For example, ray tracing in computer graphics rendering is purely the computational simulation of the optical process using laws in physics. The regular university curriculum does not provide proper preparation of the continuous mathematics for these stu- dents. Researchers and professionals in these areas have their struggles too. Unlike the students, they do not have a problem with multiplying two matrices. However, more and more modern arsenal of abstract mathemat- ics is utilized in research literature, for example, the concepts of tensors and differentiable manifolds. This book is intended to help them in these areas to better understand and apply the concepts and theories needed in their study and research. Intended Audience The intended audience is upper level undergraduate students, graduate stu- dents and researchers in computer graphics and vision and related areas. If you are a believer of the saying “A mathematician is a blind man in a dark room looking for a black cat which is not there”, probably this book is not for you. You should consider returning the book and getting a refund. If you have been looking for a book with a concise account of a certain topic but only was frustrated because the books for mathematics majors have too many details than you needed, I hope this is the book you have been looking for. Students and researchers in physics and engineering share most of the arsenal of mathematics with the computer graphics and vision community and therefore this book should help them as well. If you are a math major and want to take a break from math for a moment, you can read this book. Organization The book is organized in four parts, Part I Algebra, Part II Geometry, Part III Topology and More, and Part IV Applications. Particular emphasis was vii
    March 5, 2014 13:38 WSPC 9x6–Modern Mathematics and Appl.-8703 guo master page viii viii Preface given to tensors and manifolds. The intrinsic view of geometry, initiated by Gauss, is also emphasized. Intrinsic geometry is harder for students due to its intrinsic nature, just like the intrinsic beauty of a person is harder to discover than simple good appearance. This book is not intended to be read from the first page to the last page in the sequential order. This is a “get-what-you-want” book. There are minimal dependencies among the parts and chapters, which is illustrated in the chapter dependency chart. The reader is encouraged to start reading from any part, or chapter and he may skip around. Part of the materials of this book come from the lecture notes for two courses that I have taught, Gaming Mathematics at the un- dergraduate level, and Mathematical Methods in Computer Graphics and Vision at the graduate level, in which I taught the basics of linear algebra, tensor algebra, exterior algebra, quaternion algebra and projective geome- try with applications in computer graphics and vision. Part IV Chapter 2 is based on two of my recently published articles. This book can be used for a selected topics course at the undergraduate or graduate level with free choices of topics by the instructor. The organization of the book is influenced by the structure point of view, which is explained in Chapter 0 as a preliminary. A quick glance of this chapter will benefit in a better understanding of the organization and interrelations of different branches of modern mathematics, as well as the study of each topics. Part I covers some of the algebraic structures, which are discrete structures. Part II covers geometry, which has a mixture of discrete and continuous structures. In the topics in Part III, topological structures, or more generally, continuous structures, play an important role. The topics in Part III are more abstract, compared to Parts I and II. To help the reader, a Venn diagram illustrating the relationship of different structures is provided in the beginning of each chapter in Part III. Approach and Features A bottom-up approach is adopted in the treatment of the materials, which is different from many other books. In the history of mathematics devel- opment, concrete systems were studied first. In modern times, the focus is on abstract systems. Abstract systems are harder for students to under- stand. However, the difficulty is eased if we use the concrete systems as examples and prototypes. The bottom-up approach means always starting with a concrete system and then making a transition and generalization to abstract systems. In most of the cases, the Euclidean space, which is our
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