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Interdisciplinary Applied Mathematics Volume 26 Editors S.S. Antman J .E. Marsden L. Sirovich S. Wiggins Geophysics and Planetary Sciences Imaging, Vision, and Graphics D. Geman Mathematical Biology L. Glass, J.D. Murray Mechanics and Materials R.V. Kohn Systems and Control S.S. Sastry, P.S. Krishnaprasad Problems in engineering, computational science, and the physical and biological sciences are using increasingly sophisticated mathematical techniques. Thus, the bridge between the mathematical sciences and other disciplines is heavily trav eled. The correspondingly increased dialog between the disciplines has led to the establishment of the series: Interdisciplinary Applied Mathematics. The purpose of this series is to meet the current and future needs for the interac tion between various science and technology areas on the one hand and mathe matics on the other. This is done, firstly, by encouraging the ways that mathe matics may be applied in traditional areas, as well as point towards new and innovative areas of applications; and, secondly, by encouraging other scientific disciplines to engage in a dialog with mathematicians outlining their problems to both access new methods and suggest innovative developments within mathe matics itself. The series will consist of monographs and high-level texts from researchers working on the interplay between mathematics and other fields of science and technology.

Interdisciplinary Applied Mathematics Volumes published are listed at the end of this book. Springer Science+Business Media, LLC

An Invitation to 3-D Vision From Images to Geometric Models YiMa Stefano Soatto J ana Kosecka S. Shankar Sastry With 170 Illustrations ~ Springer Science+Business Media, LLC

Yi Ma Department of Electrical and Computer Engineering University of Illinois at Urbana.Champaign Urbana, IL 6!801 USA yima@uiuc.edu Jana KolecU Department of Computer Science George Mason University Fairfu, VA 22030 USA kosecka @cs.gmu.edu Edirors S.S. Anunan Department of Mathematics """ Institute fOf Physical Science and Technology UnivClsily of Maryland CoUege Park, MD 20742 USA ssa@math.umd.edu L. Sirovich Division of Applied Mathematics Brown University Providence, RI 02912 USA chico@camclot.mssm.cdu Stefano Soal1o Department of Computer Science University of California, Los Angeles Los Angeles, CA 90095 USA soatlo@uda.edu S. Shankar Sastry Department of Electrical Engineering and Computer Science University of California, Berkeley Berkeley, CA 94120 USA sastry@eecs.betkeley.edu lE. Marsden Conlrol and Dynamical Systems Mail Code 10Hl California Institute of Technology Pasadena, CA 9112!i USA manden@cds.caltech.edu S. Wiggins School of Mathematics University of Bristol Bristol ass ITW UK s.winins@bris.ac.uk CO~tr II/wmJli01l: MA~o-Qr t968 (180)( 180 em) by Victor VlSlUly. CopyriSht MicMle Vuarely. Ma iMmatics Subj«t Classification {2000): StUIO. 68UIO. 6SDt8 ISBN 978-1-4419-1846-8 DOI 10.1007/978-0-387-21779-6 ISBN 978-0-387-21779-6 (eBook) Printed on acid·free ~P((. CI 2004 Springer Science+Husiness Media New York Ori.inally publisbcd by Springer·Verlag New York, Inc. in 2004 Softoover repriDt of the hardcover t&t editiOD 2004 All nihil .nerved. nul work m;oy noI be translated or ~opi¢d in whole or in Part without the written pennjssiOfl of the p~bli$her. (Sprinll-" Sci~«+Business Mflli. N~ .. York) ex«opI for brief elccrplS in c(IfID«tion with revi¢wI N tcholarJy analYI;s. UK in connection with any form of informatioll .. orale and ..,I/i(~al. eteelJonic 1,uplJtion. computer lOflWlrt. or by limil.,. or diSJimilar rmthodololY now known or l\ereafLCr devdoped is fOfbidden. The use in this publicalion of tude na"'u. !ladema~s. Krv;ce ma~l. and limilar lenns. e~n if 1hey are not identified as such. is DOl to be takcn as an eaptcnion of opinion ill> 10 whether N noI theY:lre lubjectto proprietary rilhlS. 98765 432 (EB)

To my mother and my father (Y.M.) To Giuseppe Torresin, Engineer (S.S.) To my parents (J.K.) To my mother (S.S.S.)

Preface This book is intended to give students at the advanced undergraduate or introduc tory graduate level, and researchers in computer vision, robotics and computer graphics, a self-contained introduction to the geometry of three-dimensional (3- D) vision. This is the study of the reconstruction of 3-D models of objects from a collection of 2-D images. An essential prerequisite for this book is a course in linear algebra at the advanced undergraduate level. Background knowledge in rigid-body motion, estimation and optimization will certainly improve the reader's appreciation of the material but is not critical since the first few chapters and the appendices provide a review and summary of basic notions and results on these topics. Our motivation Research monographs and books on geometric approaches to computer vision have been published recently in two batches: The first was in the mid 1990s with books on the geometry of two views, see e.g. [Faugeras, 1993, Kanatani, 1993b, Maybank, 1993, Weng et aI., 1993b]. The second was more recent with books fo cusing on the geometry of multiple views, see e.g. [Hartley and Zisserman, 2000] and [Faugeras and Luong, 2001] as well as a more comprehensive book on computer vision [Forsyth and Ponce, 2002]. We felt that the time was ripe for synthesizing the material in a unified framework so as to provide a self-contained exposition of this subject, which can be used both for pedagogical purposes and by practitioners interested in this field. Although the approach we take in this book deviates from several other classical approaches, the techniques we use are mainly linear algebra and our book gives a comprehensive view of what is known

viii Preface to date on the geometry of 3-D vision. It also develops homogeneous terminology on a solid analytical foundation to enable what should be a great deal of future research in this young field. Apart from a self-contained treatment of geometry and algebra associated with computer vision, the book covers relevant aspects of the image formation process, basic image processing, and feature extraction techniques - essentially all that one needs to know in order to build a system that can automatically generate a 3-D model from a set of 2-D images. Organization of the book This book is organized as follows: Following a brief introduction, Part I provides background material for the rest of the book. Two fundamental transformations in multiple-view geometry, namely, rigid-body motion and perspective projec tion, are introduced in Chapters 2 and 3, respectively. Feature extraction and correspondence are discussed in Chapter 4. Chapters 5, 6, and 7, in Part II, cover the classic theory of two-view geometry based on the so-called epipolar constraint. Theory and algorithms are developed for both discrete and continuous motions, both general and planar scenes, both calibrated and uncalibrated camera models, and both single and multiple moving objects. Although the epipolar constraint has been very successful in the two-view case, Part III shows that a more proper tool for studying the geometry of multiple views is the so-called rank condition on the multiple-view matrix (Chapter 8), which uni fies all the constraints among multiple images that are known to date. The theory culminates in Chapter 9 with a unified theorem on a rank condition for arbitrarily mixed point, line, and plane features. It captures all possible constraints among multiple images of these geometric primitives, and serves as a key to both geomet ric analysis and algorithmic development. Chapter 10 uses the rank condition to reexamine and unify the study of single-view and multiple-view geometry given scene knowledge such as symmetry. Based on the theory and conceptual algorithms developed in the early part of the book, Chapters II and 12, in Part IV, demonstrate practical reconstruction algorithms step-by-step, as well as discuss possible extensions of the theory cov ered in this book. An outline of the logical dependency among chapters is given in Figure 1. Curriculum options Drafts of this book and the exercises in it have been used to teach a one-semester course at the University of California at Berkeley, the University of Illinois at Urbana-Champaign, Washington University in St. Louis, the George Mason University and the University of Pennsylvania, and a one-quarter course at the University of California at Los Angeles. There is apparently adequate material for two semesters or three quarters of lectures. Advanced topics suggested in Part IV or chosen by the instructor can be added to the second half of the sec-

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