Synopses & Reviews
Endowing machines with a sense of vision has been a dream of scientists and engineers alike for over half a century. Only in the past decade, however, has the geometry of vision been understood to the point where this dream becomes attainable, thanks also to the remarkable progress in imaging and computing hardware. This book addresses a central problem in computer vision -- how to recover 3-D structure and motion from a collection of 2-D images -- using techniques drawn mainly from linear algebra and matrix theory. The stress is on developing a unified framework for studying the geometry of multiple images of a 3-D scene and reconstructing geometric models from those images. The book also covers relevant aspects of image formation, basic image processing, and feature extraction. The authors bridge the gap between theory and practice by providing step-by-step instructions for the implementation of working vision algorithms and systems. Written primarily as a textbook, the aim of this book is to give senior undergraduate and beginning graduate students in computer vision, robotics, and computer graphics a solid theoretical and algorithmic foundation for future research in this burgeoning field. It is entirely self-contained with necessary background material covered in the beginning chapters and appendices, and plenty of exercises, examples, and illustrations given throughout the text.
Review
From the reviews: "Computer vision is invading our daily lives ... . Covering all the aspects would be too vast an area to cover in one book, so here, the authors concentrated on the specific goal of recovering the geometry of a 3D object ... . The 22 pages of references form a good guide to the literature. The authors found an excellent balance between a thorough mathematical treatment and the applications themselves. ... the text will be a pleasure to read for students ... ." (Adhemar Bultheel, Bulletin of the Belgian Mathematical Society, Vol. 12 (2), 2005) "This is primarily a textbook of core principles, taking the reader from the most basic concepts of machine vision ... to detailed applications, such as autonomous vehicle navigation. ... It is a clearly written book ... . Everything that is required is introduced ... . an entirely self-contained work. ... The book is aimed at graduate or advanced undergraduate students in electrical engineering, computer science, applied mathematics, or indeed anyone interested in machine vision ... . is highly recommended." (D.E. Holmgren, The Photogrammetric Record, 2004) "This very interesting book is a great book teaching how to go from two-dimensional (2D)-images to three-dimensional (3D)-models of the geometry of a scene. ... A good part of this book develops the foundations of an appropriate mathematical approach necessary for solving those difficult problems. ... Exercises (drill exercises, advanced exercises and programming exercises) are provided at the end of each chapter." (Hans-Dietrich Hecker, Zentralblatt MATH, Vol. 1043 (18), 2004) "This book gives senior undergraduate and beginning graduate students and researchers in computer vision, applied mathematics, computer graphics, and robotics a self-contained introduction to the geometry of 3D vision. That is the reconstruction of 3D models of objects from a collection of 2D images. ... Exercises are provided at the end of each chapter. Software for examples and algorithms are available on the author's website." (Daniel Leitner, Simulation News Europe, Vol. 16 (1), 2006)
Synopsis
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."
Synopsis
This book is intended to give undergraduate and beginning graduate students and researchers in computer vision, applied mathematics, robotics, and computer graphics a self-contained introduction to the geometry of 3D vision. Exercises are provided at the end of each chapter.
Synopsis
This book introduces the geometry of 3-D vision, that is, the reconstruction of 3-D models of objects from a collection of 2-D images. It details the classic theory of two view geometry and shows that a more proper tool for studying the geometry of multiple views is the so-called rank consideration of the multiple view matrix. It also develops practical reconstruction algorithms and discusses possible extensions of the theory.
Table of Contents
Preface 1 Introduction 1.1 Visual perception: from 2-D images to 3-D models 1.2 A mathematical approach 1.3 A historical perspective I Introductory material 2 Representation of a three-dimensional moving scene 2.1 Three-dimensional Euclidean space 2.2 Rigid body motion 2.3 Rotational motion and its representations 2.4 Rigid body motion and its representations 2.5 Coordinate and velocity transformations 2.6 Summary 2.7 Exercises 2.A Quaternions and Euler angles for rotations 3 Image formation 3.1 Representation of images 3.2 Lenses, light, and basic photometry 3.3 A geometric model of image formation 3.4 Summary 3.5 Exercises 3.A Basic photometry with light sources and surfaces 3.B Image formation in the language of projective geometry 4 Image primitives and correspondence 4.1 Correspondence of geometric features 4.2 Local deformation models 4.3 Matching point features 4.4 Tracking line features 4.5 Summary 4.6 Exercises 4.A Computing image gradients II Geometry of two views 5 Reconstruction from two calibrated views 5.1 Epipolar geometry 5.2 Basic reconstruction algorithms 5.3 Planar scenes and homography 5.4 Continuous motion case 5.5 Summary 5.6 Exercises 5.A Optimization subject to epipolar constraint 6 Reconstruction from two uncalibrated views 6.1 Uncalibrated camera or distorted space? 6.2 Uncalibrated epipolar geometry 6.3 Ambiguities and constraints in image formation 6.4 Stratified reconstruction 6.5 Calibration with scene knowledge 6.6 Dinner with Kruppa 6.7 Summary 6.8 Exercises 6.A From images to Fundamental matrices 6.B Properties of Kruppa's equations 7 Segmentation of multiple moving objects from two views 7.1 Multibody epipolar constraint and Fundamental matrix 7.2 A rank condition for the number of motions 7.3 Geometric properties of the multibody Fundamental matrix 7.4 Multibody motion estimation and segmentation 7.5 Multibody structure from motion 7.6 Summary 7.7 Exercises 7.A Homogeneous polynomial factorization III Geometry of multiple views 8 Multiple-view geometry of points and lines 8.1 Basic notation for (pre-)image and co-image of points and lines 8.2 Preliminary rank conditions of multiple images 8.3 Geometry of point features 8.4 Geometry of line features 8.5 Uncalibrated factorization and stratification 8.6 Summary 8.7 Exercises 8.A Proof for the properties of bilinear and trilinear constraints 9 Extension to general incidence relations 9.1 Incidence relations among points, lines, and planes 9.2 Rank conditions for incidence relations 9.3 Universal rank conditions on the multiple-view matrix 9.4 Summary 9.5 Exercises 9.A Incidence relations and rank conditions 9.B Beyond constraints among four views 9.C Examples for geometric interpretation of the rank conditions 10 Geometry and reconstruction from symmetry 10.1 Symmetry and multiple-view geometry 10.2 Symmetry-based 3-D reconstruction 10.3 Camera calibration from symmetry 10.4 Summary 10.5 Exercises IV Applications 11 Step-by-step building of a 3-D model from images 11.1 Feature selection 11.2 Feature correspondence 11.3 Projective reconstruction 11.4 Upgrade from projective to Euclidean reconstruction 11.5 Reconstruction with partial scene knowledge 11.6 Calibrated reconstruction (reality check) 11.7 Visualization 11.8 Additional techniques for image-based modeling (discussion) 12 Visual feedback 12.1 Motion and shape estimation as a filtering problem 12.2 Virtual insertion in live video 12.3 Visual feedback for autonomous car driving 12.4 Visual feedback for autonomous helicopter landing V Appendices Appendix A Basic facts from linear algebra A.1 Basic notions associated to a linear space A.2 Linear transformations and matrix groups A.3 Gram-Schmidt procedure and $QR$ decomposition A.4 Range, null, rank and eigenvectors of a matrix A.5 Symmetric matrices and skew-symmetric matrices A.6 Lyapunov map and Lyapunov equation A.7 The singular value decomposition (SVD) Appendix B Least-variance estimation and filtering B.1 Least-variance estimators of random vectors B.2 The Kalman-Bucy filter B.3 The extended Kalman filter Appendix C Basic facts from nonlinear optimization C.1 Unconstrained optimization: gradient based methods C.2 Constrained optimization: Lagrange multiplier method References Glossary of notation Index