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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.

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.

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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.

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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

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