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Multiple View Geometry in Computer Vision

Book Description

A basic problem in computer vision is to understand the structure of a real world scene given several images of it. Techniques for solving this problem are taken from projective geometry and photogrammetry. Here, the authors cover the geometric principles and their algebraic representation in terms of camera projection matrices, the fundamental matrix and the trifocal tensor. The theory and methods of computation of these entities are discussed with real examples, as is their use in the reconstruction of scenes from multiple images. The new edition features an extended introduction covering the key ideas in the book (which itself has been updated with additional examples and appendices) and significant new results which have appeared since the first edition. Comprehensive background material is provided, so readers familiar with linear algebra and basic numerical methods can understand the projective geometry and estimation algorithms presented, and implement the algorithms directly from the book.

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright
  4. Dedication
  5. Contents
  6. Foreword
  7. Preface
  8. 1. Introduction – a Tour of Multiple View Geometry
    1. 1.1 Introduction – the ubiquitous projective geometry
    2. 1.2 Camera projections
    3. 1.3 Reconstruction from more than one view
    4. 1.4 Three-view geometry
    5. 1.5 Four view geometry and n-view reconstruction
    6. 1.6 Transfer
    7. 1.7 Euclidean reconstruction
    8. 1.8 Auto-calibration
    9. 1.9 The reward I : 3D graphical models
    10. 1.10 The reward II: video augmentation
  9. Part 0: The Background: Projective Geometry, Transformations and Estimation
    1. Outline
    2. 2. Projective Geometry and Transformations of 2D
      1. 2.1 Planar geometry
      2. 2.2 The 2D projective plane
      3. 2.3 Projective transformations
      4. 2.4 A hierarchy of transformations
      5. 2.5 The projective geometry of 1D
      6. 2.6 Topology of the projective plane
      7. 2.7 Recovery of affine and metric properties from images
      8. 2.8 More properties of conics
      9. 2.9 Fixed points and lines
      10. 2.10 Closure
    3. 3. Projective Geometry and Transformations of 3D
      1. 3.1 Points and projective transformations
      2. 3.2 Representing and transforming planes, lines and quadrics
      3. 3.3 Twisted cubics
      4. 3.4 The hierarchy of transformations
      5. 3.5 The plane at infinity
      6. 3.6 The absolute conic
      7. 3.7 The absolute dual quadric
      8. 3.8 Closure
    4. 4. Estimation – 2D Projective Transformations
      1. 4.1 The Direct Linear Transformation (DLT) algorithm
      2. 4.2 Different cost functions
      3. 4.3 Statistical cost functions and Maximum Likelihood estimation
      4. 4.4 Transformation invariance and normalization
      5. 4.5 Iterative minimization methods
      6. 4.6 Experimental comparison of the algorithms
      7. 4.7 Robust estimation
      8. 4.8 Automatic computation of a homography
      9. 4.9 Closure
    5. 5. Algorithm Evaluation and Error Analysis
      1. 5.1 Bounds on performance
      2. 5.2 Covariance of the estimated transformation
      3. 5.3 Monte Carlo estimation of covariance
      4. 5.4 Closure
  10. Part I: Camera Geometry and Single View Geometry
    1. Outline
    2. 6. Camera Models
      1. 6.1 Finite cameras
      2. 6.2 The projective camera
      3. 6.3 Cameras at infinity
      4. 6.4 Other camera models
      5. 6.5 Closure
    3. 7. Computation of the Camera Matrix P
      1. 7.1 Basic equations
      2. 7.2 Geometric error
      3. 7.3 Restricted camera estimation
      4. 7.4 Radial distortion
      5. 7.5 Closure
    4. 8. More Single View Geometry
      1. 8.1 Action of a projective camera on planes, lines, and conics
      2. 8.2 Images of smooth surfaces
      3. 8.3 Action of a projective camera on quadrics
      4. 8.4 The importance of the camera centre
      5. 8.5 Camera calibration and the image of the absolute conic
      6. 8.6 Vanishing points and vanishing lines
      7. 8.7 Affine 3D measurements and reconstruction
      8. 8.8 Determining camera calibration K from a single view
      9. 8.9 Single view reconstruction
      10. 8.10 The calibrating conic
      11. 8.11 Closure
  11. Part II: Two-View Geometry
    1. Outline
    2. 9. Epipolar Geometry and the Fundamental Matrix
      1. 9.1 Epipolar geometry
      2. 9.2 The fundamental matrix F
      3. 9.3 Fundamental matrices arising from special motions
      4. 9.4 Geometric representation of the fundamental matrix
      5. 9.5 Retrieving the camera matrices
      6. 9.6 The essential matrix
      7. 9.7 Closure
    3. 10. 3D Reconstruction of Cameras and Structure
      1. 10.1 Outline of reconstruction method
      2. 10.2 Reconstruction ambiguity
      3. 10.3 The projective reconstruction theorem
      4. 10.4 Stratified reconstruction
      5. 10.5 Direct reconstruction – using ground truth
      6. 10.6 Closure
    4. 11. Computation of the Fundamental Matrix F
      1. 11.1 Basic equations
      2. 11.2 The normalized 8-point algorithm
      3. 11.3 The algebraic minimization algorithm
      4. 11.4 Geometric distance
      5. 11.5 Experimental evaluation of the algorithms
      6. 11.6 Automatic computation of F
      7. 11.7 Special cases of F-computation
      8. 11.8 Correspondence of other entities
      9. 11.9 Degeneracies
      10. 11.10 A geometric interpretation of F-computation
      11. 11.11 The envelope of epipolar lines
      12. 11.12 Image rectification
      13. 11.13 Closure
    5. 12. Structure Computation
      1. 12.1 Problem statement
      2. 12.2 Linear triangulation methods
      3. 12.3 Geometric error cost function
      4. 12.4 Sampson approximation (first-order geometric correction)
      5. 12.5 An optimal solution
      6. 12.6 Probability distribution of the estimated 3D point
      7. 12.7 Line reconstruction
      8. 12.8 Closure
    6. 13. Scene planes and homographies
      1. 13.1 Homographies given the plane and vice versa
      2. 13.2 Plane induced homographies given F and image correspondences
      3. 13.3 Computing F given the homography induced by a plane
      4. 13.4 The infinite homography H∞
      5. 13.5 Closure
    7. 14. Affine Epipolar Geometry
      1. 14.1 Affine epipolar geometry
      2. 14.2 The affine fundamental matrix
      3. 14.3 Estimating FA from image point correspondences
      4. 14.4 Triangulation
      5. 14.5 Affine reconstruction
      6. 14.6 Necker reversal and the bas-relief ambiguity
      7. 14.7 Computing the motion
      8. 14.8 Closure
  12. Part III: Three-View Geometry
    1. Outline
    2. 15. The Trifocal Tensor
      1. 15.1 The geometric basis for the trifocal tensor
      2. 15.2 The trifocal tensor and tensor notation
      3. 15.3 Transfer
      4. 15.4 The fundamental matrices for three views
      5. 15.5 Closure
    3. 16. Computation of the Trifocal Tensor T
      1. 16.1 Basic equations
      2. 16.2 The normalized linear algorithm
      3. 16.3 The algebraic minimization algorithm
      4. 16.4 Geometric distance
      5. 16.5 Experimental evaluation of the algorithms
      6. 16.6 Automatic computation of T
      7. 16.7 Special cases of T -computation
      8. 16.8 Closure
  13. Part IV: N-View Geometry
    1. Outline
    2. 17. N-Linearities and Multiple View Tensors
      1. 17.1 Bilinear relations
      2. 17.2 Trilinear relations
      3. 17.3 Quadrilinear relations
      4. 17.4 Intersections of four planes
      5. 17.5 Counting arguments
      6. 17.6 Number of independent equations
      7. 17.7 Choosing equations
      8. 17.8 Closure
    3. 18. N-View Computational Methods
      1. 18.1 Projective reconstruction – bundle adjustment
      2. 18.2 Affine reconstruction – the factorization algorithm
      3. 18.3 Non-rigid factorization
      4. 18.4 Projective factorization
      5. 18.5 Projective reconstruction using planes
      6. 18.6 Reconstruction from sequences
      7. 18.7 Closure
    4. 19. Auto-Calibration
      1. 19.1 Introduction
      2. 19.2 Algebraic framework and problem statement
      3. 19.3 Calibration using the absolute dual quadric
      4. 19.4 The Kruppa equations
      5. 19.5 A stratified solution
      6. 19.6 Calibration from rotating cameras
      7. 19.7 Auto-calibration from planes
      8. 19.8 Planar motion
      9. 19.9 Single axis rotation – turntable motion
      10. 19.10 Auto-calibration of a stereo rig
      11. 19.11 Closure
    5. 20. Duality
      1. 20.1 Carlsson–Weinshall duality
      2. 20.2 Reduced reconstruction
      3. 20.3 Closure
    6. 21. Cheirality
      1. 21.1 Quasi-affine transformations
      2. 21.2 Front and back of a camera
      3. 21.3 Three-dimensional point sets
      4. 21.4 Obtaining a quasi-affine reconstruction
      5. 21.5 Effect of transformations on cheirality
      6. 21.6 Orientation
      7. 21.7 The cheiral inequalities
      8. 21.8 Which points are visible in a third view
      9. 21.9 Which points are in front of which
      10. 21.10 Closure
    7. 22. Degenerate Configurations
      1. 22.1 Camera resectioning
      2. 22.2 Degeneracies in two views
      3. 22.3 Carlsson–Weinshall duality
      4. 22.4 Three-view critical configurations
      5. 22.5 Closure
  14. Part V : Appendices
    1. Appendix 1: Tensor Notation
    2. Appendix 2: Gaussian (Normal) and χ2 Distributions
    3. Appendix 3: Parameter Estimation
    4. Appendix 4: Matrix Properties and Decompositions
    5. Appendix 5: Least-squares Minimization
    6. Appendix 6: Iterative Estimation Methods
    7. Appendix 7: Some Special Plane Projective Transformations
  15. Bibliography
  16. Index