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"Theory of Reconstruction from Image Motion" presents the mathematics underlying the reconstruction of camera motion from the movements of points in the camera image. It describes recent work employing mathematical methods drawn from linear algebra, projective geometry, algebraic geometry, the theory of transversality and the theory of least squares approximation. Many problems in reconstruction are best tackled using methods from projective or algebraic geometry. However, these methods are not widely known to researchers in computer vision. As a consequence, purely algebraic methods are often used instead, leading to large and complicated expressions, which are difficult to interpret. Many of the arguments in this volume illustrate the speed and efficiency of geometric methods for solving certain problems that arise in reconstruction. This book is a good starting point for anyone interested in the application of different mathematical techniques to the rapidly expanding field of computer vision, especially in the areas of vehicle guidance, robotics and remote sensing.

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CONTENTS
1 Introduction = 1
 1.1 Background = 1
 1.2 Reconstruction = 3
 1.3 Concentions About the Image = 6
 1.4 Mathematical Background = 7
  1.4.1 Terminoligy = 8
  1.4.2 Euclidean Space and Projective Space = 9
  1.4.3 Algebraic Curves = 12
 References = 14
2 Reconstruction from Image Correspondences = 17
 2.1 Eucidean Framewirk for Reconstruction = 18
  2.1.1 Euclidean Treatment of Ambiguity = 22
  2.1.2 Ambiguity and Instability = 27
  2.1.3 The Maximum Number of Reconstruction = 28
 2.2 Essential Matrices = 36
  2.2.1 A Characterisation of Essential Matrices = 38
  2.2.2 The Singular Value Decomposition = 44
  2.2.3 Symmetric and Antisymmetric Parts = 45
  2.2.4 Ambiguity = 50
 2.3 Projective Framework for Reconstruction = 55
  2.3.1 The Epipolar Tramsformation = 56
  2.3.2 Ambiguity = 58
  2.3.3 The Intersection of a Critical Surface Pair = 61
 2.4 Reconstruction up to a Collineation = 62
  2.4.1 Based on the Epipolar Transformation = 63
  2.4.2 Ambiguity = 66
  2.4.3 Critical Surfaces = 68
 References = 70
3 Critical Surfaces and Horopter Curves = 73
 3.1 The Absolute Conic = 74
  3.1.1 The Absolute Conic and Camera Calibration = 79
  3.1.2 Imvolutions and the Absolute Conic = 81
 3.2 Rectangular Quadrics = 84
  3.2.1 Algebraic Characterisations of Rectangular Quadrics = 87
  3.2.2 Rigid Involutions of Rectangular Quadrics = 91
 3.3 Horopter Curves = 94
  3.3.1 Characterisations of Horopter Curves = 94
  3.3.2 Rigid Involutions of Horopter Curves = 97
  3.3.3 The Centre of a Horopter Curves = 98
  3.3.4 Examples = 100
  3.3.5 Horopter Curves on Rectangular Quadrics = 102
 3.4 Horopter Curves and Reconstruction = 106
  3.4.1 A Formula for τΨ = 108
  3.4.2 Two Cubic Constraints on Critical Surfaces = 110
  3.4.3 An Example = 112
 3.5 Reconstruction up to a Collineation = 114
 References = 118
4 Reconstruction from Image Velocities = 119
 4.1 Framework = 120
 4.2 Ambiguity = 123
  4.2.1 Preliminary Results = 124
  4.2.2 Critical Surfaces = 126
  4.2.3 Singular Critical Surfaces = 128
  4.2.4 Critical Surface Pairs = 129
  4.2.5 Cubic Polynomial Constraints on Critical Surfaces = 132
  4.2.6 The Maximum Number of Reconstructions = 133
 4.3 Algebraic Properties of Four Image Velocity Vectors = 136
  4.3.1 The Quartic Polynomial Constraint = 137
  4.3.2 Irregular Image Velocity Fields = 138
  4.3.3 The Effects of Small Perturbations = 142
  4.3.4 Symmetric Arrangements of Base Points : The Square = 144
  4.3.5 Symmetric Arrangements of Base Points : The Tripod = 150
 4.4 The Linear System of Quartics = 153
  4.4.1 The Dimension of the Linear system = 154
  4.4.2 Singular Points of Quartics = 157
  4.4.3 Base Points of the Linear System = 158
 4.5 The Derivatives of the Image Velocity Field = 162
  4.5.1 The Derivatives to Second Order = 162
  4.5.2 Polynomial Constraints on the Translational Velocity = 166
  4.5.3 Time to Contact = 169
 References = 170
5 Reconstruction from Minimal Data = 173
 5.1 Kruppa's Method = 174
  5.1.1 The Homography = 174
  5.1.2 Constraints Arising from the Camera Calibration = 178
  5.1.3 The Two Sextics = 180
 5.2 Demazure's Method = 183
  5.2.1 The Variety of Essential Matrices = 184
  5.2.2 Properties of the Variety of Essential Matrices = 185
  5.2.3 Lineat Subspaces of $$P^8$$ = 190
  5.2.4 Ten Distinct Intersections = 192
 5.3 Reconstruction up to a Collineation = 198
  5.3.1 Sturm's Method = 199
  5.3.2 An Algebraic Method = 201
 5.4 Reconstruction from Five Image Velocity Vectors = 202
  5.4.1 Preliminary Results = 203
  5.4.2 The Quartic Constraints = 206
  5.4.3 Counting the Solutions = 209
  5.4.4 Critical Surfaces = 211
  5.4.5 Counting the Critical Surfaces = 213
 References = 215
6 Algorithms = 217
 6.1 Reconstruction from Image Correspondences = 219
  6.1.1 An SVD Based Algorithm = 219
  6.1.2 Descent Algorithms = 224
 6.2 Reconstruction from Image Velocities = 232
  6.2.1 A Least Squares Algorithm = 233
  6.2.2 Properties of the Lseat Squares Error Function = 234
  6.2.3 Irregular Image Velocity Fields = 237
  6.2.4 The First Order Algorithm = 246
  6.2.5 Constraints on the Tramslational Velocity = 250
 References = 252
Subject Index = 255