Detail View

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