CONTENTS
1 Introduction = 1
1.1 The Background of Machine Vision = 1
1.2 The Aim of This Book = 4
1.3 The Organization of This Book = 6
1.3.1 Computational projective geometry = 6
1.3.2 Translational motion, stereo and 3-D rotation = 7
1.3.3 Analysis of 3-D motion and optical flow = 9
1.3.4 Analysis of conics = 11
1.3.5 Statistical analysis of geometric computation = 12
1.4 Bibliographical Notes = 13
2 Computational Projective Geometry, 1 = 15
2.1 Geometry of Perspective Projection = 15
2.1.1 Interpretation of N-vectors = 15
2.1.2 Vanishing points and vanishing lines = 19
2.2 Computation of points and Lines = 20
2.2.1 Fundamental duality of N-vectors = 20
2.2.2 Collinearity of points and concurrency of lines = 21
2.3 Geometry of 2-D Projective Space = 25
2.3.1 Collineations and camera rotation transformations = 25
2.3.2 Correlations, polarities, conjugacy, and conics = 28
2.4 Cross Ratio and Projective Coordinates = 30
2.4.1 Percpective invariance of cross ratio = 30
2.4.2 Projective invariance of cross ratio = 35
2.4.3 Harmonic Range of Points = 40
2.5 Bibliographical Notes = 42
Exercises = 43
3 Computational Projective Geometry, 2 = 51
3.1 Geometry of Standard Polarity = 51
3.1.1 The absolute conic and its polarity = 51
3.1.2 Conjugacy and orthogonality = 54
3.2 Camera Calibration = 56
3.2.1 Determination of the focal length = 56
3.2.2 Pose parameters and motion parameters = 60
3.2.3 Constraints on rectangles and squares = 63
3.3 3-D Road Shape Reconstruction = 65
3.3.1 Modelling ideal roads = 65
3.3.2 Local-flatness approximation = 67
3.3.3 3-D reconstruction by curve fitting = 69
3.4 Bibliographical Notes = 72
Exercises = 74
4 Translational Motion and Stereo = 77
4.1 Analysis of Translational Motion = 77
4.1.1 N-velosities and trajectories = 77
4.1.2 Focus of expansion = 78
4.1.3 Constant velocity motion = 80
4.1.4 Vanishing points and line orientations = 82
4.2 Motion Parallax = 84
4.2.1 Motion Prallax of a point = 84
4.2.2 Representation of a space line = 86
4.2.3 Motion parallax of a line = 89
4.2.4 Motion parallax for general motion = 91
4.3 Analysis of Stereo = 92
4.3.1 Epipolars and epipole = 92
4.3.2 Disparity maps and depth maps = 94
4.3.3 Converging stereo = 95
4.4 Bibliographical Notes = 96
Exercises = 98
5 Computation of 3-D Rotation = 100
5.1 Representation of 3-D Rotation = 100
5.1.1 Rotation matrices = 100
5.1.2 Axis and angle of rotation = 102
5.2 Optimal Estimation of 3-D Rotation = 105
5.2.1 Least-squares estimation of 3-D Rotation = 105
5.2.2 Singular value decomposition = 109
5.2.3 Polar decomposition = 112
5.2.4 Quatermion representation = 114
5.3 Orthogonality Recovery = 117
5.3.1 Orthogonality fitting = 117
5.3.2 Orthogonal frame reconstruction = 118
5.3.3 Optimal resolution = 121
5.4 Spherical Optimization Search = 123
5.4.1 Optimzation of pose and orientation = 123
5.4.2 Quadratic search = 124
5.4.3 Model update search = 128
5.5 Bibliographical Nites = 130
Exercises = 132
6 Analysis of 3-D Rigid Motion = 143
6.1 Representation of Planar Surface Motion = 143
6.1.1 Planar Surface Motion = 143
6.1.2 Collineation of Planar Surface Motion = 146
6.2 3-D Interpretation of Planar Surface Motion = 148
6.2.1 Analytical solution = 148
6.2.2 Ambiguity of Planar Surface Motion = 151
6.3 Determination of Collineation = 153
6.4 3-D Interpretation from Poimt Correspondence = 156
6.4.1 General formulation = 156
6.4.2 Optimazation search = 158
6.5 Least-Squares Point Correspondence Algorithm = 160
6.5.1 Essential matrix and eight-point Algorithm = 160
6.5.2 Rodust analytical solution = 161
6.5.3 Decomposavility and uniqueness = 165
6.6 3-D Interpretation from Line Correspondence = 166
6.6.1 General formulation = 166
6.6.2 Optimization search = 168
6.7 Least-Squares Line Correspondence Algorithm = 170
6.7.1 Essential parameters and thirteen-line Algorithm = 170
6.7.2 Robust analytical solution = 171
6.7.3 Uniqueness of the solution = 177
6.8 Ambiguity of 3-D Interpretation = 178
6.8.1 Critical surface for point correspondence = 178
6.8.2 Degeneracy into two planar surfaces = 183
6.8.3 Critical line congruence for line correspondence = 185
6.9 Bibliographical Notes = 188
Exercises = 191
7 Analysis of Optical Flow = 196
7.1 Representation of Planar Surface Optical Flow = 196
7.1.1 Infinitesimal surface motion = 196
7.1.2 Optical flow and flow matrix = 197
7.1.3 Optical flow of lines and dual flow = 199
7.2 3-D Interpretation of Planar Surface Optical Flow = 200
7.2.1 Optical flow and motion parameters = 200
7.2.2 Analytical solution = 201
7.3 Determination of the Flow Matrix = 203
7.3.1 Flow-based approach = 203
7.3.2 Contour-based approach = 206
7.4 Representation of General Optical Flow = 208
7.4.1 General optical Flow equation = 208
7.4.2 Twisted flow and the epipolar equation = 210
7.5 3-D Interpretation of General Optical Flow = 212
7.5.1 Least-squares algorithm = 212
7.5.2 Optimization search = 215
7.6 Critical Surface of Optical Flow = 217
7.6.1 Critical surface squation = 217
7.6.2 Degeneracy into two planes = 220
7.7 Bibliographical Nites = 221
Exercises = 223
8 Analysis of Conics = 228
8.1 Conics and Their Canonical Forms = 228
8.1.1 Representation of a conic = 228
8.1.2 Canonical form of a conic = 229
8.2 Polarity of a conic = 231
8.2.1 Poles, polars, and tangents = 231
8.2.2 Conjugacy of points and lines = 234
8.3 Intersections and Orthogonality = 235
8.3.1 Intersections of a conic woth a line = 235
8.3.2 Interpretation of rectangular corners = 237
8.4 Conic Fitting = 240
8.4.1 Existence and uniqueness = 240
8.4.2 Least-squares fitting = 240
8.5 3-D interpretation of a conic = 242
8.5.1 The supporting plane and the true shape = 242
8.5.2 3-D interpretation of a circle = 246
8.5.3 3-D interpretation of an ellipse = 251
8.6 Mapping of Conics and Invisible Motions = 258
8.6.1 Group of invisible motions = 258
8.6.2 Mapping of conics = 260
8.6.3 Standard circle = 262
8.7 Invisible Optical Flows = 263
8.7.1 Representation of invisible flows = 263
8.7.2 Adjoint transformation of invisible flows = 266
8.8 Deformation of a conic = 268
8.8.1 Linear space of conic deformations = 268
8.8.2 Normal flow along a conic = 271
8.9 3-D Interpretation of a Moving Conic = 271
8.9.1 Finite motion of a conic = 271
8.9.2 Infinitesimal motion of a conic = 273
8.10 Bibliographical Notes = 275
Exercises = 277
9 Statistical Analysis of Geometric Computation, 1 = 280
9.1 Stastistical Model of Noise = 280
9.1.1 Covariance matrix of an N-vector = 280
9.1.2 Model of noise = 282
9.1.3 Effective focal length = 283
9.2 Covariance Matrices of Joins and Intersections = 284
9.3 Optimal Least-Squares Estimation = 287
9.3.1 Optimal weights and optimal estimation = 287
9.3.2 Covariance matrix of optimal estimation = 289
9.3.3 Statistical bias of optimal estimation = 292
9.4 Edge Fitting, Vanishing Points, and Focuses of Espansion = 294
9.4.1 Error in edge fitting = 294
9.4.2 Error in vanishing points = 297
9.4.3 Error in focuses of expansion = 300
9.5 statistics of Rotation Fitting = 305
9.5.1 Covariance matrix of 3-D rotation = 305
9.5.2 Covariance matrix of the best fitting rotation = 305
9.6 Statistics of Depth from Stereo = 309
9.6.1 Sources of error = 309
9.6.2 Error due to image noise = 310
9.6.3 Error due to uncertainyt of camera orientation = 311
9.6.4 Error due to uncertainty of base-line = 312
9.7 Bibliographical Notes = 314
Exercises = 315
10 statistical Analysis of Geometric Computation, 2 = 318
10.1 Statistics of Focal Length Calibration = 318
10.1.1 Reliability of focal length estimation = 318
10.1.2 Optimal estimatin of focal lingth = 322
10.2 Statistical Analysis of 3-D Motion Estimation = 326
10.2.1 Statistical bias of motion parameters = 326
10.2.2 Small object approximation = 330
10.2.3 Unbiased motion parameter estimation = 333
10.3 Statistics of Conic Fitting = 336
10.3.1 Optimal conic fitting = 336
10.3.2 Covariance tensor of conic fitting = 337
10.3.3 Statistical bias of conic fitting = 340
10.3.4 Unbiased conic fitting = 346
10.4 Hypothesizing and Testing Geometric Configurations = 349
10.4.1 Gaussian approximation = 349
10.4.2 Testing edge groupings = 350
10.4.3 Testing vanishing points = 352
10.4.4 Testing focuses of expansion = 354
10.4.5 Testing vanishing lines = 355
10.5 Bibliographical Notes = 357
Exercises = 360
References = 364
Answers = 381
Index = 468
