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This book presents computational techniques for machine vision necessary for building intelligent machines which can understand the environment by vision.

Machine vision is the study of how to build intelligent machines which can understand the environment by vision. Among many existing books on this subject, this book is unique in that the entire volume is devoted to computational problems, which most books do not deal with. One of the main subjects of this book is the mathematics underlying all vision problems - projective geometry, in particular. Since projective geometry has been developed by mathematicians without any regard to machine vision applications, our first attempt is to `tune' it into the form applicable to machine vision problems. The resulting formulation is termed computational projective geometry and applied to 3-D shape analysis, camera calibration, road scene analysis, 3-D motion analysis, optical flow analysis, and conic image analysis. A salient characteristic of machine vision problems is that data are not necessarily accurate. Hence, computational procedures defined by using exact relationships may break down if blindly applied to inaccurate data. In this book, special emphasis is put on robustness, which means that the computed result is not only exact when the data are accurate but also is expected to give a good approximation in the prescence of noise. The analysis of how the computation is affected by the inaccuracy of the data is also crucial. Statistical analysis of computations based on image data is also one of the main subjects of this book.

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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