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Grouping, Matching and Reconstruction in Multiple View Geometry

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Grouping, Matching and Reconstruction in Multiple View Geometry Grouping, Matching and Reconstruction in Multiple View Geometry F. Schaffalitzky Robotics Research Group Department of Engineering Science University of Oxford, UK April 2002 Contents 1 Introduction 6 1.1 Objective and Motivation . 6 1.2 Overview . 8 2 Grouping with geometric constraints 12 2.1 Overview . 12 2.2 Objective and background . 12 2.3 Theory and geometry . 14 2.3.1 Parallel lines . 14 2.3.2 Equally spaced parallel lines . 15 2.3.3 Elations . 18 2.3.4 Regular grids . 21 2.4 Algorithms . 21 2.4.1 Vanishing point detection . 22 2.4.2 Computing the equally spaced line geometry . 24 2.4.3 Computing elations . 27 2.4.4 Computing grids . 28 2.5 Results . 30 2.5.1 Vanishing points . 30 2.5.2 Equally spaced lines . 30 2.5.3 Elations . 35 2.5.4 Grids . 40 1 2.6 Assessment . 40 3 Projective reconstruction 43 3.1 Overview . 43 3.2 Motivation and objective . 43 3.3 Theory . 45 3.3.1 Problem formulation and notation . 46 3.3.2 Pencils of cameras . 47 3.3.3 Derivation of quadratic constraints . 47 3.3.4 Five irrelevant solutions . 48 3.3.5 Factoring the constraints . 48 3.3.6 Cubic constraint . 50 3.3.7 Quasi-linear method for reconstruction . 51 3.3.8 Scaling the constraints . 53 3.3.9 Geometric error . 53 3.3.10 Related Work . 55 3.4 Algorithm details . 55 3.4.1 Computing camera pencils . 55 3.4.2 Inverting . 58 3.4.3 Robust Reconstruction Algorithm . 59 3.5 Results . 60 3.5.1 Synthetic data . 60 3.5.2 Real data I . 62 3.5.3 Real data II . 64 3.6 Assessment . 66 4 Auto-calibration 67 4.1 Overview . 67 4.2 Motivation . 67 4.3 Modulus constraints . 70 2 4.3.1 Motivation . 71 4.3.2 Algebraic nullspaces . 72 4.3.3 Horopters . 73 4.3.4 Characteristic equation of ¢¡¤£¦¥¨§ © . 76 4.3.5 Two-view (strong) modulus constraint . 77 4.3.6 Two-view (algebraic) quartic modulus constraint . 78 4.3.7 Three-view (algebraic) cubic modulus constraint . 79 4.3.8 Solving the equations . 79 4.3.9 Experimental Results. 87 4.3.10 Partial constraint . 89 4.3.11 Conclusion and further work . 90 4.4 Square pixels . 91 4.4.1 Notation . 93 4.4.2 Co-conicity constraint on § . 95 4.4.3 Pascal's theorem . 95 4.4.4 Collinearity in 3D . 97 4.4.5 Octic constraint . 98 4.4.6 Special case of 3D collinearity bracket . 99 4.4.7 Formula for sextic . 100 4.4.8 Formula for quintic . 100 4.4.9 Forty solutions . 103 4.5 Algorithms . 108 4.5.1 Recovering calibration from the plane at infinity . 108 4.5.2 Numerical Computation of Ideal Saturation . 109 4.5.3 Computing the Multiplication Operators . 111 4.5.4 Solving the Generalized Eigensystem . 111 4.6 Results . 112 4.7 Assessment . 113 3 5 Matching images 114 5.1 Objective and motivation . 114 5.1.1 What can one hope for? . 114 5.1.2 Direct methods . 116 5.1.3 Wide versus short base-line. 117 5.2 Framework of invariant descriptors . 118 5.2.1 Interest point neighbourhood descriptors . 120 5.2.2 Intensity profiles descriptors . 127 5.2.3 Region descriptors . 127 5.2.4 Texture descriptors . 128 5.2.5 Geometric features . 128 5.3 Local (affine) transformations . 129 5.4 Algorithms . 131 5.4.1 Corner detector . 132 5.4.2 Shape adaptation . 133 5.4.3 Invariants used . 134 5.4.4 Registration . 135 5.4.5 Growing new matches from old . 136 5.4.6 Constraints on multi-view geometry from local affine transformation 136 5.5 Results . 138 5.6 Assessment . 138 6 Texture 153 6.1 Objective . 153 6.2 Background . 153 6.2.1 Texture description and models . 155 6.3 Theory . 161 6.3.1 Segmentation . 161 6.3.2 The 2nd moment matrix . 162 4 6.3.3 Behaviour under affine transformations . 162 6.3.4 Weak (gradient) isotropy . 165 6.3.5 Previous use of the 2nd moment matrix . 165 6.3.6 Description in normalized frame . 166 6.4 Algorithms . 169 6.4.1 Normalized Cut Segmentation . 169 6.4.2 Practical calculation of 2nd moment matrix . ..
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