High Precision Camera Calibration (eBook)

Dr. Tobias Hanning is lecturer at the University of Passau und works as technical engineer in wheel alignment.

Contents 6
Abstract 11
Symbols 12
Chapter 1 Introduction 14
1.1 Motivation 14
1.2 Outline 15
1.3 Contribution 17
Chapter 2 Modelling the camera mapping 18
2.1 Geometric optics for computer vision 18
2.1.1 The “thin lens” assumption and first order optics 18
2.1.2 The circle of confusion 21
2.1.3 Image acquisition 22
2.1.3.1 The sensor array 22
2.1.3.2 A simplified sensor model 24
2.1.3.3 The sensor array as coordinate system 25
2.2 The pinhole camera model 26
Definition 2.2.1 26
Definition 2.2.2 (Pinhole camera) 27
2.3 Third order optics and thick lenses 28
Remark 2.3.1 (Chromatic aberrations) 29
2.4 The pinhole camera model with distortion 30
2.4.1 Definition 30
Definition 2.4.1 (Distortion model) 30
Remark 2.4.2 30
Definition 2.4.3 (Pinhole camera with distortion) 30
2.4.2 Radial distortion 31
Definition 2.4.4 (Radial distortion) 32
2.4.3 Radius transformations 32
2.4.4 Other distortion functions 34
2.4.4.1 Misaligned thin lens 34
2.4.4.2 Misaligned lens systems 35
2.5 Inverting the camera mapping 36
Definition 2.5.1 (undistortion, re-projection) 36
Remark 2.5.2 36
2.6 The pinhole camera model in homogeneous co-ordinates 38
Remark 2.6.1 38
Remark 2.6.2 39
Lemma 2.6.3 39
Chapter 3 Error functions for camera calibration and 3D reconstruction 40
3.1 Introduction 40
3.2 Projective and re-projective error 40
Definition 3.2.1 (Projective Error) 41
Definition 3.2.2 (Re-projective Error) 41
3.3 Euclidean error 42
Definition 3.3.1 (Euclidean error) 42
Definition 3.3.2 (Normalized Euclidean error) 42
Remark 3.3.3 43
Remark 3.3.4 (In front of and behind the camera) 44
3.4 Error functions for camera calibration and 3D-reconstruction 47
3.4.1 Calibration error functions 47
Remark 3.4.1 (Root mean square error) 48
Remark 3.4.2 (Existence of an optimal solution) 48
Remark 3.4.3 (Complete Calibration) 50
3.4.2 Reconstruction error functions 50
Remark 3.4.4 50
3.5 Non-linear optimization 51
Chapter 4 Initial values for camera calibration problems 53
4.1 Introduction 53
4.2 The two stage method of Tsai 55
4.3 An initial image transformation by direct linear transformation 60
Remark 4.3.1 (negative values for 62
Remark 4.3.2 62
4.4 An initial image transformation from homogra-phies 63
4.4.1 Introduction 63
4.4.2 Two necessary conditions for planar targets 63
Remark 4.4.1 64
4.4.3 Zhang’s initial value 65
Remark 4.4.2 (Minimal number of observations) 66
4.4.4 An initial image transformation with known center and zero skew 67
Remark 4.4.3 ( 67
4.4.5 An initial image transformation with known aspect ratio and no skew 68
4.4.6 An initial image transformation with known aspect ratio and unknown skew 69
Remark 4.4.4 70
4.4.7 An initial image transformation with no skew 71
4.4.7.1 A straight forward constraint 71
4.4.7.2 A solution by a linear least squares problem with Cholesky decom-position 72
4.4.8 Experimental results 73
4.4.8.1 Overview 73
4.4.8.2 Simulations 74
4.5 An initial value for the extrinsic camera param-eters 77
4.5.1 Introduction and problem statement 77
4.5.2 Standard pose estimation 77
4.5.3 An algebraic re-projective approach for regular grids 78
4.5.4 An optimal solution w.r.t. Euclidean error for 1D targets 83
4.6 An initial solution for the distortion 85
4.6.1 Introduction 85
Remark 4.6.1 (Initial distortion for the re-projective error) 85
4.6.2 Zhang’s initial solution for the radial distortion 85
4.6.3 An optimal initial solution for all distortion parameters 87
Remark 4.6.2 89
4.7 Camera calibration with distortion as a semi-linear problem 90
4.7.1 Parameter reduction by semi-linear optimization 90
4.7.2 Experimental results 91
4.7.2.1 Results for the normal setup 92
4.7.2.2 Results for the webcam setup 98
4.7.2.3 Results for the wide angle setup 102
Chapter 5 Calibration of a stereo camera system 104
5.1 Introduction 104
5.2 Epipolar geometry 104
Definition 5.2.1 (fundamental matrix, essential matrix) 105
5.3 Epipolar Curves 108
Definition 5.3.1 (Generalized Epipolar Constraint) 108
Remark 5.3.2 ( 108
5.4 Stereo camera calibration with multiple targets 110
5.5 Extrinsic stereo camera calibration with gener-alized epipolar constraints 111
5.5.1 A two step algorithm 111
i. Calculate the positions of the calibration plate with respect to the reference coordinate system 111
ii. Calculate the position of the right camera 113
5.5.2 A one step algorithm 113
5.5.3 Application and results 114
5.6 Extrinsic stereo camera calibration with respect to the projective error 116
5.7 Extrinsic and intrinsic stereo camera calibration 118
Chapter 6 Non-standard camera models 120
6.1 Introduction 120
6.2 Feature point extraction 123
6.2.1 Standard feature point extraction 123
6.2.2 Model based extraction of isolated squares 126
6.2.3 Appropriability of the feature point extraction methods 130
6.2.3.1 Appropriability with respect to the sensor model 130
6.2.3.2 Appropriability with respect to the camera model 130
6.3 The residual distortion 132
6.3.1 The point spread function by first order optics 132
6.3.2 Other sources of residual distortion 139
6.3.3 Experimental results 139
6.4 Spline correction 147
6.4.1 Motivation and related work 147
6.4.2 A depth-dependent distortion term 147
Definition 6.4.1 ( 147
6.4.3 Depth-dependent distortion correction for the projective and re-projective error function 148
6.4.4 The tensor spline space 148
Definition 6.4.2 (Tensor product) 149
Definition 6.4.3 (B-spline base) 149
Definition 6.4.4 ( 150
Definition 6.4.5 ( 150
6.4.5 Tensor splines for the re-projective depth-dependent dis-tortion 150
6.4.6 Spline correction for the Euclidean error 151
6.4.7 The viewing ray for spline corrected cameras 152
6.4.8 Spline correction for stereo reconstruction 152
6.4.9 Disadvantages of the spline correction 153
6.5 A two-plane distortion model 157
6.5.1 Motivation and related work 157
6.5.2 The plane 158
Definition 6.5.1 158
6.5.3 Distortion mappings in 159
Definition 6.5.2 (Distortion model w.r.t. the two-plane distortion) 160
6.5.4 The re-projection w.r.t. the two-plane distortion 160
Definition 6.5.3 ( 160
Remark 6.5.4 (One-plane distortion is a subset of the two-plane distortion) 161
6.5.5 Error functions for the two-plane distortion model 161
6.5.5.1 The projective error 162
6.5.5.2 The Euclidean error 162
Definition 6.5.5 (Euclidean error) 162
6.5.5.3 The projected Euclidean error 163
Definition 6.5.6 (projected Euclidean error) 163
Remark 6.5.7 (A projective error for the two-plane distortion model) 163
6.5.5.4 The normalized Euclidean error 163
Definition 6.5.8 (normalized Euclidean error) 164
6.5.5.5 Depth-dependence of the two-plane distortion model 166
6.5.6 Calibration algorithm 168
6.6 A generic multi-plane camera 169
6.6.1 Introduction and related work 169
6.6.2 From the image to a reference coordinate system 169
6.6.3 Tensor spline approximation of the coordinate transfor-mation 172
6.6.4 A calibration setup for the generic multi-plane camera 172
6.7 Experimental results 174
6.7.1 Setup 174
6.7.1.1 Calibration setup for the standard camera model 174
6.7.1.2 Calibration setup for the spline correction 174
6.7.1.3 Calibration setup for the two-plane distortion model 174
6.7.2 Results for spline corrected cameras 175
6.7.2.1 Prototype reconstruction 175
6.7.2.1.1 In-plane spline correction 175
6.7.2.1.2 3d spline correction 177
6.7.2.2 Stereo reconstruction 184
6.7.3 Results for the two-plane distortion model 186
6.7.3.1 Stereo reconstruction 186
6.7.3.2 Point to point error 186
6.7.3.3 Angles of reconstructed planes 190
6.7.3.4 Other test series 193
Standard), 193
Two-Plane) 193
One-Plane). 193
6.7.3.5 Planarity test 200
6.7.3.6 Prototype reconstruction 205
Standard), 205
One-Plane), 205
3d2dSpline, 205
Two-Plane). 205
Chapter 7 Conclusions 210
Bibliography 212
Index 224