Hanning High Precision Camera Calibration

1. Auflage 2010
ISBN: 978-3-8348-9830-2
Verlag: Vieweg & Teubner
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Deutsch, 212 Seiten, Web PDF

Reihe: Computer Science and Engineering (German Language)

A calibrated camera is the key to each optical measurement problem. This work introduces camera modelling and calibration in computer vision. Tobias Hanning explains the classic pinhole camera model with distortion derived from third order optics and shows how to obtain an optimal solution within this model. The limitations of the classic camera model in computer vision are explained and proven experimentally. To overcome these limitations two alternative approaches which augment the standard model are presented.

Zielgruppe

Research

Autoren/Hrsg.

Hanning, Tobias

Inhaltsverzeichnis

1;Contents;6
2;Abstract;11
3;Symbols;12
4;Chapter 1 Introduction;14
4.1;1.1 Motivation;14
4.2;1.2 Outline;15
4.3;1.3 Contribution;17
5;Chapter 2 Modelling the camera mapping;18
5.1;2.1 Geometric optics for computer vision;18
5.1.1;2.1.1 The “thin lens” assumption and first order optics;18
5.1.2;2.1.2 The circle of confusion;21
5.1.3;2.1.3 Image acquisition;22
5.1.3.1;2.1.3.1 The sensor array;22
5.1.3.2;2.1.3.2 A simplified sensor model;24
5.1.3.3;2.1.3.3 The sensor array as coordinate system;25
5.2;2.2 The pinhole camera model;26
5.2.1;Definition 2.2.1;26
5.2.2;Definition 2.2.2 (Pinhole camera);27
5.3;2.3 Third order optics and thick lenses;28
5.3.1;Remark 2.3.1 (Chromatic aberrations);29
5.4;2.4 The pinhole camera model with distortion;30
5.4.1;2.4.1 Definition;30
5.4.1.1;Definition 2.4.1 (Distortion model);30
5.4.1.2;Remark 2.4.2;30
5.4.1.3;Definition 2.4.3 (Pinhole camera with distortion);30
5.4.2;2.4.2 Radial distortion;31
5.4.2.1;Definition 2.4.4 (Radial distortion);32
5.4.3;2.4.3 Radius transformations;32
5.4.4;2.4.4 Other distortion functions;34
5.4.4.1;2.4.4.1 Misaligned thin lens;34
5.4.4.2;2.4.4.2 Misaligned lens systems;35
5.5;2.5 Inverting the camera mapping;36
5.5.1;Definition 2.5.1 (undistortion, re-projection);36
5.5.2;Remark 2.5.2;36
5.6;2.6 The pinhole camera model in homogeneous co-ordinates;38
5.6.1;Remark 2.6.1;38
5.6.2;Remark 2.6.2;39
5.6.3;Lemma 2.6.3;39
6;Chapter 3 Error functions for camera calibration and 3D reconstruction;40
6.1;3.1 Introduction;40
6.2;3.2 Projective and re-projective error;40
6.2.1;Definition 3.2.1 (Projective Error);41
6.2.2;Definition 3.2.2 (Re-projective Error);41
6.3;3.3 Euclidean error;42
6.3.1;Definition 3.3.1 (Euclidean error);42
6.3.2;Definition 3.3.2 (Normalized Euclidean error);42
6.3.3;Remark 3.3.3;43
6.3.4;Remark 3.3.4 (In front of and behind the camera);44
6.4;3.4 Error functions for camera calibration and 3D-reconstruction;47
6.4.1;3.4.1 Calibration error functions;47
6.4.1.1;Remark 3.4.1 (Root mean square error);48
6.4.1.2;Remark 3.4.2 (Existence of an optimal solution);48
6.4.1.3;Remark 3.4.3 (Complete Calibration);50
6.4.2;3.4.2 Reconstruction error functions;50
6.4.2.1;Remark 3.4.4;50
6.5;3.5 Non-linear optimization;51
7;Chapter 4 Initial values for camera calibration problems;53
7.1;4.1 Introduction;53
7.2;4.2 The two stage method of Tsai;55
7.3;4.3 An initial image transformation by direct linear transformation;60
7.3.1;Remark 4.3.1 (negative values for;62
7.3.2;Remark 4.3.2;62
7.4;4.4 An initial image transformation from homogra-phies;63
7.4.1;4.4.1 Introduction;63
7.4.2;4.4.2 Two necessary conditions for planar targets;63
7.4.2.1;Remark 4.4.1;64
7.4.3;4.4.3 Zhang’s initial value;65
7.4.3.1;Remark 4.4.2 (Minimal number of observations);66
7.4.4;4.4.4 An initial image transformation with known center and zero skew;67
7.4.4.1;Remark 4.4.3 (;67
7.4.5;4.4.5 An initial image transformation with known aspect ratio and no skew;68
7.4.6;4.4.6 An initial image transformation with known aspect ratio and unknown skew;69
7.4.6.1;Remark 4.4.4;70
7.4.7;4.4.7 An initial image transformation with no skew;71
7.4.7.1;4.4.7.1 A straight forward constraint;71
7.4.7.2;4.4.7.2 A solution by a linear least squares problem with Cholesky decom-position;72
7.4.8;4.4.8 Experimental results;73
7.4.8.1;4.4.8.1 Overview;73
7.4.8.2;4.4.8.2 Simulations;74
7.5;4.5 An initial value for the extrinsic camera param-eters;77
7.5.1;4.5.1 Introduction and problem statement;77
7.5.2;4.5.2 Standard pose estimation;77
7.5.3;4.5.3 An algebraic re-projective approach for regular grids;78
7.5.4;4.5.4 An optimal solution w.r.t. Euclidean error for 1D targets;83
7.6;4.6 An initial solution for the distortion;85
7.6.1;4.6.1 Introduction;85
7.6.1.1;Remark 4.6.1 (Initial distortion for the re-projective error);85
7.6.2;4.6.2 Zhang’s initial solution for the radial distortion;85
7.6.3;4.6.3 An optimal initial solution for all distortion parameters;87
7.6.3.1;Remark 4.6.2;89
7.7;4.7 Camera calibration with distortion as a semi-linear problem;90
7.7.1;4.7.1 Parameter reduction by semi-linear optimization;90
7.7.2;4.7.2 Experimental results;91
7.7.2.1;4.7.2.1 Results for the normal setup;92
7.7.2.2;4.7.2.2 Results for the webcam setup;98
7.7.2.3;4.7.2.3 Results for the wide angle setup;102
8;Chapter 5 Calibration of a stereo camera system;104
8.1;5.1 Introduction;104
8.2;5.2 Epipolar geometry;104
8.2.1;Definition 5.2.1 (fundamental matrix, essential matrix);105
8.3;5.3 Epipolar Curves;108
8.3.1;Definition 5.3.1 (Generalized Epipolar Constraint);108
8.3.2;Remark 5.3.2 (;108
8.4;5.4 Stereo camera calibration with multiple targets;110
8.5;5.5 Extrinsic stereo camera calibration with gener-alized epipolar constraints;111
8.5.1;5.5.1 A two step algorithm;111
8.5.1.1;i. Calculate the positions of the calibration plate with respect to the reference coordinate system;111
8.5.1.2;ii. Calculate the position of the right camera;113
8.5.2;5.5.2 A one step algorithm;113
8.5.3;5.5.3 Application and results;114
8.6;5.6 Extrinsic stereo camera calibration with respect to the projective error;116
8.7;5.7 Extrinsic and intrinsic stereo camera calibration;118
9;Chapter 6 Non-standard camera models;120
9.1;6.1 Introduction;120
9.2;6.2 Feature point extraction;123
9.2.1;6.2.1 Standard feature point extraction;123
9.2.2;6.2.2 Model based extraction of isolated squares;126
9.2.3;6.2.3 Appropriability of the feature point extraction methods;130
9.2.3.1;6.2.3.1 Appropriability with respect to the sensor model;130
9.2.3.2;6.2.3.2 Appropriability with respect to the camera model;130
9.3;6.3 The residual distortion;132
9.3.1;6.3.1 The point spread function by first order optics;132
9.3.2;6.3.2 Other sources of residual distortion;139
9.3.3;6.3.3 Experimental results;139
9.4;6.4 Spline correction;147
9.4.1;6.4.1 Motivation and related work;147
9.4.2;6.4.2 A depth-dependent distortion term;147
9.4.2.1;Definition 6.4.1 (;147
9.4.3;6.4.3 Depth-dependent distortion correction for the projective and re-projective error function;148
9.4.4;6.4.4 The tensor spline space;148
9.4.4.1;Definition 6.4.2 (Tensor product);149
9.4.4.2;Definition 6.4.3 (B-spline base);149
9.4.4.3;Definition 6.4.4 (;150
9.4.4.4;Definition 6.4.5 (;150
9.4.5;6.4.5 Tensor splines for the re-projective depth-dependent dis-tortion;150
9.4.6;6.4.6 Spline correction for the Euclidean error;151
9.4.7;6.4.7 The viewing ray for spline corrected cameras;152
9.4.8;6.4.8 Spline correction for stereo reconstruction;152
9.4.9;6.4.9 Disadvantages of the spline correction;153
9.5;6.5 A two-plane distortion model;157
9.5.1;6.5.1 Motivation and related work;157
9.5.2;6.5.2 The plane;158
9.5.2.1;Definition 6.5.1;158
9.5.3;6.5.3 Distortion mappings in;159
9.5.3.1;Definition 6.5.2 (Distortion model w.r.t. the two-plane distortion);160
9.5.4;6.5.4 The re-projection w.r.t. the two-plane distortion;160
9.5.4.1;Definition 6.5.3 (;160
9.5.4.2;Remark 6.5.4 (One-plane distortion is a subset of the two-plane distortion);161
9.5.5;6.5.5 Error functions for the two-plane distortion model;161
9.5.5.1;6.5.5.1 The projective error;162
9.5.5.2;6.5.5.2 The Euclidean error;162
9.5.5.3;Definition 6.5.5 (Euclidean error);162
9.5.5.4;6.5.5.3 The projected Euclidean error;163
9.5.5.5;Definition 6.5.6 (projected Euclidean error);163
9.5.5.6;Remark 6.5.7 (A projective error for the two-plane distortion model);163
9.5.5.7;6.5.5.4 The normalized Euclidean error;163
9.5.5.8;Definition 6.5.8 (normalized Euclidean error);164
9.5.5.9;6.5.5.5 Depth-dependence of the two-plane distortion model;166
9.5.6;6.5.6 Calibration algorithm;168
9.6;6.6 A generic multi-plane camera;169
9.6.1;6.6.1 Introduction and related work;169
9.6.2;6.6.2 From the image to a reference coordinate system;169
9.6.3;6.6.3 Tensor spline approximation of the coordinate transfor-mation;172
9.6.4;6.6.4 A calibration setup for the generic multi-plane camera;172
9.7;6.7 Experimental results;174
9.7.1;6.7.1 Setup;174
9.7.1.1;6.7.1.1 Calibration setup for the standard camera model;174
9.7.1.2;6.7.1.2 Calibration setup for the spline correction;174
9.7.1.3;6.7.1.3 Calibration setup for the two-plane distortion model;174
9.7.2;6.7.2 Results for spline corrected cameras;175
9.7.2.1;6.7.2.1 Prototype reconstruction;175
9.7.2.1.1;6.7.2.1.1 In-plane spline correction;175
9.7.2.1.2;6.7.2.1.2 3d spline correction;177
9.7.2.2;6.7.2.2 Stereo reconstruction;184
9.7.3;6.7.3 Results for the two-plane distortion model;186
9.7.3.1;6.7.3.1 Stereo reconstruction;186
9.7.3.2;6.7.3.2 Point to point error;186
9.7.3.3;6.7.3.3 Angles of reconstructed planes;190
9.7.3.4;6.7.3.4 Other test series;193
9.7.3.5;Standard),;193
9.7.3.6;Two-Plane);193
9.7.3.7;One-Plane).;193
9.7.3.8;6.7.3.5 Planarity test;200
9.7.3.9;6.7.3.6 Prototype reconstruction;205
9.7.3.10;Standard),;205
9.7.3.11;One-Plane),;205
9.7.3.12;3d2dSpline,;205
9.7.3.13;Two-Plane).;205
10;Chapter 7 Conclusions;210
11;Bibliography;212
12;Index;224

Über Autor(innen)

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