Contents. 1 Introduction Background Organization Features... 7

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1 Contents 1 Introduction Background Organization Features... 7 Part I Fundamental Algorithms for Computer Vision 2 Ellipse Fitting Representation of Ellipses Least-Squares Approach Noise and Covariance Matrices Algebraic Methods Iterative Reweight Renormalization and the Taubin Method Hyper-Renormalization and HyperLS Summary of Algebraic Methods Geometric Methods Geometric Distance and Sampson Error FNS Geometric Distance Minimization Hyper-Accurate Correction Ellipse-Specific Methods Ellipse Condition Method of Fitzgibbon et al Method of Random Sampling Outlier Removal Examples Supplemental Note Problems References Fundamental Matrix Computation Fundamental Matrices Covariance Matrices and Algebraic Methods vii

2 viii Contents 3.3 Geometric Distance and Sampson Error Rank Constraint A Posteriori Correction Hidden Variables Approach Extended FNS Geometric Distance Minimization Outlier Removal Examples Supplemental Note Problems References Triangulation Perspective Projection Camera Matrix and Triangulation Triangulation from Noisy Correspondence Optimal Correction of Correspondences Examples Supplemental Note Problems References D Reconstruction from Two Views Camera Modeling and Self-calibration Expression of the Fundamental Matrix Focal Length Computation Motion Parameter Computation D Shape Computation Examples Supplemental Note Problems References Homography Computation Homographies Noise and Covariance Matrices Algebraic Methods Geometric Distance and Sampson Error FNS Geometric Distance Minimization Hyperaccurate Correction Outlier Removal Examples Supplemental Note Problems References

3 Contents ix 7 Planar Triangulation Perspective Projection of a Plane Planar Triangulation Procedure of Planar Triangulation Examples Supplemental Note Problems References D Reconstruction of a Plane Self-calibration with a Plane Computation of Surface Parameters and Motion Parameters Selection of the Solution Examples Supplemental Note Problems References Ellipse Analysis and 3D Computation of Circles Intersections of Ellipses Ellipse Centers, Tangents, and Perpendiculars Projection of Circles and 3D Reconstruction Center of Circle Front Image of the Circle Examples Supplemental Note Problems References Part II Multiview 3D Reconstruction Techniques 10 Multiview Triangulation Trilinear Constraint Triangulation from Three Views Optimal Correspondence Correction Solving Linear Equations Efficiency of Computation D Position Computation Triangulation from Multiple Views Examples Supplemental Note Problems References

4 x Contents 11 Bundle Adjustment Principle of Bundle Adjustment Bundle Adjustment Algorithm Derivative Computation Gauss-Newton Approximation Derivatives with Respect to 3D Positions Derivatives with Respect to Focal Lengths Derivatives with Respect to Principal Points Derivatives with Respect to Translations Derivatives with Respect to Rotations Efficient Computation and Memory Use Efficient Linear Equation Solving Examples Supplemental Note Problems References Self-calibration of Affine Cameras Affine Cameras Factorization and Affine Reconstruction Metric Condition for Affine Cameras Description in the Camera Coordinate System Symmetric Affine Camera Self-calibration of Symmetric Affine Cameras Self-calibration of Simplified Affine Cameras Paraperspective Projection Model Weak Perspective Projection Model Orthographic Projection Model Examples Supplemental Note Problems References Self-calibration of Perspective Cameras Homogeneous Coordinates and Projective Reconstruction Projective Reconstruction by Factorization Principle of Factorization Primary Method Dual Method Euclidean Upgrading Principle of Euclidean Upgrading Computation of X Modification of K j Computation of H Procedure for Euclidean Upgrading

5 Contents xi D Reconstruction Computation Examples Supplemental Notes Problems References Part III Mathematical Foundation of Geometric Estimation 14 Accuracy of Geometric Estimation Constraint of the Problem Noise and Covariance Matrices Error Analysis Covariance and Bias Bias Elimination and Hyper-Renormalization Derivations Supplemental Note Problems References Maximum Likelihood of Geometric Estimation Maximum Likelihood Sampson Error Error Analysis Bias Analysis and Hyper-Accurate Correction Derivations Supplemental Note Problems References Theoretical Accuracy Limit Kanatani-Cramer-Rao (KCR) Lower Bound Structure of Constraints Derivation of the KCR Lower Bound Expression of the KCR Lower Bound Supplemental Note Problems References Solutions Index

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