3D Rotations : Parameter Computation and Lie Algebra based Optimization (Paperback)

Chapter 1 Introduction 1.1 3D ROTATIONS 1.2 ESTIMATION OF ROTATION 1.3 DERIVATIVE-BASED OPTIMIZATION1.4 RELIABILITY EVALUATION OF ROTATION COMPUTATION1.5 COMPUTING PROJECTS 1.6 RELATED TOPICS OF MATHEMATICS Chapter 2 ■ Geometry of Rotation 2.1 3D ROTATION 2.2 ORTHOGONAL MATRICES AND ROTATION MATRICES2.3 EULERS THEOREM 2.4 AXIAL ROTATIONS 2.5 SUPPLEMENTAL NOTE 2.6 EXERCISES Chapter 3 ■ Parameters of Rotation 3.1 ROLL, PITCH, YAW 3.2 COORDINATE SYSTEM ROTATION 153.3 EULER ANGLES 3.4 RODRIGUES FORMULA 3.5 QUATERNION REPRESENTATION 213.6 SUPPLEMENTAL NOTES 3.7 EXERCISES Chapter 4 ■ Estimation of Rotation I: Isotropic Noise 4.1 ESTIMATING ROTATION 4.2 LEAST SQUARES AND MAXIMUM LIKELIHOOD4.3 SOLUTION BY SINGULAR VALUE DECOMPOSITION4.4 SOLUTION BY QUATERNION REPRESENTATION4.5 OPTIMAL CORRECTION OF ROTATION4.6 SUPPLEMENTAL NOTE 4.7 EXERCISES Chapter 5 ■ Estimation of Rotation II: Anisotropic Noise 5.1 ANISOTROPIC GAUSSIAN DISTRIBUTIONS5.2 ROTATION ESTIMATION BY MAXIMUM LIKELIHOOD5.3 ROTATION ESTIMATION BY QUATERNION REPRESENTATION5.4 OPTIMIZATION BY FNS 5.5 METHOD OF HOMOGENEOUS CONSTRAINTS5.6 SUPPLEMENTAL NOTE 5.7 EXERCISES Chapter 6 ■ Derivative-based Optimization: Lie Algebra Method 6.1 DERIVATIVE-BASED OPTIMIZATION6.2 SMALL ROTATIONS AND ANGULAR VELOCITY6.3 EXPONENTIAL EXPRESSION OF ROTATION6.4 LIE ALGEBRA OF INFINITESIMAL ROTATIONS6.5 OPTIMIZATION OF ROTATION 6.6 ROTATION ESTIMATION BY MAXIMUM LIKELIHOOD6.7 FUNDAMENTAL MATRIX COMPUTATION6.8 BUNDLE ADJUSTMENT 6.9 SUPPLEMENTAL NOTES 6.10 EXERCISES Chapter 7 ■ Reliability of Rotation Computation 7.1 ERROR EVALUATION FOR ROTATION7.2 ACCURACY OF MAXIMUM LIKELIHOOD7.3 THEORETICAL ACCURACY BOUND7.4 KCR LOWER BOUND 7.5 SUPPLEMENTAL NOTES 7.6 EXERCISES Chapter 8 ■ Computing Projects 8.1 STEREO VISION EXPERIMENT8.2 OPTIMAL CORRECTION OF STEREO IMAGES8.3 TRIANGULATION OF STEREO IMAGES8.4 COVARIANCE EVALUATION OF STEREO RECONSTRUCTION8.5 LAND MOVEMENT COMPUTATION USING REAL GPS DATA8.6 SUPPLEMENTAL NOTES 8.7 EXERCISES Appendix A ■ Hamiltons Quaternion Algebra A.1 QUATERNIONS A.2 QUATERNION ALGEBRA A.3 CONJUGATE, NORM, AND INVERSEA.4 QUATERNION REPRESENTATION OF ROTATIONSA.5 COMPOSITION OF ROTATIONSA.6 TOPOLOGY OF ROTATIONS A.7 INFINITESIMAL ROTATIONS A.8 REPRESENTATION OF GROUP OF ROTATIONSA.9 STEREOGRAPHIC PROJECTION Appendix B ■ Topics of Linear Algebra B.1 LINEAR MAPPING AND BASISB.2 PROJECTION MATRICES B.3 PROJECTION ONTO A LINE AND A PLANEB.4 EIGENVALUES AND SPECTRAL DECOMPOSITIONB.5 MATRIX REPRESENTATION OF SPECTRAL DECOMPOSITIONB.6 SINGULAR VALUES AND SINGULAR DECOMPOSITIONB.7 COLUMN AND ROW DOMAINS Appendix C ■ Lie Groups and Lie Algebras C.1 GROUPS C.2 MAPPINGS AND GROUPS OF TRANSFORMATIONC.3 TOPOLOGY C.4 MAPPINGS OF TOPOLOGICAL SPACESC.5 MANIFOLDS C.6 LIE GROUPS C.7 LIE ALGEBRAS C.8 LIE ALGEBRAS OF LIE GROUPS Answers Bibliography Index