3D Rotations Parameter Computation and Lie Algebra based Optimization

3D rotation analysis is widely encountered in everyday problems thanks to the development of computers. Sensing 3D using cameras and sensors, analyzing and modeling 3D for computer vision and computer graphics, and controlling and simulating robot motion all require 3D rotation computation. This book focuses on the computational analysis of 3D rotation, rather than classical motion analysis. It regards noise as random variables and models their probability distributions. It also pursues statistically optimal computation for maximizing the expected accuracy, as is typical of nonlinear optimization. All concepts are illustrated using computer vision applications as examples. Mathematically, the set of all 3D rotations forms a group denoted by SO(3). Exploiting this group property, we obtain an optimal solution analytical or numerically, depending on the problem. Our numerical scheme, which we call the "Lie algebra method," is based on the Lie group structure of SO(3). This book also proposes computing projects for readers who want to code the theories presented in this book, describing necessary 3D simulation setting as well as providing real GPS 3D measurement data. To help readers not very familiar with abstract mathematics, a brief overview of quaternion algebra, matrix analysis, Lie groups, and Lie algebras is provided as Appendix at the end of the volume.

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

Kenichi Kanatani received his B.E., M.S., and Ph.D. in applied mathematics from the University of Tokyo in 1972, 1974, and 1979, respectively. After serving as Professor of computer science at Gunma University, Gunma, Japan, and Okayama University, Okayama, Japan, he retired in 2013 and is now Professor Emeritus of Okayama University.He was a visiting researcher at the University of Maryland, U.S. (1985–1986, 1988–1989, 1992), the University of Copenhagen, Denmark (1988), the University of Oxford,U.K. (1991), INRIA at Rhone Alpes, France (1988), ETH, Switzerland (2013), the Uni-versity of Paris-Est, France (2014), Link ¨oping University, Sweden (2015), and NationalTaiwan Normal University (2019).He is the author of K. Kanatani,Group-Theoretical Methods in Image Understanding(Springer, 1990), K. Kanatani,Geometric Computation for Machine Vision(Oxford Uni-versity Press, 1993), K. Kanatani,Statistical Optimization for Geometric Computation:Theory and Practice(Elsevier, 1996; reprinted Dover, 2005), K. Kanatani,Understand-ing Geometric Algebra: Hamilton, Grassmann, and Clifford for Computer Vision andGraphics(CRC Press, 2015), K. Kanatani, Y. Sugaya, Y. Kanazawa,Ellipse Fitting forComputer Vision: Implementation and Applications(Morgan-Claypool, 2016), and K.Kanatani, Y. Sugaya, Y. Kanazawa,Guide to 3D Vision Computation: Geometric Anal-ysis and Implementation(Springer, 2016).He received many awards including the best paper awards from IPSJ (1987) , IEICE(2005), and PSIVT (2009). He is a Fellow of IEEE, IAPR, and IEICE.1