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.

"A Chapman & Hall Book."

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 Rotation2.1 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 Rotation3.1 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 Noise4.1 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 Noise5.1 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 Method6.1 DERIVAT…