Symmetry in Optics and Vision Studies: A Data-Analytic Approach (Multidisciplinary and Applied Optics)

Description

This book presents an introduction to the foundations, interpretations, and data-analytic applications of symmetry studies with an emphasis on applications in optical sciences. Symmetry studies connect group theoretic and statistical methods for data summary and inference. Readers should have an understanding of calculus and linear algebra as well as introductory statistics. The book reviews finite group theory in the introductory chapters. Computational tools used in the text are available for download in the form of Mathmaticaâ notebooks or R scripts. This book: Demonstrates the usefulness of a unified view of algebra and symmetry studies to address data-analytic questions in optics and vision science Offers a brief review of finite group theory and elements of multivariate analysis Includes various examples from diverse areas of optical science Cover Half Title Series Page Title Page Copyright Page Dedication Contents Preface Authors Chapter 1: Symmetry Studies 1.1. Introduction 1.2. Bilateral Quantities 1.3. Sampling 1.4. Joint Bilateral Quantities 1.5. Covariance Structures 1.6. Visual Fields 1.7. Snellen Charts 1.8. Nystagmus 1.9. Corneal Curvature 1.10. Refraction 1.11. Extreme Values 1.12. Summary Chapter 2: Algebraic Aspects 2.1. Introduction 2.2. Main Definitions 2.3. Group Representations 2.4. Canonical Projections 2.5. Fourier Transforms 2.6. Additional Aspects 2.7. Summary Chapter 3: Dihedral Decompositions 3.1. Introduction 3.2. The Stokes Coefficients 3.3. The Group Algebra of Compensators 3.4. Elementary Optical Instruments 3.5. Coherence Matrix Formalism 3.6. Data-Analytic Considerations 3.7. Summary Chapter 4: Refraction 4.1. Introduction 4.2. Algebraic Descriptions 4.3. Finite Approximations 4.4. Conformal Analysis 4.5. The Refraction Study 4.6. Summary Chapter 5: Dihedral Polynomials 5.1. Introduction 5.2. Dihedral Polynomials 5.3. Between-Class Relations 5.4. Applications 5.5. Summary Chapter 6: Visual Perception of Symmetry 6.1. Introduction 6.2. A Symmetry Perception Experiment 6.3. The Elementary Orbits 6.4. Entropy Invariants 6.5. Product Orbit 6.6. Summary Chapter 7: Cyclic Reduction of Symbolic Sequences 7.1. Introduction 7.2. Symbolic Sequences 7.3. Injective Sequences 7.4. Scalar Quantifications 7.5. Orbit Invariance 7.6. Invariant Reductions 7.7. Entropy Invariants 7.8. Dihedral Fourier Analysis 7.9. Summary Chapter 8: Symmetrically Dependent Observations 8.1. Introduction 8.2. Models and Assumptions 8.3. The Joint Covariance Structure 8.4. Maximum Likelihood and Large-Sample Estimates 8.5. Visual Acuity Data Applications 8.6. Block-Equicorrelated Observations 8.7. Intraocular Pressure Data Applications 8.8. Linearizations of Ordered Symmetric Observations 8.9. Concomitants of Order Statistics 8.10. Bilateral Hemispherical Data Applications 8.11. Summary Chapter 9: Additional Aspects and Applications 9.1. Chromatic-Spatial Decompositions 9.2. Dihedral Summaries of Visual Field Data 9.3. Actions on Mappings 9.4. Local and Global Symmetries 9.5. Entropy and Symmetry 9.6. Matrix and Transformation Groups Appendix Bibliography Index