A vivid, hands-on discussion of the statistical methods in imaging, optics, and photonics applications In the field of imaging science, there is a growing need for students and practitioners to be equipped with the necessary knowledge and tools to carry out quantitative analysis of data. Providing a self-contained approach that is not too heavily statistical in nature, Statistics for Imaging, Optics, and Photonics presents necessary analytical techniques in the context of real examples from various areas within the field, including remote sensing, color science, printing, and astronomy. Bridging the gap between imaging, optics, photonics, and statistical data analysis, the author uniquely concentrates on statistical inference, providing a wide range of relevant methods. Brief introductions to key probabilistic terms are provided at the beginning of the book in order to present the notation used, followed by discussions on multivariate techniques such as: Linear regression models, vector and matrix algebra, and random vectors and matrices Multivariate statistical inference, including inferences about both mean vectors and covariance matrices Principal components analysis Canonical correlation analysis Discrimination and classification analysis for two or more populations and spatial smoothing Cluster analysis, including similarity and dissimilarity measures and hierarchical and nonhierarchical clustering methods Intuitive and geometric understanding of concepts is emphasized, and all examples are relatively simple and include background explanations. Computational results and graphs are presented using the freely available R software, and can be replicated by using a variety of software packages. Throughout the book, problem sets and solutions contain partial numerical results, allowing readers to confirm the accuracy of their approach; and a related website features additional resources including the book's datasets and figures. Statistics for Imaging, Optics, and Photonics is an excellent book for courses on multivariate statistics for imaging science, optics, and photonics at the upper-undergraduate and graduate levels. The book also serves as a valuable reference for professionals working in imaging, optics, and photonics who carry out data analyses in their everyday work.

Table of Contents:
Preface xiii 1 Introduction 1 1.1 Who Should Read This Book 6 1.2 How This Book is Organized 6 1.3 How to Read This Book and Learn from It 7 1.4 Note for Instructors 8 1.5 Book Web Site 9 2 Fundamentals of Statistics 11 2.1 Statistical Thinking 11 2.2 Data Format 13 2.3 Descriptive Statistics 14 2.3.1 Measures of Location 14 2.3.2 Measures of Variability 16 2.4 Data Visualization 17 2.4.1 Dot Plots 17 2.4.2 Histograms 19 2.4.3 Box Plots 23 2.4.4 Scatter Plots 24 2.5 Probability and Probability Distributions 26 2.5.1 Probability and Its Properties 26 2.5.2 Probability Distributions 30 2.5.3 Expected Value and Moments 33 2.5.4 Joint Distributions and Independence 34 2.5.5 Covariance and Correlation 38 2.6 Rules of Two and Three Sigma 40 2.7 Sampling Distributions and the Laws of Large Numbers 41 2.8 Skewness and Kurtosis 44 3 Statistical Inference 51 3.1 Introduction 51 3.2 Point Estimation of Parameters 53 3.2.1 Definition and Properties of Estimators 53 3.2.2 The Method of the Moments and Plug-In Principle 56 3.2.3 The Maximum Likelihood Estimation 57 3.3 Interval Estimation 60 3.4 Hypothesis Testing 63 3.5 Samples From Two Populations 71 3.6 Probability Plots and Testing for Population Distributions 73 3.6.1 Probability Plots 74 3.6.2 Kolmogorov–Smirnov Statistic 75 3.6.3 Chi-Squared Test 76 3.6.4 Ryan–Joiner Test for Normality 76 3.7 Outlier Detection 77 3.8 Monte Carlo Simulations 79 3.9 Bootstrap 79 4 Statistical Models 85 4.1 Introduction 85 4.2 Regression Models 85 4.2.1 Simple Linear Regression Model 86 4.2.2 Residual Analysis 94 4.2.3 Multiple Linear Regression and Matrix Notation 96 4.2.4 Geometric Interpretation in an n-Dimensional Space 99 4.2.5 Statistical Inference in Multiple Linear Regression 100 4.2.6 Prediction of the Response and Estimation of the Mean Response 104 4.2.7 More on Checking the Model Assumptions 107 4.2.8 Other Topics in Regression 110 4.3 Experimental Design and Analysis 111 4.3.1 Analysis of Designs with Qualitative Factors 116 4.3.2 Other Topics in Experimental Design 124 Supplement 4A. Vector and Matrix Algebra 125 Vectors 125 Matrices 127 Eigenvalues and Eigenvectors of Matrices 130 Spectral Decomposition of Matrices 130 Positive Definite Matrices 131 A Square Root Matrix 131 Supplement 4B. Random Vectors and Matrices 132 Sphering 134 5 Fundamentals of Multivariate Statistics 137 5.1 Introduction 137 5.2 The Multivariate Random Sample 139 5.3 Multivariate Data Visualization 143 5.4 The Geometry of the Sample 148 5.4.1 The Geometric Interpretation of the Sample Mean 148 5.4.2 The Geometric Interpretation of the Sample Standard Deviation 149 5.4.3 The Geometric Interpretation of the Sample Correlation Coefficient 150 5.5 The Generalized Variance 151 5.6 Distances in the p-Dimensional Space 159 5.7 The Multivariate Normal (Gaussian) Distribution 163 5.7.1 The Definition and Properties of the Multivariate Normal Distribution 163 5.7.2 Properties of the Mahalanobis Distance 166 6 Multivariate Statistical Inference 173 6.1 Introduction 173 6.2 Inferences About a Mean Vector 173 6.2.1 Testing the Multivariate Population Mean 173 6.2.2 Interval Estimation for the Multivariate Population Mean 175 6.2.3 T 2 Confidence Regions 179 6.3 Comparing Mean Vectors from Two Populations 183 6.3.1 Equal Covariance Matrices 184 6.3.2 Unequal Covariance Matrices and Large Samples 185 6.3.3 Unequal Covariance Matrices and Samples Sizes Not So Large 186 6.4 Inferences About a Variance–Covariance Matrix 187 6.5 How to Check Multivariate Normality 188 7 Principal Component Analysis 193 7.1 Introduction 193 7.2 Definition and Properties of Principal Components 195 7.2.1 Definition of Principal Components 195 7.2.2 Finding Principal Components 196 7.2.3 Interpretation of Principal Component Loadings 200 7.2.4 Scaling of Variables 207 7.3 Stopping Rules for Principal Component Analysis 209 7.3.1 Fair-Share Stopping Rules 210 7.3.2 Large-Gap Stopping Rules 213 7.4 Principal Component Scores 217 7.5 Residual Analysis 220 7.6 Statistical Inference in Principal Component Analysis 227 7.6.1 Independent and Identically Distributed Observations 227 7.6.2 Imaging Related Sampling Schemes 228 7.7 Further Reading 238 8 Canonical Correlation Analysis 241 8.1 Introduction 241 8.2 Mathematical Formulation 242 8.3 Practical Application 245 8.4 Calculating Variability Explained by Canonical Variables 246 8.5 Canonical Correlation Regression 251 8.6 Further Reading 256 Supplement 8A. Cross-Validation 256 9 Discrimination and Classification – Supervised Learning 261 9.1 Introduction 261 9.2 Classification for Two Populations 264 9.2.1 Classification Rules for Multivariate Normal Distributions 267 9.2.2 Cross-Validation of Classification Rules 277 9.2.3 Fisher’s Discriminant Function 280 9.3 Classification for Several Populations 284 9.3.1 Gaussian Rules 284 9.3.2 Fisher’s Method 286 9.4 Spatial Smoothing for Classification 291 9.5 Further Reading 293 10 Clustering – Unsupervised Learning 297 10.1 Introduction 297 10.2 Similarity and Dissimilarity Measures 298 10.2.1 Similarity and Dissimilarity Measures for Observations 298 10.2.2 Similarity and Dissimilarity Measures for Variables and Other Objects 304 10.3 Hierarchical Clustering Methods 304 10.3.1 Single Linkage Algorithm 305 10.3.2 Complete Linkage Algorithm 312 10.3.3 Average Linkage Algorithm 315 10.3.4 Ward Method 319 10.4 Nonhierarchical Clustering Methods 320 10.4.1 K-Means Method 320 10.5 Clustering Variables 323 10.6 Further Reading 325 Appendix A Probability Distributions 329 Appendix B Data Sets 349 Appendix C Miscellanea 355 References 365 Index 371