Textbook
Statistics for Imaging, Optics, and PhotonicsOctober 2011, ©2011

A vivid, handson 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 selfcontained 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 upperundergraduate 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.
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 PlugIn 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 ChiSquared 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 nDimensional 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 pDimensional 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 T2 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 FairShare Stopping Rules, 210
7.3.2 LargeGap 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. CrossValidation, 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 CrossValidation 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 KMeans 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
PETER BAJORSKI, PhD, is Associate Professor in the Graduate Statistics Department at Rochester Institute of Technology, where he is also a core member of the graduate program faculty at the Center for Imaging Science. The author of numerous published articles on statistics and imaging, Dr. Bajorski's areas of statistical expertise include regression techniques, multivariate analysis, design of experiments, nonparametric methods, and visualization methods. A senior member of the IEEE and SPIE, his research in imaging includes unmixing and target detection in spectral images.
“In a word, this is a wellstructured volume which will meet the demand visualised by the author.” (Contemporary Physics, 6 December 2013)
"The monograph is applicable for courses on multivariate statistics for imaging science, optics, and photonics at the upperundergraduate and graduate levels. It also serves as a valuable reference for professionals working in imaging, optics, and photonics who carry out data analyses in their everyday work." (Zentralblatt MATH, 1 August 2013)
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