Statistics for Imaging, Optics, and Photonics

Wiley Series in Probability and Statistics
Peter Bajorski

There is a growing need for statistical analysis of data in imaging, optics, and photonics applications. Although there is a vast literature explaining statistical methods needed for such applications, there are two major difficulties for practitioners using these statistical resources. The first difficulty is that most statistical books are written in a formal statistical and mathematical language, which an occasional user of statistics may find difficult to understand. The second difficulty is that the needed material is scattered among many statistical books.

The purpose of this book is to bridge the gap between imaging, optics, and photonics, and statistics and data analysis. The statistical techniques are explained in the context of real examples from remote sensing, color science, printing, astronomy, and other related disciplines. I emphasize intuitive and geometric understanding of concepts and provide many (198) graphs for their illustration.

The scope of the material is very broad. It starts with rudimentary data analysis and ends with sophisticated multivariate statistical methods. Necessarily, the presentation is brief and does not cover all aspects of the discussed methods. I concentrate on teaching the skills of statistical thinking, and providing the tools needed the most in imaging, optics, and photonics. There are many details in this book that you will not find in most statistical textbooks. They enhance the reader’s understanding and answer the usual questions asked by students of the subject.

In the following, I provide brief descriptions of the book chapters and list some highlights that are either original contributions, or interesting applications, or other features that distinguish this book from other books on statistics. The listed highlights are not intended as the list of the most important topics.
You can find a full table of contents by clicking here .


Chapter 1. An “appetizer” chapter with examples of data and types of analyses performed in this book

Chapter 2. A brief review of descriptive statistics, graphs, and probability for readers with some prior experience with those topics. Some distinguishing features:

  • In Example 2.3, I combine the use of a histogram and a definition of a connected set in order to solve a practical imaging problem.
  • Property 2.3 is an original contribution.
  • Section 2.8 includes some examples that shed a new light on the properties of kurtosis.

Chapter 3. A brief review of inferential statistics (confidence intervals and hypothesis testing) for readers with some prior experience with those topics. Some distinguishing features:

  • For the Ryan-Joiner test of normality presented in Section 3.6.4, correct critical values are provided (in Appendix C.4). Some books and software provide very imprecise critical values for that test.
  • In Section 3.7, I explain the importance of the sample size in outlier detection. This topic is often explained incorrectly in other sources. Many software packages use incorrect methods for assessing outliers.
  • Sections 3.8 and 3.9 provide a brief overview of Monte Carlo simulations and bootstrapping.

Chapter 4. A brief review of linear regression models and design and analysis of experiments. Supplement 4A provides basics of matrix algebra, and Supplement 4B discusses random vectors. Some distinguishing features:

  • In Section 4.2.7, I advocate use of externally studentized residuals and provide guidelines on the outlier detection. The outlier threshold should be at least 3.5, but usually much higher. The outlier detection in regression is often done incorrectly in other sources, including some statistical software.
  • In Section 4.3 (p. 113),I discuss the scientific rule of changing one factor at a time in experiments. I explain what goes wrong when this rule is applied too rigidly, resulting in the so-called one-factor-at-a-time experiments. I then show a better implementation of this rule in the full factorial designs.

Chapter 5. Introduction to fundamental concepts of multivariate statistics. Some distinguishing features:
In Section 5.5, I introduce k-dimensional generalized sample variance, particularly useful for small samples from high-dimensional data

  • In Section 5.7.2, I discuss what happens with two- and three-sigma rules in multidimensional spaces and how this relates to the so-called “curse of dimensionality.” I also provide formulas and graphs for detection of multivariate outliers with the use of the Mahalanobis distance.

Chapter 6. Statistical inference for one and two multivariate samples. The case of more than two samples is covered when testing equality of the variance-covariance matrices. Some distinguishing features:

  • Section 6.2.2 provides a thorough coverage of Bonferroni confidence intervals, including some illustrative comparisons to the single confidence intervals.
  • Example 6.4 and Figure 6.3 illustrate the relationship between the univariate confidence intervals and the elliptical confidence regions.

Chapter 7. Principal component analysis. Some distinguishing features:

  • A new tool of impact plots is introduced in Section 7.2.3
  • Some new stopping rules that are particularly suited for high-dimensional data are introduced in Section 7.3.
  • Residual analysis of imaging spectral data (Section 7.5) reveals spatial structure in high-order principal components that are often believed to contain only noise.
  • Section 7.6.2 shows examples of statistical inference in sampling schemes relevant in imaging data. I also explain why the classic statistical inference based on the independent and identically distributed sampling schemes is typically not suitable for imaging data.

Chapter 8. Canonical Correlation Analysis. Some distinguishing features:

  • Section 8.5 covers the canonical correlation regression, which is rarely discussed in statistical books.
  • Supplement 8A explains the concept of cross-validation.

Chapter 9. Discrimination and Classification – Supervised Learning. Some distinguishing features:

  • Calculation of misclassification rates on p. 271 is very illustrative for educational purposes.
  • Section 9.4 shows how to use the spatial information to enhance a classification rule.

Chapter 10. Clustering – Unsupervised Learning. Some distinguishing features:

  • In Section 10.2.1, I give some guidelines on how to decide which Lp metric is most appropriate in a given context.
  • In Section 10.3.1, I discuss the minimum spanning tree and the minimum forest, and then I use those tools in graphical representation of clusters.


Appendix A. A list of probability distributions with a brief description of their properties. Tabulation of some distributions is also provided.

Appendix B. Background information about 14 case studies used throughout the book.

Appendix C. Miscellaneous topics:

  • Singular values decomposition
  • An alternative approach to canonical correlation analysis
  • List of abbreviations and mathematical symbols used throughout the book
 
 

 

 

ISBN: 978-0-470-50945-6
Hardcover
408 pages
October 2011

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