This new edition incorporates corrections of all known typographical errors in the first edition, as well as some more substantive changes. Chief among the latter is the addition of Chap. 17, on methods of estimation. As with the rest of the text, most applications and examples cited in the new chapter are from the optical perspective. The intention behind this new chapter is to empower the optical researcher with a yet broader range of research tools. Certainly a basic knowledge of estimation methods should be among these. In particular, the sections on likelihood theory and Fisher information prepare readers for the problems of optical parameter estimation and probability law estimation. Physicists and optical scientists might find this material particularly useful, since the subject of Fisher information is generally not covered in standard physical science curricula. Since the words "statistical optics" are prominent in the title of this book, their meaning needs to be clarified. There is a general tendency to overly emphasize the statistics of photons as the sine qua non of statistical optics. In view is taken, which equally emphasizes the random medium this text a wider that surrounds the photon, be it a photographic emulsion, the turbulent atmo­ sphere, a vibrating lens holder, etc. Also included are random interpretations of ostensibly deterministic phenomena, such as the Hurter-Driffield (H and D) curve of photography. Such a "random interpretation" sometimes breaks new ground, as in Chap.

1. Introduction.- 1.1 What Is Chance, and Why Study It?.- 1.1.1 Chance vs Determinism.- 1.1.2 Probability Problems in Optics.- 1.1.3 Statistical Problems in Optics.- 2. The Axiomatic Approach.- 2.1 Notion of an Experiment; Events.- 2.1.1 Event Space; The Space Event.- 2.1.2 Disjoint Events.- 2.1.3 The Certain Event.- Exercise 2.1.- 2.2 Definition of Probability.- 2.3 Relation to Frequency of Occurrence.- 2.4 Some Elementary Consequences.- 2.4.1 Additivity Property.- 2.4.2 Normalization Property.- 2.5 Marginal Probability.- 2.6 The "Traditional" Definition of Probability.- 2.7 Illustrative Problem: A Dice Game.- 2.8 Illustrative Problem: Let's (Try to) Take a Trip.- 2.9 Law of Large Numbers.- 2.10 Optical Objects and Images as Probability Laws.- 2.11 Conditional Probability.- Exercise 2.2.- 2.12 The Quantity of Information.- 2.13 Statistical Independence.- 2.13.1 Illustrative Problem: Let's (Try to) Take a Trip (Continued).- 2.14 Informationless Messages.- 2.15 A Definition of Noise.- 2.16 "Additivity" Property of Information.- 2.17 Partition Law.- 2.18 Illustrative Problem: Transmittance Through a Film.....- 2.19 How to Correct a Success Rate for Guesses.- Exercise 2.3.- 2.20 Bayes' Rule.- 2.21 Some Optical Applications.- 2.22 Information Theory Application.- 2.23 Application to Markov Events.- 2.24 Complex Number Events.- Exercise 2.4.- 3. Continuous Random Variables.- 3.1 Definition of a Random Variable.- 3.2 Probability Density Function, Basic Properties.- 3.3 Information Theory Application: Continuous Limit.....- 3.4 Optical Application: Continuous Form of Imaging Law...- 3.5 Expected Values, Moments.- 3.6 Optical Application: Moments of the Slit Diffraction Pattern.- 3.7 Information Theory Application.- 3.8 Case of Statistical Independence.- 3.9 Mean of a Sum.- 3.10 Optical Application.- 3.11 Deterministic Limit; Representations of the Dirac ?-Function.- 3.12 Correspondence Between Discrete and Continuous Cases..- 3.13 Cumulative Probability.- 3.14 The Means of an Algebraic Expression: A Simplified Approach.- 3.15 A Potpourri of Probability Laws.- 3.15.1 Poisson.- 3.15.2 Binomial.- 3.15.3 Uniform.- 3.15.4 Exponential.- 3.15.5 Normal (One-Dimensional).- 3.15.6 Normal (Two-Dimensional).- 3.15.7 Normal (Multi-Dimensional).- 3.15.8 Skewed Gaussian Case; Gram-Charlier Expansion..- 3.15.9 Optical Application.- 3.15.10 Geometric Law.- 3.15.11 Cauchy Law.- 3.15.12 Sinc2 Law.- Exercise 3.1.- 4. Fourier Methods in Probability.- 4.1 Characteristic Function Defined.- 4.2 Use in Generating Moments.- 4.3 An Alternative to Describing RV x.- 4.4 On Optical Applications.- 4.5 Shift Theorem.- 4.6 Poisson Case.- 4.7 Binomial Case.- 4.8 Uniform Case.- 4.9 Exponential Case.- 4.10 Normal Case (One Dimension).- 4.11 Multidimensional Cases.- 4.12 Normal Case (Two Dimensions).- 4.13 Convolution Theorem, Transfer Theorem.- 4.14 Probability Law for the Sum of Two Independent RV's.- 4.15 Optical Applications.- 4.15.1 Imaging Equation as the Sum of Two Random Displacements.- 4.15.2 Unsharp Masking.- 4.16 Sum of n Independent RV's; the "Random Walk" Phenomenon.- Exercise 4.1.- 4.17 Resulting Mean and Variance: Normal, Poisson, and General Cases.- 4.18 Sum of n Dependent RV's.- 4.19 Case of Two Gaussian Bivariate RV's.- 4.20 Sampling Theorems for Probability.- 4.21 Case of Limited Range of x, Derivation.- 4.22 Discussion.- 4.23 Optical Application.- 4.24 Case of Limited Range of ?.- 4.25 Central Limit Theorem.- 4.26 Derivation.- Exercise 4.2.- 4.27 How Large Does n Have to be?.- 4.28 Optical Applications.- 4.28.1 Cascaded Optical Systems.- 4.28.2 Laser Resonator.- 4.28.3 Atmospheric Turbulence.- 4.29 Generating Normally Distributed Numbers from Uniformly Random Numbers.- 4.30 The Error Function.- Exercise 4.3.- 5. Functions of Random Variables.- 5.1 Case of a Single Random Variable.- 5.2 Unique Root.- 5.3 Application from Geometrical Optics.- 5.4 Multiple Roots.- 5.5 Illustrative Example.- 5.6 Case of n Random Variables, r Roots.- 5.7 Optical Applications.- 5.8 Statistical Modeling.- 5.9 Application of Transformation Theory to Laser Speckle.- 5.9.1 Physical Layout.- 5.9.2 Plan.- 5.9.3 Statistical Model.- 5.9.4 Marginal Probabilities for Light Amplitudes Ure, Uim.- 5.9.5 Correlation Between Ure and Uim.- 5.9.6 Joint Probability Law for Ure, Uim.- 5.9.7 Probability Laws for Intensity and Phase; Transformation of the RV's.- 5.9.8 Marginal Law for Intensity and Phase.- 5.9.9 Signal-to-Noise (S/N) Ratio in the Speckle Image.- 5.10 Speckle Reduction by Use of a Scanning Aperture.- 5.10.1 Statistical Model.- 5.10.2 Probability Density for Output Intensity pI (v).- 5.10.3 Moments and S/N Ratio.- 5.10.4 Standard Form for the Chi-Square Distribution.- 5.11 Calculation of Spot Intensity Profiles Using Transformation Theory.- 5.11.1 Illustrative Example.- 5.11.2 Implementation by Ray-Trace.- 5.12 Application of Transformation Theory to a Satellite-Ground Communication Problem.- Exercise 5.1.- 6. Bernoulli Trials and Limiting Cases.- 6.1 Analysis.- 6.2 Illustrative Problems.- 6.2.1 Illustrative Problem: Let's (Try to) Take a Trip: The Last Word.- 6.2.2 Illustrative Problem: Mental Telepathy as a Communication Link?.- 6.3 Characteristic Function and Moments.- 6.4 Optical Application: Checkerboard Model of Granularity.- 6.5 The Poisson Limit.- 6.5.1 Analysis.- 6.5.2 Example of Degree of Approximation.- 6.6 Optical Application: The Shot Effect.- 6.7 Optical Application: Combined Sources.- 6.8 Poisson Joint Count for Two Detectors - Intensity Interferometry.- 6.9 The Normal Limit (DeMoivre-Laplace Law).- 6.9.1 Derivation.- 6.9.2 Conditions of Use.- 6.9.3 Use of the Error Function.- Exercise 6.1.- 7. The Monte Carlo Calculation.- 7.1 Producing Random Numbers That Obey a Prescribed Probability Law.- 7.1.1 Illustrative Case.- 7.1.2 Normal Case.- 7.2 Analysis of the Photographic Emulsion by Monte Carlo Calculation.- 7.3 Application of the Monte Carlo Calculation to Remote Sensing.- 7.4 Monte Carlo Formation of Optical Images.- 7.4.1 An Example.- 7.5 Monte Carlo Simulation of Speckle Patterns.- Exercise 7.1.- 8. Stochastic Processes.- 8.1 Definition of a Stochastic Process.- 8.2 Definition of Power Spectrum.- 8.2.1 Some Examples of Power Spectra.- 8.3 Definition of Autocorrelation Function; Kinds of Stationarity.- 8.4 Fourier Transform Theorem.- 8.5 Case of a "White" Power Spectrum.- 8.6 Application: Average Transfer Function Through Atmospheric Turbulence.- 8.6.…