Probability, Statistical Optics, and Data Testing

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ISBN: 9783540417088 | 3540417087
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Copyright: 2/1/2002

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Scientists in optics are increasingly confronted with problems that are of a random nature and that require a working knowledge of probability and statistics for their solution. This textbook develops these subjects within the context of optics using a problem-solving approach. All methods are explicitly derived and can be traced back to three simple axioms given at the outset. Students with some previous exposure to Fourier optics or linear theory will find the material particularly absorbing and easy to understand.This third edition contains many new applications to optical and physical phenomena. This includes a method of estimating probability laws exactly, by regarding them as laws of physics to be determined using a new variational principle.

Introduction

(6)

What Is Chance, and Why Study It?

(4)

Chance vs. Determinism

(1)

Probability Problems in Optics

(1)

Statistical Problems in Optics

(1)

Historical Notes

(2)

The Axiomatic Approach

(32)

Notion of an Experiment; Events

(2)

Event Space; The Space Event

(1)

Disjoint Events

(1)

The Certain Event

(1)

Definition of Probability

(1)

Relation to Frequency of Occurrence

(1)

Some Elementary Consequences

(2)

Additivity Property

(1)

Normalization Property

(1)

Marginal Probability

(1)

The ``Traditional'' Definition of Probability

(1)

Illustrative Problem: A Dice Game

(1)

Illustrative Problem: Let's (Try to) Take a Trip

(1)

Law of Large Numbers

(1)

Optical Objects and Images as Probability Laws

(2)

Conditional Probability

(1)

The Quantity of Information

(2)

Statistical Independence

(2)

Illustrative Problem: Let's (Try to) Take a Trip (Continued)

(1)

Informationless Messages

(1)

A Definition of Noise

(1)

``Additivity'' Property of Information

(1)

Partition Law

(1)

Illustrative Problem: Transmittance Through a Film

(1)

How to Correct a Success Rate for Guesses

(1)

Bayes' Rule

(1)

Some Optical Applications

(1)

Information Theory Application

(1)

Application to Markov Events

(1)

Complex Number Events

(1)

What is the Probability of Winning a Lottery Jackpot?

(1)

What is the Probability of a Coincidence of Birthdays at a Party?

(8)

Continuous Random Variables

(40)

Definition of a Random Variable

(1)

Probability Density Function, Basic Properties

(2)

Information Theory Application: Continuous Limit

(1)

Optical Application: Continuous Form of Imaging Law

(1)

Expected Values, Moments

(1)

Optical Application: Moments of the Slit Diffraction Pattern

(1)

Information Theory Application

(1)

Case of Statistical Independence

(1)

Mean of a Sum

(1)

Optical Application

(1)

Deterministic Limit; Representations of the Dirac δ-Function

(1)

Correspondence Between Discrete and Continuous Cases

(1)

Cumulative Probability

(1)

The Means of an Algebraic Expression: A Simplified Approach

(1)

A Potpourri of Probability Laws

(18)

Poisson

(1)

Binomial

(1)

Uniform

(1)

Exponential

(1)

Normal (One-Dimensional)

(1)

Normal (Two-Dimensional)

(2)

Normal (Multi-Dimensional)

(1)

Skewed Gaussian Case; Gram-Charlier Expansion

(1)

Optical Application

(2)

Geometric Law

(1)

Cauchy Law

(1)

sinc2 Law

(10)

Derivation of Heisenberg Uncertainty Principle

(2)

Schwarz Inequality for Complex Functions

(1)

Fourier Relations

(1)

Uncertainty Product

(1)

Hirschman's Form of the Uncertainty Principle

(1)

Measures of Information

(3)

Kullback-Leibler Information

(1)

Renyi Information

(1)

Wootters Information

(1)

Hellinger Information

(1)

Tsallis Information

(1)

Fisher Information

(1)

Fisher Information Matrix

(6)

Fourier Methods in Probability

(28)

Characteristic Function Defined

(1)

Use in Generating Moments

(1)

An Alternative to Describing RV x

(1)

On Optical Applications

(1)

Shift Theorem

(1)

Poisson Case

(1)

Binomial Case

(1)

Uniform Case

(1)

Exponential Case

(1)

Normal Case (One Dimension)

(1)

Multidimensional Cases

(1)

Normal Case (Two Dimensions)

(1)

Convolution Theorem, Transfer Theorem

(1)

Probability Law for the Sum of Two Independent RV's

(1)

Optical Applications

(2)

Imaging Equation as the Sum of Two Random Displacements

(1)

Unsharp Masking

(2)

Sum of n Independent RV's; The ``Random Walk'' Phenomenon

(2)

Resulting Mean and Variance: Normal, Poisson, and General Cases

(1)

Sum of n Dependent RV's

(1)

Case of Two Gaussian Bivariate RV's

(1)

Sampling Theorems for Probability

(1)

Case of Limited Range of x, Derivation

(1)

Discussion

(1)

Optical Application

(1)

Case of Limited Range of ω

(1)

Central Limit Theorem

(1)

Derivation

(2)

How Large Does n Have To Be?

(1)

Optical Applications

(3)

Cascaded Electro-Optical Systems

(1)

Laser Resonator

(1)

Atmospheric Turbulence

(1)

Generating Normally Distributed Numbers from Uniformly Random Numbers

100

(2)

The Error Function

102

(5)

Functions of Random Variables

107

(40)

Case of a Single Random Variable

107

(1)

Unique Root

108

(1)

Application from Geometrical Optics

109

(1)

Multiple Roots

110

(1)

Illustrative Example

111

(1)

Case of n Random Variables, r Roots

111

(1)

Optical Applications

112

(1)

Statistical Modeling

112

(1)

Application of Transformation Theory to Laser Speckle

113

(6)

Physical Layout

113

(1)

Plan

114

(1)

Statistical Model

114

(1)

Marginal Probabilities for Light Amplitudes Ure, Uim

115

(1)

Correlation Between Ure and Uim

116

(1)

Joint Probability Law for Ure, Uim

117

(1)

Probability Laws for Intensity and Phase; Transformation of the RV's

117

(1)

Marginal Laws for Intensity and Phase

118

(1)

Signal-to-Noise (S/N) Ratio in the Speckle Image

118

(1)

Speckle Reduction by Use of a Scanning Aperture

119

(4)

Statistical Model

119

(1)

Probability Density for Output Intensity p1(v)

120

(1)

Moments and S/N Ratio

121

(1)

Standard Form for the Chi-Square Distribution

122

(1)

Calculation of Spot Intensity Profiles Using Transformation Theory

123

(3)

Illustrative Example

124

(1)

Implementation by Ray-Trace

125

(1)

Application of Transformation Theory to a Satellite-Ground Communication Problem

126

(14)

Unequal Numbers of Input and Output Variables: ``Helper Variables''

140

(2)

Probability Law for a Quotient of Random Variables

140

(1)

Probability Law for a Product of Independent Random Variables

141

(1)

More Complicated Transformation Problems

142

(1)

Use of an Invariance Principle to Find a Probability Law

142

(2)

Probability Law for Transformation of a Discrete Random Variable

144

(3)

Bernoulli Trials and Limiting Cases

147

(28)

Analysis

147

(2)

Illustrative Problems

149

(3)

Illustrative Problem: Let's (Try to) Take a Trip: The Last Word

149

(1)

Illustrative Problem: Mental Telepathy as a Communication Link?

150

(2)

Characteristic Function and Moments

152

(1)

Optical Application: Checkerboard Model of Granularity

152

(2)

The Poisson Limit

154

(3)

Analysis

154

(1)

Example of Degree of Approximation

155

(1)

Normal Limit of Poisson Law

156

(1)

Optical Application: The Shot Effect

157

(1)

Optical Application: Combined Sources

158

(1)

Poisson Joint Count for Two Detectors-Intensity Interferometry

158

(4)

The Normal Limit (De Moivre-Laplace Law)

162

(13)

Derivation

162

(1)

Conditions of Use

163

(1)

Use of the Error Function

164

(11)

The Monte Carlo Calculation

175

(16)

Producing Random Numbers That Obey a Prescribed Probability Law

176

(2)

Illustrative Case

177

(1)

Normal Case

177

(1)

Analysis of the Photographic Emulsion by Monte Carlo Calculation

178

(2)

Application of the Monte Carlo Calculation to Remote Sensing

180

(1)

Monte Carlo Formation of Optical Images

181

(2)

An Example

182

(1)

Monte Carlo Simulation of Speckle Patterns

183

(8)

Stochastic Processes

191

(52)

Definition of a Stochastic Process

191

(1)

Definition of Power Spectrum

192

(2)

Some Examples of Power Spectra

194

(1)

Definition of Autocorrelation Function; Kinds of Stationarity

194

(1)

Fourier Transform Theorem

195

(1)

Case of a ``White'' Power Spectrum

196

(1)

Application: Average Transfer Function Through Atmospheric Turbulence

197

(4)

Statistical Model for Phase Fluctuations

198

(1)

A Transfer Function for Turbulence

199

(2)

Transfer Theorems for Power Spectra

201

(7)

Determining the MTF Using Random Objects

201

(1)

Speckle Interferometry of Labeyrie

202

(1)

Resolution Limits of Speckle Interferometry

203

(5)

Transfer Theorem for Autocorrelation: The Knox-Thompson Method

208

(3)

Additive Noise

211

(1)

Random Noise

212

(1)

Ergodic Property

213

(4)

Optimum Restoring Filter

217

(4)

Definition of Restoring Filter

217

(1)

Model

218

(1)

Solution

219

(2)

Information Content in the Optical Image

221

(5)

Statistical Model

222

(1)

Analysis

223

(1)

Noise Entropy

223

(1)

Data Entropy

224

(1)

The Answer

225

(1)

Interpretation

225

(1)

Data Information and Its Ability to be Restored

226

(1)

Superposition Processes; the Shot Noise Process

227

(16)

Probability Law for i

229

(1)

Some Important Averages

229

(1)

Mean Value (i(x0))

230

(1)

Shot Noise Case

231

(1)

Second Moment (i2(x0))

231

(1)

Variance σ2

232

(1)

Shot Noise Case

232

(1)

Signal-to-Noise (S/N) Ratio

233

(1)

Autocorrelation Function

234

(1)

Shot Noise Case

235

(1)

Application: An Overlapping Circular Grain Model for the Emulsion

236

(1)

Application: Light Fluctuations due to Randomly Tilted Waves, the ``Swimming Pool'' Effect

237

(6)

Introduction to Statistical Methods: Estimating the Mean, Median, Variance, S/N, and Simple Probability

243

(34)

Estimating a Mean from a Finite Sample

244

(1)

Statistical Model

244

(1)

Analysis

245

(1)

Discussion

246

(1)

Error in a Discrete, Linear Processor: Why Linear Methods Often Fail

246

(2)

Estimating a Probability: Derivation of the Law of Large Numbers

248

(1)

Variance of Error

249

(1)

Illustrative Uses of the Error Expression

250

(2)

Estimating Probabilities from Empirical Rates

250

(1)

Aperture Size for Required Accuracy in Transmittance Readings

251

(1)

Probability Law for the Estimated Probability; Confidence Limits

252

(1)

Calculation of the Sample Variance

253

(5)

Unbiased Estimate of the Variance

253

(2)

Expected Error in the Sample Variance

255

(1)

Illustrative Problems

256

(2)

Estimating the Signal-to-Noise Ratio; Student's Probability Law

258

(3)

Probability Law for SNR

258

(1)

Moments of SNR

259

(2)

Limit c → 0; A Student Probability Law

261

(1)

Properties of a Median Window

261

(2)

Statistics of the Median

263

(6)

Probability Law for the Median

264

(1)

Laser Speckle Case: Exponential Probability Law

264

(5)

Dominance of the Cauchy Law in Diffraction

269

(8)

Estimating an Optical Slit Position: An Optical Central Limit Theorem

270

(1)

Analysis by Characteristic Function

270

(2)

Cauchy Limit, Showing Independence to Aberrations

272

(1)

Widening the Scope of the Optical Central Limit Theorem

273

(4)

Introduction to Estimating Probability Laws

277

(30)

Estimating Probability Densities Using Orthogonal Expansions

278

(3)

Karhunen-Loeve Expansion

281

(1)

The Multinomial Probability Law

282

(1)

Derivation

282

(1)

Illustrative Example

283

(1)

Estimating an Empirical Occurrence Law as the Maximum Probable Answer

283

(24)

Principle of Maximum Probability (MP)

284

(1)

Maximum Entropy Estimate

285

(1)

The Search for ``Maximum Prior Ignorance''

286

(1)

Other Types of Estimates (Summary)

287

(1)

Return to Maximum Entropy Estimation, Discrete Case

288

(1)

Transition to a Continuous Random Variable

289

(1)

Solution

290

(1)

Maximized H

290

(1)

Illustrative Example: Significance of the Normal Law

290

(1)

The Smoothness Property; Least Biased Aspect

291

(1)

A Well Known Distribution Derived

292

(1)

When Does the Maximum Entropy Estimate Equal the True Law?

293

(1)

Maximum Probable Estimates of Optical Objects

294

(2)

Case of Nearly Featureless Objects

296

(11)

The Chi-Square Test of Significance

307

(14)

Forming the χ2 Statistic

308

(1)

Probability Law for χ2 Statistic

309

(2)

When is a Coin Fixed?

311

(1)

Equivalence of Chi-Square to Other Statistics; Sufficient Statistics

312

(1)

When Is a Vote Decisive?

313

(1)

Generalization to N Voters

314

(1)

Use as an Image Detector

315

(6)

The Student t-Test on the Mean

321

(12)

Cases Where Data Accuracy is Unknown

322

(1)

Philosophy of the Approach: Statistical Inference

322

(1)

Forming the Statistic

323

(2)

Student's t-Distribution: Derivation

325

(1)

Some Properties of Student's t-Distribution

326

(1)

Application to the Problem: Student's t-Test

327

(1)

Illustrative Example

327

(2)

Other Applications

329

(4)

The F-Test on Variance

333

(8)

Snedecor's F-Distribution; Derivation

333

(1)

Some Properties of Snedecor's F-Distribution

334

(1)

The F-Test

335

(1)

Illustrative Example

336

(1)

Application to Image Detection

336

(5)

Least-Squares Curve Fitting-Regression Analysis

341

(22)

Summation Model for the Physical Effect

341

(2)

Linear Regression Model for the Noise

343

(2)

Equivalence of ML and Least-Squares Solutions

345

(1)

Solution

346

(1)

Return to Film Problem

347

(1)

``Significant'' Factors; The R-Statistic

347

(2)

Example: Was T2 an Insignificant Factor?

349

(1)

Accuracy of the Estimated Coefficients

350

(13)

Absorptance of an Optical Fiber

350

(1)

Variance of Error in the General Case

351

(2)

Error in the Estimated Absorptance of an Optical Fiber

353

(10)

Principal Components Analysis

363

(12)

A Photographic Problem

363

(1)

Equivalent Eigenvalue Problem

364

(2)

The Eigenvalues as Sample Variances

366

(1)

The Data in Terms of Principal Components

366

(1)

Reduction in Data Dimensionality

367

(1)

Return to the H-D Problem

368

(1)

Application to Multispectral Imagery

368

(3)

Error Analysis

371

(4)

The Controversy Between Bayesians and Classicists

375

(12)

Bayesian Approach to Confidence Limits for an Estimated Probability

376

(4)

Probability Law for the Unknown Probability

377

(1)

Assumption of a Uniform Prior

377

Irrelevance of Choice of Prior Statistic p0(x) if N is Large

376

(3)

Limiting Form for N Large

379

(1)

Illustrative Problem

379

(1)

Laplace's Rule of Succession

380

(7)

Derivation

380

(3)

Role of the Prior

383

(1)

Bull Market, Bear Market

384

(3)

Introduction to Estimation Methods

387

(64)

Deterministic Parameters: Likelihood Theory

388

(10)

Unbiased Estimators

388

(1)

Maximum Likelihood Estimators

389

(2)

Cramer-Rao Lower Bound on Error

391

(2)

Achieving the Lower Bound

393

(1)

Testing for Efficient Estimators

394

(1)

Can a Bound to the Error be Known if an Efficient Estimator Does Not Exist?

395

(2)

When can the Bhattacharyya Bound be Achieved?

397

(1)

Random Parameters: Bayesian Estimation Theory

398

(15)

Cost Functions

399

(1)

Risk

400

(5)

MAP Estimates

405

(8)

Exact Estimates of Probability Laws: The Principle of Extreme Physical Information

413

(38)

A Knowledge-Based View of Nature

415

(1)

Fisher Information as a Bridge Between Noumenon and Phenomenon

416

(3)

No Absolute Origins

419

(1)

Invariance of the Fisher Information Length to Unitary Transformation

420

(1)

Multidimensional Form of I

421

(1)

Lorentz Transformation of Special Relativity

422

(1)

What Constants Should be Regarded as Universal Physical Constants?

423

(1)

Transition to Complex Probability Amplitudes

424

(1)

Space-Time Measurement: Information Capacity in Fourier Space

424

(1)

Relation Among Energy, Mass and Momentum

425

(1)

Ultimate Resolution

426

(1)

Bound Information J and Efficiency Constant k

426

(1)

Perturbing Effect of the Probe Particle

427

(1)

Equality of the Perturbed Informations

428

(3)

EPI Variational Principle, and Framework

431

(2)

The Measurement Process in Detail

433

(2)

Euler-Lagrange Solutions

435

(1)

Free-Field Klein-Gordon Equation

435

(1)

Dirac Equations

436

(1)

Schroedinger Wave Equation (SWE)

437

(1)

Dimensionality, and Plato's Cave

438

(1)

Wheeler's ``Participatory Universe''

439

(1)

Exhaustivity Property, and Future Research

439

(1)

Can EPI be Used in a Design Mode?

440

(1)

EPI as a Knowledge Acquisition Game

440

(1)

Ultimate Uses of EPI

441

(10)

Appendix

451

(18)

Appendix A

451

(2)

Appendix B

453

(2)

Appendix C

455

(1)

Appendix D

456

(3)

Appendix E

459

(1)

Appendix F

460

(4)

Appendix G

464

(5)

References

469

(10)

Index

479

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Probability, Statistical Optics, and Data Testing

Summary

Table of Contents

Supplemental Materials