Springer Information Sciences

A Problem Solving Approach

By: Roy Frieden

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Paperback | 17 July 2001 | Edition Number 3

At a Glance

Paperback
524 Pages

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Revised

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23.5 x 14.61 x 1.91

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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.

Industry Reviews

From the reviews of the third edition:

"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 solutions. This textbook develops these subjects within the context of optics using a problem-solving approach. ... The book is exclusively wealthy in contents. The author generously shares his reflections and assumptions with the reader and puts the unsolved problems yet, what makes this book an especially interesting one." (Dmitry Ostrouchov, Zentralblatt MATH, Vol. 978, 2002)

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Introduction	p. 1
What Is Chance, and Why Study It?	p. 3
Chance vs. Determinism	p. 3
Probability Problems in Optics	p. 4
Statistical Problems in Optics	p. 5
Historical Notes	p. 5
The Axiomatic Approach	p. 7
Notion of an Experiment;Events	p. 7
Event Space; The Space Event	p. 8
Disjoint Events	p. 8
The Certain Event	p. 9
Definition of Probability	p. 9
Relation to Frequency of Occurrence	p. 10
Some Elementary Consequences	p. 10
Additivity Property	p. 11
Normalization Property	p. 11
Marginal Probability	p. 12
The "Traditional" Definition of Probability	p. 12
Illustrative Problem: A Dice Game	p. 13
Illustrative Problem: Let's (Try to) Take a Trip	p. 14
Law of Large Numbers	p. 15
Optical Objects and Images as Probability Laws	p. 15
Conditional Probability	p. 17
The Quantity of Information	p. 18
Statistical Independence	p. 20
Illustrative Problem: Let's (Try to) Take a Trip (Continued)	p. 21
Informationless Messages	p. 22
A Definition of Noise	p. 22
"Additivity" Property of Information	p. 23
Partition Law	p. 23
Illustrative Problem: Transmittance Through a Film	p. 24
How to Correct a Success Rate for Guesses	p. 25
Bayes' Rule	p. 26
Some Optical Applications	p. 27
Information Theory Application	p. 28
Applicationto Markov Events	p. 28
Complex Number Events	p. 29
What is the Probability of Winning a Lottery Jackpot?	p. 30
Whatis the Probability of a Coincidence of Birthdaysata Party?	p. 31
Continuous Random Variables	p. 39
Definition of a Random Variable	p. 39
Probability Density Function, Basic Properties	p. 39
Information Theory Application: Continuous Limit	p. 41
Optical Application: Continuous Form of Imaging Law	p. 411
Expected Values,Moments	p. 42
Optical Application: Moments of the Slit Diffraction Pattern	p. 43
Information Theory Application	p. 44
Case of Statistical Independence	p. 45
Mean of a Sum	p. 45
Optical Application	p. 46
Deterministic Limit; Representations of the Dirac 8-Function	p. 47
Correspondence Between Discrete and Continuous Cases	p. 48
Cumulative Probability	p. 48
The Means of an Algebraic Expression: A Simplified Approach	p. 49
A Potpourri of Probability Laws	p. 50
Poisson	p. 50
Binomial	p. 51
Uniform	p. 51
Exponential	p. 52
Normal(One-Dimensional)	p. 53
Normal(Two-Dimensional)	p. 53
Normal(Multi-Dimensional)	p. 55
Skewed Gaussian Case; Gram-Charlier Expansion	p. 56
Optical Application	p. 56
Geometric Law	p. 58
Cauchy Law	p. 58
sinc2 Law	p. 58
Derivation of Heisenberg Uncertainty Principle	p. 68
Schwarz Inequality for Complex Functions	p. 68
Fourier Relations	p. 68
Uncertainty Product	p. 69
Hirschman's Form of the Uncertainty Principle	p. 70
Measures of Information	p. 70
Kullback-Leibler Information	p. 70
Renyi Information	p. 71
Wootters Information	p. 71
Hellinger Information	p. 72
Tsallis Information	p. 72
Fisher Information	p. 72
Fisher Information Matrix	p. 73
Fourier Methods in Probability	p. 79
Characteristic Function Defined	p. 79
Usein Generating Moments	p. 80
An Alternative to Describing RV x	p. 80
On Optical Applications	p. 80
Shift Theorem	p. 81
Poisson Case	p. 81
Binomial Case	p. 82
Uniform Case	p. 82
Exponential Case	p. 82
Normal Case (One Dimension)	p. 83
Multidimensional Cases	p. 83
Normal Case(Two Dimensions)	p. 83
Convolution Theorem,Transfer Theorem	p. 83
Probability Lawforthe Sumof Two Independent RV's	p. 84
Optical Applications	p. 85
Imaging Equationas the Sum of Two Random Displacements	p. 85
Unsharp Masking	p. 85
Sum of n Independent RV's; The "Random Walk" Phenomenon	p. 87
Resulting Mean and Variance: Normal,Poisson, and General Cases	p. 89
Sum of n Dependent RV's	p. 89
Case of Two Gaussian Bivariate RV's	p. 90
Sampling Theorems for Probability	p. 91
Case of Limited Range of x, Derivation	p. 91
Discussion	p. 92
Optical Application	p. 93
Case of Limited Range of	p. 94
Central Limit Theorem	p. 94
Derivation	p. 95
How Large Does n Have To Be?	p. 97
Optical Applications	p. 97
Cascaded Electro-Optical Systems	p. 97
Laser Resonator	p. 98
Atmospheric Turbulence	p. 99
Generating Normally Distributed Numbers from Uniformly Random Numbers	p. 100
The Error Function	p. 102
Functions of Random Variables	p. 107
Caseofa Single Random Variable	p. 107
Unique Root	p. 108
Applicationfrom Geometrical Optics	p. 109
Multiple Roots	p. 110
Illustrative Example	p. 111
Case of n Random Variables, r Roots	p. 111
Optical Applications	p. 112
Statistical Modeling	p. 112
Application of Transformation Theory to Laser Speckle	p. 113
Physical Layout	p. 113
Plan	p. 114
Statistical Model	p. 114
Marginal Probabilities for Light Amplitudes Ure, Uim	p. 115
Correlation Between Ure and Uim	p. 116
Joint Probability Law for Ure, Uim	p. 117
Probability Laws for Intensityand Phase; Transformation of the RV's	p. 117
Marginal Laws for Intensity and Phase	p. 118
Signal-to-Noise (S/N) Ratio in the Speckle Image118
Speckle Reduction by Use of a Scanning Aperture	p. 119
Statistical Model	p. 119
Probability Density for Output Intensity pI(v)120
Moments and S/N Ratio	p. 121
Standard Formforthe Chi-Square Distribution .122
Calculation of Spot Intensity Profiles Using Transformation Theory	p. 123
Illustrative Example	p. 124
Implementation by Ray-Trace	p. 125
Application of Transformation Theory to a Satellite-Ground Communication Problem	p. 126
Unequal Numbers of Input and Output Variables: "Helper Variables"	p. 140
Probability Law for a Quotient of Random Variables	p. 140
Probability Lawfora Product of Independent Random Variables	p. 141
More Complicated Transformation Problems	p. 142
Use of an Invariance Principle to Find a Probability Law	p. 142
Probability Law for Transformation of a Discrete Random Variable	p. 144
Bernoulli Trials and Limiting Cases	p. 147
Analysis	p. 147
Illustrative Problems	p. 149
Illustrative Problem: Let's (Try to) Take a Trip: The Last Word	p. 149
Illustrative Problem: Mental Telepathy asa Communication Link	p. 150
Characteristic Functionand Moments	p. 152
Optical Application: Checkerboard Model of Granularity	p. 152
The Poisson Limit	p. 154
Analysis	p. 154
Example of Degree of Approximation	p. 155
Normal Limit of Poisson Law	p. 156
Optical Application:The Shot Effect	p. 157
Optical Application:Combined Sources	p. 158
Poisson Joint Count for Two Detectors - Intensity Interferometry	p. 158
The Normal Limit (De Moivre-Laplace Law)	p. 162
Derivation	p. 162
Conditions of Use	p. 163
Use of the Error Function	p. 164
The Monte Carlo Calculation	p. 175
Producing Random Numbers That Obey a Prescribed Probability Law	p. 176
Illustrative Case	p. 177
Normal Case	p. 177
Analysis of the Photographic Emulsion by Monte Carlo Calculation	p. 178
Application of the Monte Carlo Calculation to Remote Sensing	p. 180
Monte Carlo Formationof Optical Images	p. 181
An Example	p. 182
Monte Carlo Simulationof Speckle Patterns	p. 183
Stochastic Processes	p. 191
Definition of a Stochastic Process	p. 191
Definition of Power Spectrum	p. 192
Some Examplesof Power Spectra	p. 194
Definition of Autocorrelation Function; Kinds of Stationarity	p. 194
Fourier Transform Theorem	p. 195
Case of a "White" Power Spectrum	p. 196
Application: Average Transfer Function Through Atmospheric Turbulence	p. 197
Statistical Model for Phase Fluctuations	p. 198
ATransfer function for Turbulence	p. 199
Transfer Theorems for Power Spectra	p. 201
Determiningthe MTFUsing Random Objects	p. 201
Speckle Interferometry of Labeyrie	p. 202
Resolution Limitsof Speckle Interferometry	p. 203
Transfer Theoremfor Autocorrelation: The Knox-Thompson Method	p. 208
Additive Noise	p. 211
Random Noise	p. 212
Ergodic Property	p. 213
Optimum Restoring Filter	p. 217
Definition of Restoring Filter	p. 217
Model	p. 218
Solution	p. 219
Information Content in the Optical Image	p. 221
Statistical Model	p. 222
Analysis	p. 223
Noise Entropy	p. 223
Data Entropy	p. 224
The Answer	p. 225
Interpretation	p. 225
Data Informationand Its Ability tobe Restored	p. 226
Superposition Processes; the Shot Noise Process	p. 227
Probability Law for i	p. 229
Some Important Averages	p. 229
Mean Value < i(x0) >	p. 230
Shot Noise Case	p. 231
Second Moment < i2(x0) >	p. 231
Variance a2fe)	p. 232
Shot Noise Case	p. 232
Signal-to-Noise (S/N) Ratio	p. 233
Autocorrelation Function	p. 234
Shot Noise Case	p. 235
Application: An Overlapping Circular Grain Model forthe Emulsion	p. 236
Application: Light Fluctuations due to Randomly Tilted Waves, the "Swimming Pool" Effect	p. 237
Introduction to Statistical Methods: Estimating the Mean, Median, Variance, S/N, and Simple Probability	p. 243
Estimatinga Meanfroma Finite Sample	p. 244
Statistical Model	p. 244
Analysis	p. 245
Discussion	p. 246
Error in a Discrete, Linear Processor: Why Linear Methods Often Fail	p. 246
Estimating a Probability: Derivation of the Law of Large Numbers	p. 248
Variance of Error	p. 249
Illustrative Uses of the Error Expression	p. 250
Estimating Probabilities from Empirical Rates	p. 250
Aperture Size for Required Accuracy in Transmittance Readings	p. 251
Probability Law for the Estimated Probability; Confidence Limits	p. 252
Calculation of the Sample Variance	p. 253
Unbiased Estimate of the Variance	p. 253
Expected Error in the Sample Variance	p. 255
Illustrative Problems	p. 256
Estimating the Signal-to-Noise Ratio; Student's Probability Law	p. 258
Probability Law for SNR	p. 258
Moments of SNR	p. 259
Limit c -> 0; A Student Probability Law	p. 261
Properties of a Median Window	p. 261
Statistics of the Median	p. 263
Probability Law for the Median	p. 264
Laser Speckle Case: Exponential Probability Law	p. 264
Dominance of the Cauchy Law in Diffraction	p. 269
Estimating an Optical Slit Position: An Optical Central Limit Theorem	p. 270
Analysis by Characteristic Function	p. 270
Cauchy Limit, Showing Independence to Aberrations	p. 272
Widening the Scope of the Optical Central Limit Theorem. 273
Introduction to Estimating Probability Laws	p. 277
Estimating Probability Densities Using Orthogonal Expansions	p. 278
Table of Contents provided by Publisher. All Rights Reserved.

Springer Information Sciences

A Problem Solving Approach

At a Glance

Paperback

Industry Reviews

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