OPTI 508

7/06

OPTI 508. Probability and Statistics in Optics (3) II. Probability theory; stochastic processes; noise; statistical optics; information theory; hypothesis testing; estimation; restoration. P, Opti 501.

Course Outline (75-minute lectures)

Foundations of Probability Theory:
  1. Concepts of randomness and probability; Kolmogoroff's axioms;
    additivity, normalization.
  2. Marginal probability; conditional probability; Laplace method of computing probabilities.
  3. Bayes' theorem; law of total probability; independent events.
  4. Markov events; complex events; frequency of occurrence; law of large numbers.
  5. Def. of communication channel; Shannon information in a message; additivity.
  6. Probability density function (p.d.f.), distribution function; expectation, moments.
  7. Special p.d.f.'s (normal, Poisson, binomial, etc.)
  8. Characteristic function; moment generation; p.d.f. for sum of random variables; random walk.
  9. Central limit theorem, normal p.d.f.; derivation, restrictions; error function; examples: atmospheric turbulence, cascaded MTF's.
  10. Function of a random variable; single root-, multiple root-cases; functions of random variables.
  11. Continuation from #10. Application to laser speckle (exponential) p.d.f. for intensity; Chi-square p.d.f. for aperture-averaged speckle.
  12. Bernoulli trials sequence; binomial probability law; photographic
    application, film grain noise.
  13. Bernoulli trials continued: Poisson limit (Shot effect); normal limit(DeMoivre-Laplace law).
  14. Monte Carlo calculation; basis, limitations; application to film grain noise.
Stochastic Processes:
  1. Def. stochastic process; ensemble average; power spectrum.
  2. Autocorrelation function; power spectrum; Fourier transform theorem.
  3. Stationarity; linear filtering of process; matched filter.
  4. Derivation of OTF for long-term atmospheric turbulence.
  5. White noise, additive noise, random noise, thermal noise; ergodicity.
  6. Wiener restoring/smoothing filter; information in an image.
  7. Mandel's formula for photoelectron occurrences; optical shot noise process.

Hypothesis Testing and Estimation:

  1. Unbiased estimate; estimating the mean from a data sample; rms error in estimate.
  2. Estimating median and single probability from a sample; rms errors.
  3. Estimating a p.d.f. using maximum entropy; application to image
    restoration.
  4. Determining the significance of data; Chi-square test.
  5. Student t-test on the mean; confidence levels.
  6. Regression analysis; least-square curve fitting; significant factors; R-statistic; example: absorptance of optical fiber.
  7. Maximum likelihood estimates; Cramer-Rao inequality; Fisher
    information.
  8. Bayesian binary decision; risk; cost function matrix; likelihood ratio; receiver operating characteristics.
  9. Bayesian estimation; role of prior p.d.f.; cost, risk examples.
  10. MAP, mmse, median estimators; application to gamma camera.

Grading Criteria:

  • Equal weights are given to five inputs:
    1. a first-quarter exam (closed book)
    2. a mid-term exam (closed book)
    3. a final exam (open summary sheet)
    4. homeworks
    5. term project – a Monte Carlo computer calculation.

Required Text:

  • B.R. Frieden: Probability, Statistical Optics, and Data Testing, 3rd ed. (Springer Verlag, 2001)

 Supplementary Reading on reserve in the OSC Library:

  • J.W. Goodman: Statistical Optics (Wiley)
  • A. Papoulis: Probability, Random Variables and Stochastic Processes (McGraw-Hill)