OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Vol. 6, Iss. 1 — Jan. 1, 1989
  • pp: 80–91

Multiply stochastic representations for K distributions and their Poisson transforms

Malvin C. Teich and Paul Diament  »View Author Affiliations


JOSA A, Vol. 6, Issue 1, pp. 80-91 (1989)
http://dx.doi.org/10.1364/JOSAA.6.000080


View Full Text Article

Enhanced HTML    Acrobat PDF (1434 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

The K distribution is used in a number of areas of scientific endeavor. In optics, it provides a useful statistical description for fluctuations of the irradiance (and the electric field) of light that has been scattered or transmitted through random media (e.g., the turbulent atmosphere). The Poisson transform of the K distribution describes the photon-counting statistics of light whose irradiance is K distributed. The K-distribution family can be represented in a multiply stochastic (compound) form whereby the mean of a gamma distribution is itself stochastic and is described by a member of the gamma family of distributions. Similarly, the family of Poisson transforms of the K distributions can be represented as a family of negative-binomial transforms of the gamma distributions or as Whittaker distributions. The K distributions have heretofore had their origins in random-walk models; the multiply stochastic representations provide an alternative interpretation of the genesis of these distributions and their Poisson transforms. By multiple compounding, we have developed a new transform pair as a possibly useful addition to the K-distribution family. All these distributions decay slowly and are difficult to calculate accurately by conventional formulas. A recursion relation, together with a generalized method of steepest descent, has been developed to evaluate numerically the photon-counting distributions and their factorial moments with excellent accuracy.

© 1989 Optical Society of America

History
Original Manuscript: March 10, 1988
Manuscript Accepted: July 25, 1988
Published: January 1, 1989

Citation
Malvin C. Teich and Paul Diament, "Multiply stochastic representations for K distributions and their Poisson transforms," J. Opt. Soc. Am. A 6, 80-91 (1989)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-6-1-80


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. S. R. Broadbent, D. G. Kendall, “The random walk of Trichostrongylus retortaeformis,” Biometrics 9, 460–466 (1953). [CrossRef]
  2. E. J. Williams, “The distribution of larvae of randomly moving insects,” Austral. J. Biol. Sci. 14, 598–604 (1961). Williams’s calculation is virtually identical to that of Broadbent and Kendall.1
  3. N. Yasuda, “The random walk model of human migration,” Theor. Popul. Biol. 7, 156–167 (1975). [CrossRef] [PubMed]
  4. G. Malécot, “Identical loci and relationship,” in Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability; Biology and Problems of Health, L. LeCam, J. Neyman, eds. (U. California Press, Berkeley, Calif., 1967), Vol. 4, pp. 317–332.
  5. M. Bertolotti, B. Crosignani, P. DiPorto, “On the statistics of Gaussian light scattered by a Gaussian medium,”J. Phys. A 3, L37–L38 (1970). [CrossRef]
  6. P. N. Pusey, “Statistical properties of scattered radiation,” in Photon Correlation Spectroscopy and Velocimetry, H. Z. Cummins, E. R. Pike, eds. (Plenum, New York, 1976), pp. 45–141.
  7. G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “Focussing by a random phase screen,” Opt. Commun. 22, 195–201 (1977). [CrossRef]
  8. E. Jakeman, P. N. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978). [CrossRef]
  9. G. Parry, P. N. Pusey, “K distributions in atmospheric propagation of laser light,”J. Opt. Soc. Am. 69, 796–798 (1979). [CrossRef]
  10. E. Jakeman, “On the statistics of K-distributed noise,”J. Phys. A 13, 31–48 (1980). [CrossRef]
  11. E. Jakeman, “Speckle statistics with a small number of scatterers,” in Applications of Speckle Phenomena, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.243, 9–19 (1980). [CrossRef]
  12. K. A. O’Donnell, “Speckle statistics of doubly scattered light,”J. Opt. Soc. Am. 72, 1459–1463 (1982). [CrossRef]
  13. E. Jakeman, P. N. Pusey, “A model for non-Rayleigh sea echo,”IEEE Trans. Antennas Propag. AP-24, 806–814 (1976). [CrossRef]
  14. D. J. Lewinski, “Nonstationary probabilistic target and clutter scattering models,”IEEE Trans. Antennas Propag. AP-31, 490–498 (1983). [CrossRef]
  15. S. Watts, K. D. Ward, “Spatial correlation in K-distributed sea clutter,” Proc. Inst. Electr. Eng. Part F 134, 526–532 (1987).
  16. L. C. Andrews, R. L. Phillips, “I–K distribution as a universal propagation model of laser beams in atmospheric turbulence,” J. Opt. Soc. Am. A 2, 160–163 (1985). [CrossRef]
  17. L. C. Andrews, R. L. Phillips, “Mathematical genesis of the I–K distribution for random optical fields,” J. Opt. Soc. Am. A 3, 1912–1919 (1986). [CrossRef]
  18. R. Barakat, “Weak-scatterer generalization of the K-density function with application to laser scattering in atmospheric turbulence,” J. Opt. Soc. Am. A 3, 401–409 (1986). [CrossRef]
  19. E. Jakeman, R. J. A. Tough, “Generalized K distribution: a statistical model for weak scattering,” J. Opt. Soc. Am. A 4, 1764–1772 (1987). [CrossRef]
  20. E. B. Rockower, “Quantum derivation of K-distributed noise for finite 〈N〉,” J. Opt. Soc. Am. A 5, 730–734 (1988). [CrossRef]
  21. G. Parry, “Measurement of atmospheric turbulence induced intensity fluctuations in a laser beam,” Opt. Acta 28, 715–728 (1981). [CrossRef]
  22. A. Consortini, R. J. Hill, “Reduction of the moments of intensity fluctuations caused by amplifier saturation for both the K and the log-normally modulated exponential probability densities,” Opt. Lett. 12, 304–306 (1987). [CrossRef] [PubMed]
  23. K. D. Ward, “Compound representation of high resolution sea clutter,” Electron. Lett. 17, 561–563 (1981). [CrossRef]
  24. R. J. A. Tough, “Fokker–Planck description of K-distributed noise,”J. Phys. A 20, 551–567 (1987). [CrossRef]
  25. C. J. Oliver, R. J. A. Tough, “On the simulation of correlated K-distributed random clutter,” Opt. Acta 33, 223–250 (1986). [CrossRef]
  26. E. Conte, M. Longo, “Characterisation of radar clutter as a spherically invariant random process,” Proc. Inst. Electr. Eng. Part F 134, 191–197 (1987).
  27. B. E. A. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978). [CrossRef]
  28. E. Jakeman, P. N. Pusey, “Photon-counting statistics of optical scintillation,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed., Vol. 20 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 75–116. [CrossRef]
  29. Y. M. Lure, M. Gao, C. C. Yang, “Probability distribution for the photocount associated with a K distribution for laser intensity,” J. Opt. Soc. Am. A 4(13), P84 (1987).
  30. M. Greenwood, G. U. Yule, “An inquiry into the nature of frequency distributions representative of multiple happenings with particular reference to the occurrence of multiple attacks of disease or of repeated accidents,”J. R. Stat. Soc. A 83, 255–279 (1920). [CrossRef]
  31. J. Gurland, “Some interrelations among compound and generalized distributions,” Biometrika 44, 265–268 (1957).
  32. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions, Vol. 55 of National Bureau of Standards Applied Mathematics Series (U.S. Government Printing Office, Washington, D.C., 1964).
  33. R. G. Laha, “On some properties of the Bessel function distributions,” Bull. Calcutta Math. Soc. 46, 59–72 (1954).
  34. J. Peřina, “Superposition of coherent and incoherent fields,” Phys. Lett. A 24, 333–334 (1967); “Superposition of thermal and coherent fields,” Acta Univ. Palack. Olomuc. Fac. Rerum Nat. 27, 227–234 (1968); G. Lachs, “Quantum statistics of multiple-mode, superposed coherent and chaotic radiation,” J. Appl. Phys. 38, 3439–3448 (1967). [CrossRef]
  35. W. J. McGill, “Neural counting mechanisms and energy detection in audition,”J. Math. Psychol. 4, 351–376 (1967). [CrossRef]
  36. M. C. Teich, W. J. McGill, “Neural counting and photon counting in the presence of dead time,” Phys. Rev. Lett. 36, 754–758 (1976). [CrossRef]
  37. S. H. Ong, P. A. Lee, “The non-central negative binomial distribution,” Biom. J. 21, 611–627 (1979). [CrossRef]
  38. E. T. Whittaker, G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge U. Press, Cambridge, 1962).
  39. M. S. Bartlett, An Introduction to Stochastic Processes, 3rd ed. (Cambridge U. Press, Cambridge, 1978).
  40. W. J. McGill, M. C. Teich, “Signal discrimination in an amplifying auditory transmission system,” in Quantitative Analyses of Behavior, Vol. 10 of Signal Detection, M. L. Commons, J. A. Nevins, eds. (Erlbaum, Hillsdale, N.J., to be published).
  41. J. Gurland, “A generalized class of contagious distributions,” Biometrics 14, 229–249 (1958). [CrossRef]
  42. M. T. Boswell, G. P. Patil, “Chance mechanisms generating the negative binomial distributions,” in Random Counts in Models and Structures, Vol. 1 of Random Counts in Scientific Work, G. P. Patil, ed., Penn State Statistics Series (Pennsylvania State U. Press, University Park, Pa., 1970), pp. 3–27.

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited