OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 21, Iss. 11 — Nov. 1, 2004
  • pp: 2124–2134

Entropy of partially polarized light and application to statistical processing techniques

Philippe Réfrégier, François Goudail, Pierre Chavel, and Ari Friberg  »View Author Affiliations

JOSA A, Vol. 21, Issue 11, pp. 2124-2134 (2004)

View Full Text Article

Enhanced HTML    Acrobat PDF (399 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



We have analyzed entropy properties of coherent and partially polarized light in an arbitrary number of spatial dimensions. We show that for Gaussian fields, the Shannon entropy is a simple function of the intensity and of the Barakat degree of polarization. In particular, we provide a probabilistic interpretation of this definition of the degree of polarization. Using information theory results, we also deduce some physical properties of partially polarized light such as additivity of the entropy and depolarization effects induced by mixing partially polarized states of light. Finally, we demonstrate that entropy measures can play an important role in segmentation and detection tasks.

© 2004 Optical Society of America

OCIS Codes
(030.0030) Coherence and statistical optics : Coherence and statistical optics
(030.4280) Coherence and statistical optics : Noise in imaging systems
(260.5430) Physical optics : Polarization

Original Manuscript: February 25, 2004
Revised Manuscript: June 10, 2004
Manuscript Accepted: June 10, 2004
Published: November 1, 2004

Philippe Réfrégier, François Goudail, Pierre Chavel, and Ari Friberg, "Entropy of partially polarized light and application to statistical processing techniques," J. Opt. Soc. Am. A 21, 2124-2134 (2004)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. M. Floc’h, G. Le Brun, C. Kieleck, J. Cariou, J. Lotrian, “Polarimetric considerations to optimize lidar detection of immersed targets,” Pure Appl. Opt. 7, 1327–1340 (1998). [CrossRef]
  2. S. Breugnot, Ph. Clémenceau, “Modeling and performances of a polarization active imager at lambda=806 nm,” in Laser Radar Technology and Applications IV, G. W. Kamerman, C. Werner, eds., Proc. SPIE3707, 449–460 (1999). [CrossRef]
  3. A. Gleckler, A. Gelbart, “Multiple-slit streak tube imaging lidar MS-STIL applications,” in Laser Radar Technology and Applications V, G. W. Kamerman, U. N. Singh, C. H. Werner, V. V. Molebny, eds., Proc. SPIE4035, 266–278 (2000). [CrossRef]
  4. L. B. Wolff, “Polarization camera for computer vision with a beam splitter,” J. Opt. Soc. Am. A 11, 2935–2945 (1994). [CrossRef]
  5. J. E. Solomon, “Polarization imaging,” Appl. Opt. 20, 1537–1544 (1981). [CrossRef] [PubMed]
  6. W. G. Egan, W. R. Johnson, V. S. Whitehead, “Terrestrial polarization imagery obtained from the space shuttle: characterization and interpretation,” Appl. Opt. 30, 435–442 (1991). [CrossRef] [PubMed]
  7. J. L. Pezzaniti, R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. 34, 1558–1568 (1995). [CrossRef]
  8. J. S. Tyo, M. P. Rowe, E. N. Pugh, N. Engheta, “Target detection in optical scattering media by polarization-difference imaging,” Appl. Opt. 35, 1855–1870 (1996). [CrossRef] [PubMed]
  9. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 116–156.
  10. C. Brosseau, Fundamentals of Polarized Light–A Statistical Approach (Wiley, New York, 1998), pp. 138–164.
  11. Ref. 9, pp. 237–285.
  12. J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, Heidelberg, Germany, 1975).
  13. T. Setälä, M. Kaivola, A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88, 123902 (2002). [CrossRef] [PubMed]
  14. R. S. Cloude, E. Pottier, “Concept of polarization entropy in optical scattering,” Opt. Eng. 34, 1599–1610 (1995). [CrossRef]
  15. T. M. Cover, J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991), pp. 12–49.
  16. J. C. Samson, “Descriptions of the polarization states of vector processes: applications to ULF magnetic fields,” Geophys. J. R. Astron. Soc. 34, 403–419 (1973). [CrossRef]
  17. R. Barakat, “N-fold polarization measures and associated thermodynamic entropy of N partially coherent pencils of radiation,” Opt. Acta 30, 1171–1182 (1983). [CrossRef]
  18. Ref. 10, pp. 165–175.
  19. M. D. Esteban, D. A. Morales, “A summary of entropy statistics,” Kybernetica 31, 337–346 (1995).
  20. R. Baraniuk, P. Flandrin, O. Michel, “Measuring time frequency information content using the Renyi entropies,” IEEE Trans. Inf. Theory 47, 1391–1409 (2001). [CrossRef]
  21. Ref. 15, pp. 279–335.
  22. C. W. Therrien, Decision Estimation and Classification (Wiley, New York, 1989), pp. 139–155.
  23. C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948). [CrossRef]
  24. Ref. 15, pp. 266–278.
  25. A. Firooz, A. Sadjadi, “Passive infrared automatic target recognition,” in Image Recognition and Classification: Algorithm, System and Applications, B. Javidi, ed., (Marcel Dekker, New York, 2002), pp. 37–60.
  26. A. F. Sadjadi, C. S. L. Chun, “Automatic detection of small objects from their infrared state-of-polarization vectors,” Opt. Lett. 28, 531–533 (2003). [CrossRef] [PubMed]
  27. R. J. Muirhead, Aspects of Multivariate Statistical Theory (Wiley, New York, 1982).
  28. T. S. Ferguson, Mathematical Statistics, a Decision Theoretic Approach (Academic, New York, 1967), pp. 112–119.
  29. J. Rissanen, Stochastic Complexity in Statistical Inquiry (World Scientific, Singapore, 1989).
  30. O. Ruch, Ph. Réfrégier, “Minimal-complexity segmentation with a polygonal snake adapted to different optical noise models,” Opt. Lett. 41, 977–979 (2001). [CrossRef]
  31. S. C. Zhu, A. Yuille, “Region competition: unifying snakes, region growing, and Bayes/MDL for multiband image segmentation,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 884–900 (1996). [CrossRef]
  32. M. Figueiredo, J. Leitão, A. K. Jain, “Unsupervised contour representation and estimation using B-splines and a minimum description length criterion,” IEEE Trans. Image Process. 9, 1075–1087 (2000). [CrossRef]
  33. C. Chesnaud, Ph. Réfrégier, V. Boulet, “Statistical region snake-based segmentation adapted to different physical noise models,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 1145–1157 (1999). [CrossRef]
  34. L. Ferro-Famil, E. Pottier, J. S. Lee, “Unsupervised classification of multifrequency and full polarimetric SAR images based on the H/A/alpha–Wishart classifier,” IEEE Trans. Geosci. Remote Sens. 39, 2332–2342 (2001). [CrossRef]
  35. F. Goudail, F. Galland, Ph. Réfrégier, “A general framework for designing image processing algorithms for coherent polarimetric images,” in Proceedings of IEEE 2003 International Conference on Image Processing (IEEE Press, Piscataway, N.J.2003), pp. 153–156.
  36. S. M. Kay, Fundamentals of Statistical Signal Processing—Volume II: Detection Theory (Prentice Hall, Upper Saddle River, N.J., 1998), pp. 186–247.

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.


Fig. 1 Fig. 2 Fig. 3
Fig. 4

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited