Abstract

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

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  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.

2003 (1)

2002 (1)

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]

2001 (3)

O. Ruch, Ph. Réfrégier, “Minimal-complexity segmentation with a polygonal snake adapted to different optical noise models,” Opt. Lett. 41, 977–979 (2001).
[CrossRef]

R. Baraniuk, P. Flandrin, O. Michel, “Measuring time frequency information content using the Renyi entropies,” IEEE Trans. Inf. Theory 47, 1391–1409 (2001).
[CrossRef]

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]

2000 (1)

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]

1999 (1)

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]

1998 (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]

1996 (2)

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]

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]

1995 (3)

M. D. Esteban, D. A. Morales, “A summary of entropy statistics,” Kybernetica 31, 337–346 (1995).

R. S. Cloude, E. Pottier, “Concept of polarization entropy in optical scattering,” Opt. Eng. 34, 1599–1610 (1995).
[CrossRef]

J. L. Pezzaniti, R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. 34, 1558–1568 (1995).
[CrossRef]

1994 (1)

1991 (1)

1983 (1)

R. Barakat, “N-fold polarization measures and associated thermodynamic entropy of N partially coherent pencils of radiation,” Opt. Acta 30, 1171–1182 (1983).
[CrossRef]

1981 (1)

1973 (1)

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]

1948 (1)

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).
[CrossRef]

Barakat, R.

R. Barakat, “N-fold polarization measures and associated thermodynamic entropy of N partially coherent pencils of radiation,” Opt. Acta 30, 1171–1182 (1983).
[CrossRef]

Baraniuk, R.

R. Baraniuk, P. Flandrin, O. Michel, “Measuring time frequency information content using the Renyi entropies,” IEEE Trans. Inf. Theory 47, 1391–1409 (2001).
[CrossRef]

Boulet, V.

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]

Breugnot, S.

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]

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light–A Statistical Approach (Wiley, New York, 1998), pp. 138–164.

Cariou, J.

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]

Chesnaud, C.

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]

Chipman, R. A.

J. L. Pezzaniti, R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. 34, 1558–1568 (1995).
[CrossRef]

Chun, C. S. L.

Clémenceau, Ph.

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]

Cloude, R. S.

R. S. Cloude, E. Pottier, “Concept of polarization entropy in optical scattering,” Opt. Eng. 34, 1599–1610 (1995).
[CrossRef]

Cover, T. M.

T. M. Cover, J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991), pp. 12–49.

Dainty, J. C.

J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, Heidelberg, Germany, 1975).

Egan, W. G.

Engheta, N.

Esteban, M. D.

M. D. Esteban, D. A. Morales, “A summary of entropy statistics,” Kybernetica 31, 337–346 (1995).

Ferguson, T. S.

T. S. Ferguson, Mathematical Statistics, a Decision Theoretic Approach (Academic, New York, 1967), pp. 112–119.

Ferro-Famil, L.

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]

Figueiredo, M.

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]

Firooz, A.

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.

Flandrin, P.

R. Baraniuk, P. Flandrin, O. Michel, “Measuring time frequency information content using the Renyi entropies,” IEEE Trans. Inf. Theory 47, 1391–1409 (2001).
[CrossRef]

Floc’h, M.

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]

Friberg, A. T.

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]

Galland, F.

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.

Gelbart, A.

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]

Gleckler, A.

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]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 116–156.

Goudail, F.

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.

Jain, A. K.

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]

Johnson, W. R.

Kaivola, M.

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]

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing—Volume II: Detection Theory (Prentice Hall, Upper Saddle River, N.J., 1998), pp. 186–247.

Kieleck, C.

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]

Le Brun, G.

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]

Lee, J. S.

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]

Leitão, J.

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]

Lotrian, J.

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]

Michel, O.

R. Baraniuk, P. Flandrin, O. Michel, “Measuring time frequency information content using the Renyi entropies,” IEEE Trans. Inf. Theory 47, 1391–1409 (2001).
[CrossRef]

Morales, D. A.

M. D. Esteban, D. A. Morales, “A summary of entropy statistics,” Kybernetica 31, 337–346 (1995).

Muirhead, R. J.

R. J. Muirhead, Aspects of Multivariate Statistical Theory (Wiley, New York, 1982).

Pezzaniti, J. L.

J. L. Pezzaniti, R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. 34, 1558–1568 (1995).
[CrossRef]

Pottier, E.

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]

R. S. Cloude, E. Pottier, “Concept of polarization entropy in optical scattering,” Opt. Eng. 34, 1599–1610 (1995).
[CrossRef]

Pugh, E. N.

Réfrégier, Ph.

O. Ruch, Ph. Réfrégier, “Minimal-complexity segmentation with a polygonal snake adapted to different optical noise models,” Opt. Lett. 41, 977–979 (2001).
[CrossRef]

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]

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.

Rissanen, J.

J. Rissanen, Stochastic Complexity in Statistical Inquiry (World Scientific, Singapore, 1989).

Rowe, M. P.

Ruch, O.

O. Ruch, Ph. Réfrégier, “Minimal-complexity segmentation with a polygonal snake adapted to different optical noise models,” Opt. Lett. 41, 977–979 (2001).
[CrossRef]

Sadjadi, A.

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.

Sadjadi, A. F.

Samson, J. C.

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]

Setälä, T.

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]

Shannon, C. E.

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).
[CrossRef]

Solomon, J. E.

Therrien, C. W.

C. W. Therrien, Decision Estimation and Classification (Wiley, New York, 1989), pp. 139–155.

Thomas, J. A.

T. M. Cover, J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991), pp. 12–49.

Tyo, J. S.

Whitehead, V. S.

Wolff, L. B.

Yuille, A.

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]

Zhu, S. C.

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]

Appl. Opt. (3)

Bell Syst. Tech. J. (1)

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).
[CrossRef]

Geophys. J. R. Astron. Soc. (1)

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]

IEEE Trans. Geosci. Remote Sens. (1)

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]

IEEE Trans. Image Process. (1)

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]

IEEE Trans. Inf. Theory (1)

R. Baraniuk, P. Flandrin, O. Michel, “Measuring time frequency information content using the Renyi entropies,” IEEE Trans. Inf. Theory 47, 1391–1409 (2001).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (2)

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]

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]

J. Opt. Soc. Am. A (1)

Kybernetica (1)

M. D. Esteban, D. A. Morales, “A summary of entropy statistics,” Kybernetica 31, 337–346 (1995).

Opt. Acta (1)

R. Barakat, “N-fold polarization measures and associated thermodynamic entropy of N partially coherent pencils of radiation,” Opt. Acta 30, 1171–1182 (1983).
[CrossRef]

Opt. Eng. (2)

R. S. Cloude, E. Pottier, “Concept of polarization entropy in optical scattering,” Opt. Eng. 34, 1599–1610 (1995).
[CrossRef]

J. L. Pezzaniti, R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. 34, 1558–1568 (1995).
[CrossRef]

Opt. Lett. (2)

O. Ruch, Ph. Réfrégier, “Minimal-complexity segmentation with a polygonal snake adapted to different optical noise models,” Opt. Lett. 41, 977–979 (2001).
[CrossRef]

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]

Phys. Rev. Lett. (1)

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]

Pure Appl. Opt. (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]

Other (17)

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]

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]

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 116–156.

C. Brosseau, Fundamentals of Polarized Light–A Statistical Approach (Wiley, New York, 1998), pp. 138–164.

Ref. 9, pp. 237–285.

J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, Heidelberg, Germany, 1975).

T. M. Cover, J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991), pp. 12–49.

Ref. 10, pp. 165–175.

Ref. 15, pp. 279–335.

C. W. Therrien, Decision Estimation and Classification (Wiley, New York, 1989), pp. 139–155.

R. J. Muirhead, Aspects of Multivariate Statistical Theory (Wiley, New York, 1982).

T. S. Ferguson, Mathematical Statistics, a Decision Theoretic Approach (Academic, New York, 1967), pp. 112–119.

J. Rissanen, Stochastic Complexity in Statistical Inquiry (World Scientific, Singapore, 1989).

Ref. 15, pp. 266–278.

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.

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.

S. M. Kay, Fundamentals of Statistical Signal Processing—Volume II: Detection Theory (Prentice Hall, Upper Saddle River, N.J., 1998), pp. 186–247.

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Figures (4)

Fig. 1
Fig. 1

Illustration of two different types of mixing: top, random choice between E(a) and E(b); bottom additive mixing of E(a) and E(b).

Fig. 2
Fig. 2

Segmentation of a polarimetric image extracted from an image acquired with the NASA–JPL AIRSAR system: (a), (b), (c) modulus squares of each channel of the polarimetric image; (d) segmentation result on the intensity channel (a) with the MDL polygonal snake (a gamma PDF has been assumed); (e) segmentation result on the 3-complex channel polarimetric image with the MDL polygonal snake. For both segmentation examples, the initial contour was the white square represented in (a).

Fig. 3
Fig. 3

AUC in function of (a) NaD and (b) (Na+Nb)C(α), where C(α) is the Chernoff distance and α=Na/(Na+Nb). Each point corresponds to a different configuration of parameters (Γa, Γb, Na, α). The AUC for each configuration has been estimated on 104 Monte Carlo trials.

Fig. 4
Fig. 4

AUC in function of (Na+Nb)C(α) for different values of α and Na=20. Inset, same results for Na=4.

Equations (48)

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Γ=|E1|2E1E2*E2E1*|E2|2=μ1ρρ*μ2,
PΓ(E)=1π2det(Γ)   exp(-EΓ-1E),
PΓ(E)=1πddet(Γ)   exp(-EΓ-1E) .
S2=log[π2 det(Γ)]+(EΓ-1E)PΓ(E)dE,
Sd=log[πded det(Γ)].
Sd=d log(πI0)+log(δ)+d.
Sd=d+d log(πI0)+logλ1λ2λd(λ1+λ2++λd)d.
Sd=d1+logπd+d log(I0)+log(1-Pd2).
S33+3 log(πI0)+log(λ1+λ2)+logλ1λ2(λ1+λ2)2,
K(QP)=   logQ(E)P(E)Q(E)dE.
PN(Q|P)exp[-NK(QP)].
K(PΓaPΓb)=   logPΓa(E)PΓb(E)PΓa(E)dE=logdet(Γb)det(Γa)+tr(ΓaΓb-1)-d,
K(PΓaPΓb)=-log(1-Pa2)
PN(PΓa|PΓb)=(1-Pa2)N.
MN(a)MN(b)=1-Pa21-Pb2N,
SαRSα=log{πded det[αΓa+(1-α)Γb]}.
-H[αPΓa(E)+(1-α)PΓb(E)]
-αH[PΓa(E)]-(1-α)H[PΓb(E)];
E(α)=(α)1/2E(a)+(1-α)1/2   exp(iφ)E(b),
ΔSαA=logdet[αΓa+(1-α)Γb]det(Γa)αdet(Γb)(1-α).
E(α)=αE(a)+(1-α)1/2   exp(iφ)E(b),
Q*(E)=[PΓa(E)]1-s*[PΓb(E)]s*[PΓa(E)]1-s*[PΓb(E)]s*dE,
C(s)=-log[PΓa(E)]1-s[PΓb(E)]sdE.
SαSi-2α log(1-P2),
log[1-P(α)2](1-2α)log(1-P2),
E(x, y)=Ea(x, y)waθ(x, y)+Eb(x, y)wbθ(x, y)
L(Γu)=-Nuθ   log πd-Nuθ   log[det(Γu)]-(x,y)ΩuθE(x, y)Γu-1E(x, y).
Γuθ^=1Nuθ   (x,y)ΩuθE(x, y)E(x, y).
L^u(θ)=-Nuθ log[det(Γuθ^)]-dNuθ(1+log π).
L^u(θ)=-Nuθd log(I^u)+log[1-P^d2(u)]+d logπed,
ΔNaSa+NbSb+k log(N).
Δapp=Naθ   log{I^ad[1-P^d2(a)]}+Nbθ   log{I^bd[1-P^d2(b)]}+k log(N)-Cst,
Rˆ=-Na   log[det(Γ^a)]-Nb   log[det(Γ^b)]+NF   log[det(Γ^F)],
Rˆ=(Na+Nb)logdet[αΓ^a+(1-α)Γ^b]det(Γ^a)αdet(Γ^b)(1-α),
Rˆ=-Nad log[I^a]-Nbd log[I^b]+NFd log[I^F]-Na   log[1-P^d2(a)]-Nb   log[1-P^d2(b)]+NF   log[1-P^d2(F)],
D=K[PpaPpb]+K[PpbPpa].
dds U(s)=logPΓb(E)PΓa(E)PΓa(E)1-sPΓb(E)sdE;
dds U(0)=logPΓb(E)PΓa(E)PΓa(E)dE=-K(PΓaPΓb),
dds U(1)=logPΓb(E)PΓa(E)PΓb(E)dE=K(PΓbPΓa).
dds C(s)=-ddsln U(s)=-1U(s)dds U(s),
d2ds2 U(s)=logPΓb(E)PΓa(E)2PΓa(E)1-sPΓb(E)sdE,
PΓa(E)=1πddet(Γa)exp(-EΓa-1E),
PΓb(E)=1πddet(Γb)exp(-EΓb-1E).
ΔSα=logdet[αΓa+(1-α)Γb]det(Γa)αdet(Γb)(1-α).
U(s)=(1/A)exp{-E[(1-s)Γa-1+sΓb-1]E}dE
U(s)=det[(1-s)Γa-1+sΓb-1]-1det(Γa)1-sdet(Γb)s.
U(s)=det(Γa)sdet(Γb)1-sdet[(1-s)Γb+sΓa],
C(s)=logdet[(1-s)Γb+sΓa]det(Γa)sdet(Γb)1-s,

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