Abstract

We propose and analyze a statistical method to estimate the degree of polarization of light from a single speckle intensity image by analyzing the statistical distribution of the light intensity. The optimal precision of such an estimation method is evaluated by computing the Cramer–Rao bounds for several speckle degrees. Two moment-based estimators of the square degree of polarization are introduced and characterized. For the first time to our knowledge, it is shown theoretically and through simulations that the estimators are almost efficient for high orders of speckle. The robustness of the method is discussed for the case when the intensity fluctuations do not follow the standard speckle model.

© 2008 Optical Society of America

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References

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  1. L. B. Wolff, "Polarization-based material classification from specular reflection," IEEE Trans. Pattern Anal. Mach. Intell. 12, 1059-1071 (1990).
    [CrossRef]
  2. R. A. Chipman, "Polarization diversity active imaging," Proc. SPIE 3170, 68-73 (1997).
    [CrossRef]
  3. S. Breugnot and Ph. Clémenceau, "Modeling and performances of a polarization active imager at lambda=806nm," Proc. SPIE 3707, 449-460 (1999).
    [CrossRef]
  4. S. L. Jacques, J. C. Ramella-Roman, and K. Lee, "Imaging skin pathology with polarized light," J. Biomed. Opt. 7, 329-340 (2002).
    [CrossRef] [PubMed]
  5. M. Roche, J. Fade, and Ph. Réfrégier, "Parametric estimation of the square degree of polarization from two intensity images degraded by fully developed speckle noise," J. Opt. Soc. Am. A 24, 2719-2727 (2007).
    [CrossRef]
  6. J. W. Goodman, Statistical Optics (Wiley, 1985).
  7. P. Réfrégier, J. Fade, and M. Roche, "Estimation precision of the degree of polarization from a single speckle intensity image," Opt. Lett. 32, 739-741 (2007).
    [CrossRef] [PubMed]
  8. Ph. Réfrégier, Noise Theory and Application to Physics: From Fluctuations to Information (Springer, 2004).
  9. S. Huard, Polarization of Light (Wiley, 1997).
  10. B. R. Frieden, Probability, Statistical Optics and Data Testing (Springer Verlag, 2001).
    [CrossRef]
  11. P. H. Garthwaite, I. T. Jolliffe, and B. Jones, Statistical Inference (Prentice Hall, 1995).
  12. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980).
  13. M. Evans, N. Hastings, and B. Peacock, Statistical Distributions, Wiley Series in Probability and Statistics (Wiley, 2000).
  14. F. Le Chevalier, Principles of Radar and Sonar Signal Processing (Artech House, Thales Airborne Systems, 2002).
  15. J. K. Jao, "Amplitude distribution of composite terrain radar clutter and the K-distribution," IEEE Trans. Antennas Propag. AP-32, 1049-1062 (1984).
    [CrossRef]
  16. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 1991).

2007 (2)

2002 (1)

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, "Imaging skin pathology with polarized light," J. Biomed. Opt. 7, 329-340 (2002).
[CrossRef] [PubMed]

1999 (1)

S. Breugnot and Ph. Clémenceau, "Modeling and performances of a polarization active imager at lambda=806nm," Proc. SPIE 3707, 449-460 (1999).
[CrossRef]

1997 (1)

R. A. Chipman, "Polarization diversity active imaging," Proc. SPIE 3170, 68-73 (1997).
[CrossRef]

1990 (1)

L. B. Wolff, "Polarization-based material classification from specular reflection," IEEE Trans. Pattern Anal. Mach. Intell. 12, 1059-1071 (1990).
[CrossRef]

1984 (1)

J. K. Jao, "Amplitude distribution of composite terrain radar clutter and the K-distribution," IEEE Trans. Antennas Propag. AP-32, 1049-1062 (1984).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

J. K. Jao, "Amplitude distribution of composite terrain radar clutter and the K-distribution," IEEE Trans. Antennas Propag. AP-32, 1049-1062 (1984).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

L. B. Wolff, "Polarization-based material classification from specular reflection," IEEE Trans. Pattern Anal. Mach. Intell. 12, 1059-1071 (1990).
[CrossRef]

J. Biomed. Opt. (1)

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, "Imaging skin pathology with polarized light," J. Biomed. Opt. 7, 329-340 (2002).
[CrossRef] [PubMed]

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

Opt. Lett. (1)

Proc. SPIE (2)

R. A. Chipman, "Polarization diversity active imaging," Proc. SPIE 3170, 68-73 (1997).
[CrossRef]

S. Breugnot and Ph. Clémenceau, "Modeling and performances of a polarization active imager at lambda=806nm," Proc. SPIE 3707, 449-460 (1999).
[CrossRef]

Other (9)

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 1991).

Ph. Réfrégier, Noise Theory and Application to Physics: From Fluctuations to Information (Springer, 2004).

S. Huard, Polarization of Light (Wiley, 1997).

B. R. Frieden, Probability, Statistical Optics and Data Testing (Springer Verlag, 2001).
[CrossRef]

P. H. Garthwaite, I. T. Jolliffe, and B. Jones, Statistical Inference (Prentice Hall, 1995).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980).

M. Evans, N. Hastings, and B. Peacock, Statistical Distributions, Wiley Series in Probability and Statistics (Wiley, 2000).

F. Le Chevalier, Principles of Radar and Sonar Signal Processing (Artech House, Thales Airborne Systems, 2002).

J. W. Goodman, Statistical Optics (Wiley, 1985).

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Figures (4)

Fig. 1
Fig. 1

CRBs of the estimation of β plotted as a function of P for different orders L of speckle noise (dashed curves). The “vectorial” CRB in the case of a fully developed speckle ( L = 1 ) is plotted in the dotted curve. The light solid curve represents the asymptotic CRB for the Gaussian limit case.

Fig. 2
Fig. 2

Case of gamma PDFs: the variance of the estimators is plotted as a function of the gamma law order L for different polarization degrees. Triangles, variance of β ̂ 1 ; squares, variance of β ̂ 2 ; dashed (resp. solid) curves, theoretical variance of β ̂ 1 (resp. β ̂ 2 ).

Fig. 3
Fig. 3

Comparison between the CRB and the variance of the estimator β ̂ 2 for different speckle orders: (a) L = 2 ; bold dashed curve, CRB; dotted curve, theoretical variance; squares, simulation results. The light solid curve (blue in color) represents the asymptotic CRB for the Gaussian limit case. (b) Same as (a) with L = 10 .

Fig. 4
Fig. 4

(a) Experimental variance of β ̂ 1 as a function of P for different cases of marginal PDFs: triangles, Weibull law of order 2; squares, Weibull law of order 5; diamonds, K -law of order 1; crosses, K -law of order 10. The dashed curves correspond to the theoretical variances. (b) Same as (a) for the estimator β ̂ 2 .

Tables (1)

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Table 1 Normalized Cumulants of Marginal Probability Laws

Equations (40)

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p X ( x ) = 1 μ X exp ( x μ X ) ,
P 1 4 det ( Γ ) tr ( Γ ) 2 ,
p I ( I ) = 1 P μ I [ exp ( 2 I ( 1 + P ) μ I ) exp ( 2 I ( 1 P ) μ I ) ] .
var ( I ) = I 2 I 2 = 1 + P 2 2 μ I 2 .
p X ( x ) = ( L μ X ) L x L 1 Γ ( L ) exp ( L x μ X ) .
var ( P ̂ ) C R B N ( P ) = 1 I F N ( P ) ,
I F N ( P ) = . . 2 [ ln L ( χ P ) ] P 2 p I ( I 1 ) . . p I ( I N ) d I 1 . . d I N ,
C R B N 1 ( β ) = 4 P 4 ( 1 P 2 ) N ( 1 + P 2 ) [ 1 1 + P 2 2 P 2 ζ ( 3 , 1 + P 2 P ) ] 1 ,
CRB N 1 , v ( β ) = 2 P 4 ( 1 P 2 ) N [ ( 1 P 2 ) ζ ( 3 , 1 + P 2 P ) + 2 P 2 ] [ ( 1 + P 2 ) ζ ( 3 , 1 + P 2 P ) 2 P 2 ] .
p I ( I ) N ( I , μ I , 1 + β 2 L μ I 2 ) .
l ( χ β ) = j = 1 N [ 1 2 ln ( 1 + β ) + 1 2 ln L π μ I 2 L ( I j μ I ) 2 μ I 2 ( 1 + β ) ] ,
C R B N g a u s s ( β ) = 2 ( 1 + β ) 2 N .
C I , 1 = I = μ I = μ X + μ Y ,
C I , 2 = var ( I ) = κ 2 ( 1 + P 2 ) μ I 2 2 .
β = P 2 = 2 κ 2 μ I 2 var ( I ) 1 .
β ̂ 1 = 2 κ 2 μ I 2 [ S ̂ 1 ] 1 , with S ̂ 1 = 1 N i = 1 N I i 2 μ I 2 ;
β ̂ 2 = 2 κ 2 μ ̂ I 2 [ S ̂ 2 ] 1 , with S ̂ 2 = 1 N i = 1 N I i 2 ( μ ̂ I ) 2 ,
var ( β ̂ 1 ) = 2 ( 1 + β ) 2 N + γ 2 2 N ( 1 + 6 β + β 2 ) ,
var ( β ̂ 2 ) var ( β ̂ 1 ) + 2 κ 2 ( 1 + β ) 3 N 2 γ 1 κ 2 ( 1 + β ) ( 1 + 3 β ) N ,
var ( β ̂ 1 ) = 2 ( 1 + β ) 2 N + 3 N L ( 1 + 6 β + β 2 ) ,
var ( β ̂ 2 ) var ( β ̂ 1 ) 2 ( 1 + β ) ( 1 + 4 β β 2 ) N L .
l ( χ μ I , P ) = j = 1 N ln 2 μ I ln P 2 I j ( 1 P 2 ) μ I + ln { sinh [ 2 I j P ( 1 P 2 ) μ I ] } .
u , u C N ( P ̂ , μ ̂ I ) u u I F N 1 ( P , μ I ) u ,
I F N ( P , μ I ) = [ 2 l ( χ μ I , P ) P 2 2 l ( χ μ I , P ) P μ I 2 l ( χ μ I , P ) μ I P 2 l ( χ μ I , P ) μ I 2 ] = [ I 1 , 1 I 2 , 1 I 1 , 2 I 2 , 2 ] .
α = 0 x e x 1 P 2 cosh ( P x 1 P 2 ) d x = 1 + P 2 ,
β = 0 x 2 e x 1 P 2 1 sinh ( P x 1 P 2 ) d x = ( 1 P 2 ) 3 2 P 3 ζ ( 3 , 1 + P 2 P ) ,
I F N ( P , μ I ) = N [ 1 + P 2 P 2 ( 1 P 2 ) [ 1 1 + P 2 2 P 2 ζ ( 3 , 1 + P 2 P ) ] 1 μ I P [ 1 1 + P 2 2 P 2 ζ ( 3 , 1 + P 2 P ) ] 1 μ I P [ 1 1 + P 2 2 P 2 ζ ( 3 , 1 + P 2 P ) ] 1 μ I 2 [ 1 + 1 P 2 2 P 2 ζ ( 3 , 1 + P 2 P ) ] ] .
I F N 1 ( P , μ I ) = 1 N [ P 2 ( 1 P 2 ) 2 [ ( 1 P 2 ) ζ ( 3 , 1 + P 2 P ) + 2 P 2 ( 1 + P 2 ) ζ ( 3 , 1 + P 2 P ) 2 P 2 ] μ I P ( 1 P 2 ) 2 μ I P ( 1 P 2 ) 2 μ I 2 ( 1 + P 2 ) 2 ] .
CRB N 1 , v ( β ) = 4 P 2 CRB N 1 , v ( P ) = u 1 I F N 1 ( P , μ I ) u 1 = 2 P 4 ( 1 P 2 ) N [ ( 1 P 2 ) ζ ( 3 , 1 + P 2 P ) + 2 P 2 ] [ ( 1 + P 2 ) ζ ( 3 , 1 + P 2 P ) 2 P 2 ] .
C 1 = μ I ; C 2 = κ 2 1 + β 2 μ I 2 ; C 3 = κ 3 1 + 3 β 4 μ I 3 ;
C 4 = κ 4 1 + 6 β + β 2 8 μ I 4 .
var ( S ̂ 1 ) = 1 N var [ ( I μ I ) 2 ] = m 4 m 2 2 N = C 4 + 2 C 2 2 N .
var ( β ̂ 1 ) = 2 ( 1 + β ) 2 N + κ 4 2 N κ 2 2 ( 1 + 6 β + β 2 ) .
β ̂ 2 β + β μ I [ μ ̂ I μ I ] + β C 2 [ S ̂ 2 C 2 ] ,
β ̂ 2 β β μ I μ ̂ I μ I + β C 2 S ̂ 2 C 2 = 2 C 2 N κ 2 μ I 2 = 1 + β N ,
var ( β ̂ 2 ) ( β μ I ) 2 var ( μ ̂ I ) + ( β C 2 ) 2 var ( S ̂ 2 ) + 2 β μ I β C 2 cov ( μ ̂ I , S ̂ 2 ) ,
cov ( μ ̂ I , S ̂ 2 ) = ( μ ̂ I μ ̂ I ) ( S ̂ 2 S ̂ 2 ) = μ ̂ I S ̂ 2 μ ̂ I S ̂ 2 = 1 N 2 i = 1 N I i j = 1 N ( I j 1 N k = 1 N I k ) 2 N 1 N μ I C 2 = 1 N 2 i = 1 N j = 1 N I i I j 2 1 N 3 i = 1 N j = 1 N k = 1 N I i I j I k N 1 N μ I C 2 = C 3 ( N 1 ) N 2 .
var ( S ̂ 2 ) = ( N 1 ) 2 N 2 var ( v ̂ ) = N 1 N 3 [ ( N 1 ) m 4 ( N 3 ) m 2 2 ] = C 4 + 2 C 2 2 N 2 C 4 + 2 C 2 2 N 2 + C 4 N 3 .
var ( β ̂ 2 ) 4 N κ 2 2 μ I 4 [ C 4 + 2 C 2 2 + 4 C 2 3 μ I 2 4 C 2 C 3 μ I + 1 N ( 4 C 2 C 3 μ I 2 C 4 2 C 2 2 ) + C 4 N 2 ] .
var ( β ̂ 2 ) var ( β ̂ 1 ) + 2 κ 2 N ( 1 + β ) 3 2 κ 3 N κ 2 ( 1 + β ) ( 1 + 3 β ) .

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