Abstract

We address the problem of the estimation of the degree of polarization from a single intensity image. For that purpose, one considers the case of coherent active imagery that leads to speckle fluctuations and assumes that the measured intensity image corresponds to a fully developed speckle for each polarized component of the electric field. In particular, we determine the Cramer–Rao bound of the degree of polarization estimation and propose to illustrate this result by analyzing the variance of different simple estimators.

© 2007 Optical Society of America

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References

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  1. L. B. Wolff, IEEE Trans. Pattern Anal. Mach. Intell. 12, 1059 (1990).
    [CrossRef]
  2. R. B. Holmes, Proc. SPIE 1871, 306 (1993).
    [CrossRef]
  3. R. A. Chipman, Proc. SPIE 3170, 68 (1997).
    [CrossRef]
  4. S. Breugnot and P. Clémenceau, Proc. SPIE 3707, 449 (1999).
    [CrossRef]
  5. J. W. Goodman, Statistical Optics (Wiley, 1985).
  6. C. Brosseau, Fundamentals of Polarized Light--A Statistical Approach (Wiley, 1998).
  7. P. H. Garthwaite, I. T. Jolliffe, and B. Jones, Statistical Inference (Prentice Hall Europe, 1995).
  8. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980).
  9. P. Réfrégier, F. Goudail, and N. Roux, J. Opt. Soc. Am. A 21, 2292 (2004).
    [CrossRef]
  10. N. Roux, F. Goudail, and P. Réfrégier, J. Opt. Soc. Am. A 22, 2532 (2005).
    [CrossRef]

2005 (1)

2004 (1)

1999 (1)

S. Breugnot and P. Clémenceau, Proc. SPIE 3707, 449 (1999).
[CrossRef]

1997 (1)

R. A. Chipman, Proc. SPIE 3170, 68 (1997).
[CrossRef]

1993 (1)

R. B. Holmes, Proc. SPIE 1871, 306 (1993).
[CrossRef]

1990 (1)

L. B. Wolff, IEEE Trans. Pattern Anal. Mach. Intell. 12, 1059 (1990).
[CrossRef]

Breugnot, S.

S. Breugnot and P. Clémenceau, Proc. SPIE 3707, 449 (1999).
[CrossRef]

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light--A Statistical Approach (Wiley, 1998).

Chipman, R. A.

R. A. Chipman, Proc. SPIE 3170, 68 (1997).
[CrossRef]

Clémenceau, P.

S. Breugnot and P. Clémenceau, Proc. SPIE 3707, 449 (1999).
[CrossRef]

Garthwaite, P. H.

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

Goodman, J. W.

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

Goudail, F.

Gradshteyn, I. S.

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

Holmes, R. B.

R. B. Holmes, Proc. SPIE 1871, 306 (1993).
[CrossRef]

Jolliffe, I. T.

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

Jones, B.

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

Réfrégier, P.

Roux, N.

Ryzhik, I. M.

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

Wolff, L. B.

L. B. Wolff, IEEE Trans. Pattern Anal. Mach. Intell. 12, 1059 (1990).
[CrossRef]

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

L. B. Wolff, IEEE Trans. Pattern Anal. Mach. Intell. 12, 1059 (1990).
[CrossRef]

J. Opt. Soc. Am. A (2)

Proc. SPIE (3)

R. B. Holmes, Proc. SPIE 1871, 306 (1993).
[CrossRef]

R. A. Chipman, Proc. SPIE 3170, 68 (1997).
[CrossRef]

S. Breugnot and P. Clémenceau, Proc. SPIE 3707, 449 (1999).
[CrossRef]

Other (4)

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

C. Brosseau, Fundamentals of Polarized Light--A Statistical Approach (Wiley, 1998).

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

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

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Figures (2)

Fig. 1
Fig. 1

Comparison of the variance of some estimators of the square value of the degree of polarization to the CRB. Solid curve, CRB; triangles, variance of β ̂ 1 ; squares, variance of β ̂ 2 ; crosses, variance of the ML estimator. Inset, Bias of some estimators of the square value of the degree of polarization. Triangles, bias of β ̂ 1 ; squares, bias of β ̂ 2 ; crosses, bias of the ML estimator. The dashed curves are only guides for the eyes.

Fig. 2
Fig. 2

Base 10 logarithm of the CRB in the case of a single measurement of an intensity image of the total reflected light (solid curve) and in the case of two intensity measurements performed with orthogonal polarizers (dashed curve). Base 10 logarithm of the variance of the ML estimator in the case of a single measurement (crosses) and of the ML estimator in the case of two measurements (diamonds).

Equations (7)

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p I ( I ) = 1 P I T { exp ( 2 I ( 1 + P ) I T ) exp ( 2 I ( 1 P ) I T ) } ,
I F N ( P ) = 2 P 2 ln [ P I ( I 1 ) P I ( I N ) ] × P I ( I 1 ) P I ( I N ) d I 1 d I N .
I F N ( P ) = N [ 1 P 2 + 2 1 P 2 ( 1 + P 2 ) 2 2 P 4 ( 1 P 2 ) ζ ( 3 , 1 + P 2 P ) ] ,
CRB N ( β ) = 4 P 2 N [ 1 P 2 + 2 1 P 2 ( 1 + P 2 ) 2 2 P 4 ( 1 P 2 ) ζ ( 3 , 1 + P 2 P ) ] 1 .
β ̂ 1 = 2 I T 2 [ 1 N i = 1 N I i 2 I T 2 ] 1
β ̂ 2 = 2 ( 1 N i = 1 N I i ) 2 [ 1 N i = 1 N I i 2 ( 1 N i = 1 N I i ) 2 ] 1 .
CRB N ( 2 ) ( β ) = 2 P 2 N ( 1 P 2 ) 2 1 + P 2 .

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