Abstract

Active polarimetric imagery systems allow one to reveal polarimetric characteristics of the scene. Among them, the degree of polarization allows one to have information about the polarizing nature of an imaged object. Its estimation is standardly done from four images of the scene. Reducing this number of images can be of great interest for industrial applications, allowing in particular reduction of cost in terms of money and acquisition time. We propose a parametric method to estimate the square degree of polarization from only two measurements when coherent illumination is considered and when the images are corrupted with fully developed speckle, and we characterize the performances of the estimation.

© 2007 Optical Society of America

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References

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  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]
  2. L. B. Wolff, "Polarization-based material classification from specular reflection," IEEE Trans. Pattern Anal. Mach. Intell. 12, 1059-1071 (1990).
    [CrossRef]
  3. L. B. Wolff, "Polarization camera for computer vision with a beam splitter," J. Opt. Soc. Am. A 11, 2935-2945 (1994).
  4. B. Johnson, R. Joseph, M. L. Nischan, A. Newbury, J. P. Kerekes, H. T. Barclay, B. C. Willard, and J. J. Zayhowski, "Compact active hyperspectral imaging system for the detection of concealed targets," Proc. SPIE 3710, 144-153 (1999).
  5. W. G. Egan, W. R. Johnson, and V. S. Whitehead, "Terrestrial polarization imagery obtained from the space shuttle: characterization and interpretation," Appl. Opt. 30, 435-442 (1991).
  6. J. W. Goodman, Statistical Optics (Wiley, 1985).
  7. F. Goudail, P. Terrier, Y. Takakura, L. Bigué, F. Galland, and V. de Vlaminck, "Target detection with a liquid crystal-based passive Stokes polarimeter," Appl. Opt. 43, 274-282 (2004).
    [CrossRef]
  8. C. Brosseau, Fundamentals of Polarized Light: A Statistical Approach (Wiley, 1998).
  9. J. E. Solomon, "Polarization imaging," Appl. Opt. 20, 1537-1544 (1981).
  10. S. Breugnot and Ph. Clémenceau, "Modeling and performances of a polarization active imager at lambda=806 nm," Proc. SPIE 3707, 449-460 (1999).
  11. M. P. Rowe, E. N. Pugh, J. S. Tyo, and N. Engheta, "Polarization-difference imaging: a biologically inspired technique for observation through scattering media," Opt. Lett. 20, 608-610 (1995).
  12. F. Goudail and Ph. Réfrégier, Statistical Image Processing Techniques for Noisy Images: An Application Oriented Approach (Kluwer, 2004).
  13. F. Goudail and Ph. Réfrégier, "Statistical techniques for target detection in polarization diversity images," Opt. Lett. 26, 644-646 (2001).
  14. S. Huard, Polarization of Light (Wiley, Masson, 1997).
  15. Ph. Réfrégier, Noise Theory and Application to Physics: From Fluctuations to Information (Springer, 2004).
  16. C. L. Mehta, Lectures in Theoretical Physics VIIC, W.E.Brittin, ed. (U. Colorado Press, 1965), p. 345.

2004 (1)

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]

2001 (1)

1999 (2)

B. Johnson, R. Joseph, M. L. Nischan, A. Newbury, J. P. Kerekes, H. T. Barclay, B. C. Willard, and J. J. Zayhowski, "Compact active hyperspectral imaging system for the detection of concealed targets," Proc. SPIE 3710, 144-153 (1999).

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

1995 (1)

1994 (1)

1991 (1)

1990 (1)

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

1981 (1)

Barclay, H. T.

B. Johnson, R. Joseph, M. L. Nischan, A. Newbury, J. P. Kerekes, H. T. Barclay, B. C. Willard, and J. J. Zayhowski, "Compact active hyperspectral imaging system for the detection of concealed targets," Proc. SPIE 3710, 144-153 (1999).

Bigué, L.

Breugnot, S.

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

Brosseau, C.

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

Clémenceau, Ph.

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

de Vlaminck, V.

Egan, W. G.

Engheta, N.

Galland, F.

Goodman, J. W.

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

Goudail, F.

Huard, S.

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

Jacques, S. L.

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

Johnson, B.

B. Johnson, R. Joseph, M. L. Nischan, A. Newbury, J. P. Kerekes, H. T. Barclay, B. C. Willard, and J. J. Zayhowski, "Compact active hyperspectral imaging system for the detection of concealed targets," Proc. SPIE 3710, 144-153 (1999).

Johnson, W. R.

Joseph, R.

B. Johnson, R. Joseph, M. L. Nischan, A. Newbury, J. P. Kerekes, H. T. Barclay, B. C. Willard, and J. J. Zayhowski, "Compact active hyperspectral imaging system for the detection of concealed targets," Proc. SPIE 3710, 144-153 (1999).

Kerekes, J. P.

B. Johnson, R. Joseph, M. L. Nischan, A. Newbury, J. P. Kerekes, H. T. Barclay, B. C. Willard, and J. J. Zayhowski, "Compact active hyperspectral imaging system for the detection of concealed targets," Proc. SPIE 3710, 144-153 (1999).

Lee, K.

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

Mehta, C. L.

C. L. Mehta, Lectures in Theoretical Physics VIIC, W.E.Brittin, ed. (U. Colorado Press, 1965), p. 345.

Newbury, A.

B. Johnson, R. Joseph, M. L. Nischan, A. Newbury, J. P. Kerekes, H. T. Barclay, B. C. Willard, and J. J. Zayhowski, "Compact active hyperspectral imaging system for the detection of concealed targets," Proc. SPIE 3710, 144-153 (1999).

Nischan, M. L.

B. Johnson, R. Joseph, M. L. Nischan, A. Newbury, J. P. Kerekes, H. T. Barclay, B. C. Willard, and J. J. Zayhowski, "Compact active hyperspectral imaging system for the detection of concealed targets," Proc. SPIE 3710, 144-153 (1999).

Pugh, E. N.

Ramella-Roman, J. C.

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

Réfrégier, Ph.

F. Goudail and Ph. Réfrégier, "Statistical techniques for target detection in polarization diversity images," Opt. Lett. 26, 644-646 (2001).

F. Goudail and Ph. Réfrégier, Statistical Image Processing Techniques for Noisy Images: An Application Oriented Approach (Kluwer, 2004).

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

Rowe, M. P.

Solomon, J. E.

Takakura, Y.

Terrier, P.

Tyo, J. S.

Whitehead, V. S.

Willard, B. C.

B. Johnson, R. Joseph, M. L. Nischan, A. Newbury, J. P. Kerekes, H. T. Barclay, B. C. Willard, and J. J. Zayhowski, "Compact active hyperspectral imaging system for the detection of concealed targets," Proc. SPIE 3710, 144-153 (1999).

Wolff, L. B.

L. B. Wolff, "Polarization camera for computer vision with a beam splitter," J. Opt. Soc. Am. A 11, 2935-2945 (1994).

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

Zayhowski, J. J.

B. Johnson, R. Joseph, M. L. Nischan, A. Newbury, J. P. Kerekes, H. T. Barclay, B. C. Willard, and J. J. Zayhowski, "Compact active hyperspectral imaging system for the detection of concealed targets," Proc. SPIE 3710, 144-153 (1999).

Appl. Opt. (3)

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]

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

Opt. Lett. (2)

Proc. SPIE (2)

B. Johnson, R. Joseph, M. L. Nischan, A. Newbury, J. P. Kerekes, H. T. Barclay, B. C. Willard, and J. J. Zayhowski, "Compact active hyperspectral imaging system for the detection of concealed targets," Proc. SPIE 3710, 144-153 (1999).

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

Other (6)

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

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

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

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

C. L. Mehta, Lectures in Theoretical Physics VIIC, W.E.Brittin, ed. (U. Colorado Press, 1965), p. 345.

F. Goudail and Ph. Réfrégier, Statistical Image Processing Techniques for Noisy Images: An Application Oriented Approach (Kluwer, 2004).

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Figures (7)

Fig. 1
Fig. 1

Standard experimental setup of a polarimetric coherent imagery system. The scene is illuminated by coherent light. A CCD camera registers the light backscaterred by the scene after it has been transmitted by a polarizer or the combination of a polarizer and a retardation plate. This setup allows one to obtain four images, I X , I Y , I 45 ° , I ( 45 ° , π 2 ) , that are acquired with four different configurations of the coupled polarizer and retardation plate.

Fig. 2
Fig. 2

Evolution of the variance of P 4 2 ̂ and P 2 2 ̂ as a function of P . The polarization matrices are Γ i v with growing c i . Number of realizations R = 10 4 and N = 10 3 . The bold dashed curve and bold symbols correspond to the cases with two measures. The dashed curves correspond to the asymptotic expressions of the variances. Squares and diamonds represent the numerical experimental results.

Fig. 3
Fig. 3

Same as Fig. 2 with covariance matrices of the form Γ i d for R = 10 4 and N = 10 2 .

Fig. 4
Fig. 4

Mask of the shape of the different simulated regions.

Fig. 5
Fig. 5

Polarimetric image simulated from the mask of Fig. 4 with the covariance matrices of Table 1, corresponding to (a) I X , (b) I Y , (c) I 45 ° , and (d) I ( 45 ° , π 2 ) .

Fig. 6
Fig. 6

(a) Image of the true P 2 ; the higher the square DOP is, the brighter is the gray level. (b) Image of the estimation of P 2 with a window of size 11 × 11 and the method with four measures. (c) Same as (b) but with a 21 × 21 window. (d) Same as (b) but for the OSCI method. (e) Same as (b) with the method using two images. (f) Same as (c) for the method with two measures.

Fig. 7
Fig. 7

(a) Homogeneous regions considered to estimate P 2 . The true values of the DOP correspond to that of Fig. 6a. (b) Map of the estimated square DOP with the regions defined by (a), using the method with four measures. (c) Same as (b) but with the proposed method with two intensity images.

Tables (1)

Tables Icon

Table 1 Properties of Regions Composing the Simulated Polarimetric Image

Equations (42)

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E ( r , t ) = [ A X ( r , t ) e X + A Y ( r , t ) e Y ] e i 2 π ν t ,
Γ = ( A X ( r , t ) A X * ( r , t ) A X ( r , t ) A Y * ( r , t ) A Y ( r , t ) A X * ( r , t ) A Y ( r , t ) A Y * ( r , t ) ) ( I X c c * I Y ) ,
P 2 = 1 4 det ( Γ ) [ tr ( Γ ) ] 2 = 1 4 I X I Y ρ ( I X + I Y ) 2 ,
p A ( A ) = 1 π 2 det ( Γ ) e A Γ 1 A ,
OSCI ( i ) = I X ( i ) I Y ( i ) I X ( i ) + I Y ( i ) ,
P k 2 ̂ = 1 4 I X ̂ I Y ̂ ρ ̂ k ( I X ̂ + I Y ̂ ) 2 , k { 2 , 4 } ,
I X ̂ = 1 N i = 1 N I X ( i ) , I Y ̂ = 1 N i = 1 N I Y ( i ) .
Var ( I X ̂ ) = I X 2 N , Var ( I Y ̂ ) = I Y 2 N .
ρ ̂ 4 = 1 N i = 1 N A X ( i ) A Y * ( i ) 2 .
δ ( ρ ̂ 4 ) ρ ̂ 4 ρ = I X I Y N .
Var ( ρ ̂ 4 ) = I X 2 I Y 2 ( N + 2 ) + 2 I X I Y ρ ( 2 + 4 N + N 2 ) + ρ 2 N ( 2 N + 1 ) N 3 ,
Var a ( ρ ̂ 4 ) = 2 ρ ( I X I Y + ρ ) N .
I X I Y = A X A X * A Y A Y * = A X A X * A Y A Y * + A X A Y * A Y A X * = I X I Y + ρ .
ρ ̂ 2 = 1 N i = 1 N I X ( i ) I Y ( i ) I X ̂ I Y ̂ = 1 N i = 1 N I X ( i ) I Y ( i ) [ 1 N j = 1 N I X ( j ) ] [ 1 N k = 1 N I Y ( k ) ] .
δ ( ρ ̂ 2 ) ρ ̂ 2 ρ = ρ N ,
Var ( ρ ̂ 2 ) = N ( N 1 ) I X 2 I Y 2 + 4 ( N 2 2 N + 1 ) I X I Y ρ + ( 3 N 2 5 N + 2 ) ρ 2 N 3 .
Var a ( ρ ̂ 2 ) = I X 2 I Y 2 + 4 I X I Y ρ + 3 ρ 2 N ,
Var a ( ρ ̂ 2 ) = Var a ( ρ ̂ 4 ) + [ det ( Γ ) ] 2 N + 4 I X I Y ρ N .
δ ( P k 2 ̂ ) P k 2 ̂ P 2 4 ( I X + I Y ) 2 ( ρ ̂ k ρ k ) ,
δ ( P 4 2 ̂ ) = P 4 2 ̂ P 2 4 I X I Y N ( I X + I Y ) 2 ,
δ ( P 2 2 ̂ ) = P 2 2 ̂ P 2 4 ρ N ( I X + I Y ) 2 .
Var a ( P 4 2 ̂ ) = 2 ( 1 P 2 ) 2 P 2 N ,
Var a ( P 2 2 ̂ ) = Var a ( P 4 2 ̂ ) + ( 1 P 2 ) 2 N + 64 I X I Y ρ N I T 4 .
Γ i v = ( 1 c i c i * 1 )
Γ i d = ( 1 0 0 d i )
P i = { 0 ; 0.33 ; 0.66 ; 0.82 ; 0.98 } .
ρ ̂ 4 2 = 1 N 4 i = 1 N A X i A Y i * 2 j = 1 N A Y j A X j * 2 = 1 N 4 i = 1 N j = 1 N k = 1 N l = 1 N A X i A Y i * A X j A Y j * A X k * A Y k A X l * A Y l .
Var ( ρ ̂ 4 ) = ρ ̂ 4 2 ρ ̂ 4 2 = I X 2 I Y 2 ( N + 2 ) + 2 I X I Y ρ ( 2 + 4 N + N 2 ) + ρ 2 N ( 2 N + 1 ) N 3 .
ρ ̂ 2 = 1 N i = 1 N I X ( i ) I Y ( i ) 1 N 2 j = 1 N k = 1 N I X ( j ) I Y ( k ) = I X I Y 1 N 2 ( N I X I Y + N ( N 1 ) I X I Y ) = ( 1 1 N ) ρ ,
Var ( ρ ̂ 2 ) = ρ ̂ 2 2 ρ ̂ 2 2 .
ρ ̂ 2 2 = 1 N 2 i = 1 N j = 1 N I X ( i ) I Y ( i ) I X ( j ) I Y ( j ) + [ 1 N 2 i = 1 N j = 1 N I X ( i ) I X ( j ) ] [ 1 N 2 k = 1 N l = 1 N I Y ( k ) I Y ( l ) ] 2 N 3 i = 1 N I X ( i ) j = 1 N I Y ( j ) k = 1 N I X ( k ) I Y ( k ) .
Var ( ρ ̂ 2 ) = ρ ̂ 2 2 ρ ̂ 2 2
= N ( N 1 ) I X 2 I Y 2 + 4 ( N 2 2 N + 1 ) I X I Y ρ + ( 3 N 2 5 N + 2 ) ρ 2 N 3 ,
Var a ( ρ ̂ 2 ) = I X 2 I Y 2 + 4 I X I Y ρ + 3 ρ 2 N .
P k 2 ̂ = P 2 + f I X ( I X ̂ I X ) + f I Y ( I Y ̂ I Y ) + f ρ ( ρ ̂ k ρ ) + o ( ( I X ̂ I X ) + ( I Y ̂ I Y ) + ( ρ ̂ k ρ ) ) ,
f I X = P 2 I X , f I Y = P 2 I Y , f ρ = P 2 ρ
f I X = 4 ( I X I Y I Y 2 2 ρ ) ( I X + I Y ) 3 ,
f I Y = 4 ( I X I Y I X 2 2 ρ ) ( I X + I Y ) 3 ,
f ρ = 4 ( I X + I Y ) 2 .
Var ( P k 2 ̂ ) = ( P k 2 ̂ P k 2 ̂ ) 2 = ( P k 2 ̂ P 2 ) 2 δ ( P k 2 ̂ ) 2 [ f I X ( I X ̂ I X ) + f I Y ( I Y ̂ I Y ) + f ρ ( ρ ̂ k ρ ) ] 2 f ρ 2 δ ( ρ ̂ k ) 2 f I X 2 Var ( I X ̂ ) + f I Y 2 Var ( I Y ̂ ) + f ρ 2 Var ( ρ ̂ k ) + f ρ 2 δ ( ρ ̂ k ) 2 + 2 f I X f I Y Covar ( I X ̂ , I Y ̂ ) + 2 f I X f ρ Covar ( I X ̂ , ρ ̂ k ) + 2 f I Y f ρ Covar ( I Y ̂ , ρ ̂ k ) f ρ 2 δ ( ρ ̂ k ) 2 f I X 2 Var ( I X ̂ ) + f I Y 2 Var ( I Y ̂ ) + f ρ 2 Var ( ρ ̂ k ) + 2 f I X f I Y Covar ( I X ̂ , I Y ̂ ) + 2 f I X f ρ Covar ( I X ̂ , ρ ̂ k ) + 2 f I Y f ρ Covar ( I Y ̂ , ρ ̂ k ) ,
Var a ( P 4 2 ̂ ) = 2 ( 1 P 2 ) 2 P 2 N ,
Var a ( P 2 2 ̂ ) = Var a ( P 4 2 ̂ ) + ( 1 P 2 ) 2 N + 64 I X I Y ρ N I T 4 .

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