Abstract

In this paper, we introduce a general Bayesian approach to estimate polarization parameters in the Stokes imaging framework. We demonstrate that this new approach yields a neat solution to the polarimetric data reduction problem that preserves the physical admissibility constraints and provides a robust clustering of Stokes images in regard to image noises. The proposed approach is extensively evaluated by using synthetic simulated data and applied to cluster and retrieves the Stokes image issuing from a set of real measurements.

© 2007 Optical Society of America

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References

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  1. D. Miyazaki, M. Saito, Y. Sato, and K. Ikeuchi, "Determining surface orientations of transparent objects based on polarization degrees in visible and infrared wavelengths," J. Opt. Soc. Am. A 19, 687-694 (2002).
    [CrossRef]
  2. D. Miyazaki, M. Kagesawa, and K. Ikeuchi, "Transparent surface modeling from a pair of polarization images," IEEE Trans. PAMI 26,920-932 (2004).
    [CrossRef]
  3. J. M. Bueno and P. Artal, "Double-pass imaging polarimetry in the human eye," Opt. Letters. 2464-66 (1999).
    [CrossRef]
  4. S. D. Giattina,  et al., "Assessment of coronary plaque collagen with polarization sensitive optical coherence tomography (PS-OCT)," Int. J. Cardiol. 107, 400-409 (2006).
    [CrossRef] [PubMed]
  5. D. H. Goldstein, D. B. Chenault, and Society of Photo-optical Instrumentation Engineers, Polarization: measurement, analysis, and remote sensing II, 19-21 July, 1999, Denver, Colorado. 1999, Bellingham, Washington: SPIE. ix, 426 p.
  6. M. H. Smith, "Optimizing a dual-rotating-retarder Mueller matrix polarimeter," in Polarization Analysis and Measurements IV, SPIE (2001).
  7. J. S. Tyo, "Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error," Appl. Opt. 41619-630 (2002).
    [CrossRef] [PubMed]
  8. S. Ainouz, J. Zallat, A. de Martino, and C. Collet., "Physical interpretation of polarization-encoded images by color preview," Opt. Express 14, 5916-5927 (2006).
    [CrossRef] [PubMed]
  9. S. N. Savenkov, "Optimization and structuring of the instrument matrix for polarimetric measurements," Opt. Eng. 41, 965-972 (2002).
    [CrossRef]
  10. J. Bernardo and A. Smith, Bayesian Theory, (Wiley, 2000).
  11. A. Gelman, J. Carlin, H. Stern, and D. Rubin, Bayesian data analysis, Second ed., (CRC Press, 2003).
  12. S. Z. Li, Markov random field modeling in image analysis, Second ed., (Springer, 2001).
  13. A. Gray, J. Kay, and D. Titterington, "An empirical study of the simulation of various models used for images," IEEE Trans. PAMI,  16, 507-513 (1994).
    [CrossRef]
  14. A. Dunmur and D. Titterington, "Computational Bayesian analysis of hidden Markov mesh models," IEEE PAMI. 19, 1296-1300 (1997).
    [CrossRef]
  15. S. Richardson and P. Green, "On Bayesian analysis of mixtures with an unknown number of components (with discussion)," J. R. Stat. Soc. Ser. B. 59, 731-792 (1997).
    [CrossRef]
  16. J. Besag, "On the statistical analysis of dirty pictures (with discussion)," J. R. Stat. Soc. B 48, 259-302 (1986).

2006 (2)

S. D. Giattina,  et al., "Assessment of coronary plaque collagen with polarization sensitive optical coherence tomography (PS-OCT)," Int. J. Cardiol. 107, 400-409 (2006).
[CrossRef] [PubMed]

S. Ainouz, J. Zallat, A. de Martino, and C. Collet., "Physical interpretation of polarization-encoded images by color preview," Opt. Express 14, 5916-5927 (2006).
[CrossRef] [PubMed]

2004 (1)

D. Miyazaki, M. Kagesawa, and K. Ikeuchi, "Transparent surface modeling from a pair of polarization images," IEEE Trans. PAMI 26,920-932 (2004).
[CrossRef]

2002 (3)

1999 (1)

J. M. Bueno and P. Artal, "Double-pass imaging polarimetry in the human eye," Opt. Letters. 2464-66 (1999).
[CrossRef]

1997 (2)

A. Dunmur and D. Titterington, "Computational Bayesian analysis of hidden Markov mesh models," IEEE PAMI. 19, 1296-1300 (1997).
[CrossRef]

S. Richardson and P. Green, "On Bayesian analysis of mixtures with an unknown number of components (with discussion)," J. R. Stat. Soc. Ser. B. 59, 731-792 (1997).
[CrossRef]

1994 (1)

A. Gray, J. Kay, and D. Titterington, "An empirical study of the simulation of various models used for images," IEEE Trans. PAMI,  16, 507-513 (1994).
[CrossRef]

1986 (1)

J. Besag, "On the statistical analysis of dirty pictures (with discussion)," J. R. Stat. Soc. B 48, 259-302 (1986).

Ainouz, S.

Artal, P.

J. M. Bueno and P. Artal, "Double-pass imaging polarimetry in the human eye," Opt. Letters. 2464-66 (1999).
[CrossRef]

Besag, J.

J. Besag, "On the statistical analysis of dirty pictures (with discussion)," J. R. Stat. Soc. B 48, 259-302 (1986).

Bueno, J. M.

J. M. Bueno and P. Artal, "Double-pass imaging polarimetry in the human eye," Opt. Letters. 2464-66 (1999).
[CrossRef]

Collet, C.

de Martino, A.

Dunmur, A.

A. Dunmur and D. Titterington, "Computational Bayesian analysis of hidden Markov mesh models," IEEE PAMI. 19, 1296-1300 (1997).
[CrossRef]

Giattina, S. D.

S. D. Giattina,  et al., "Assessment of coronary plaque collagen with polarization sensitive optical coherence tomography (PS-OCT)," Int. J. Cardiol. 107, 400-409 (2006).
[CrossRef] [PubMed]

Gray, A.

A. Gray, J. Kay, and D. Titterington, "An empirical study of the simulation of various models used for images," IEEE Trans. PAMI,  16, 507-513 (1994).
[CrossRef]

Green, P.

S. Richardson and P. Green, "On Bayesian analysis of mixtures with an unknown number of components (with discussion)," J. R. Stat. Soc. Ser. B. 59, 731-792 (1997).
[CrossRef]

Ikeuchi, K.

Kagesawa, M.

D. Miyazaki, M. Kagesawa, and K. Ikeuchi, "Transparent surface modeling from a pair of polarization images," IEEE Trans. PAMI 26,920-932 (2004).
[CrossRef]

Kay, J.

A. Gray, J. Kay, and D. Titterington, "An empirical study of the simulation of various models used for images," IEEE Trans. PAMI,  16, 507-513 (1994).
[CrossRef]

Miyazaki, D.

Richardson, S.

S. Richardson and P. Green, "On Bayesian analysis of mixtures with an unknown number of components (with discussion)," J. R. Stat. Soc. Ser. B. 59, 731-792 (1997).
[CrossRef]

Saito, M.

Sato, Y.

Savenkov, S. N.

S. N. Savenkov, "Optimization and structuring of the instrument matrix for polarimetric measurements," Opt. Eng. 41, 965-972 (2002).
[CrossRef]

Titterington, D.

A. Dunmur and D. Titterington, "Computational Bayesian analysis of hidden Markov mesh models," IEEE PAMI. 19, 1296-1300 (1997).
[CrossRef]

A. Gray, J. Kay, and D. Titterington, "An empirical study of the simulation of various models used for images," IEEE Trans. PAMI,  16, 507-513 (1994).
[CrossRef]

Tyo, J. S.

Zallat, J.

Appl. Opt. (1)

IEEE PAMI. (1)

A. Dunmur and D. Titterington, "Computational Bayesian analysis of hidden Markov mesh models," IEEE PAMI. 19, 1296-1300 (1997).
[CrossRef]

IEEE Trans. PAMI (2)

A. Gray, J. Kay, and D. Titterington, "An empirical study of the simulation of various models used for images," IEEE Trans. PAMI,  16, 507-513 (1994).
[CrossRef]

D. Miyazaki, M. Kagesawa, and K. Ikeuchi, "Transparent surface modeling from a pair of polarization images," IEEE Trans. PAMI 26,920-932 (2004).
[CrossRef]

Int. J. Cardiol. (1)

S. D. Giattina,  et al., "Assessment of coronary plaque collagen with polarization sensitive optical coherence tomography (PS-OCT)," Int. J. Cardiol. 107, 400-409 (2006).
[CrossRef] [PubMed]

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

J. R. Stat. Soc. B (1)

J. Besag, "On the statistical analysis of dirty pictures (with discussion)," J. R. Stat. Soc. B 48, 259-302 (1986).

J. R. Stat. Soc. Ser. B. (1)

S. Richardson and P. Green, "On Bayesian analysis of mixtures with an unknown number of components (with discussion)," J. R. Stat. Soc. Ser. B. 59, 731-792 (1997).
[CrossRef]

Opt. Eng. (1)

S. N. Savenkov, "Optimization and structuring of the instrument matrix for polarimetric measurements," Opt. Eng. 41, 965-972 (2002).
[CrossRef]

Opt. Express (1)

Opt. Letters. (1)

J. M. Bueno and P. Artal, "Double-pass imaging polarimetry in the human eye," Opt. Letters. 2464-66 (1999).
[CrossRef]

Other (5)

D. H. Goldstein, D. B. Chenault, and Society of Photo-optical Instrumentation Engineers, Polarization: measurement, analysis, and remote sensing II, 19-21 July, 1999, Denver, Colorado. 1999, Bellingham, Washington: SPIE. ix, 426 p.

M. H. Smith, "Optimizing a dual-rotating-retarder Mueller matrix polarimeter," in Polarization Analysis and Measurements IV, SPIE (2001).

J. Bernardo and A. Smith, Bayesian Theory, (Wiley, 2000).

A. Gelman, J. Carlin, H. Stern, and D. Rubin, Bayesian data analysis, Second ed., (CRC Press, 2003).

S. Z. Li, Markov random field modeling in image analysis, Second ed., (Springer, 2001).

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Figures (11)

Fig. 1.
Fig. 1.

The directed acyclic graph (DAG) that represents the dependence relations between the variables. According to usual conventions, square boxes represent fixed or observed quantities whereas circles represent quantities to be estimated.

Fig. 2.
Fig. 2.

(a). The label map that was used to generate the simulated Stokes image. The s1 Stokes vector was assigned to black pixels (1215 pixels), s2 was assigned to heavy gray pixels (436 pixels), s3 was assigned to light gray pixels (997 pixels), and S4 was assigned to white pixels (1448 pixels). (b) Red circles indicate the positions of the four polarization states corresponding to the si vectors on the Poincaré sphere.

Fig. 3.
Fig. 3.

From upper left to bottom right, the four Stokes channels (s0 (total intensity), s1, s2, and s3) obtained by the min-norm solution Ŝ LS for a noise variance of 0.01.

Fig. 4.
Fig. 4.

Polarization states locations corresponding to Ŝ LS over the Poincaré sphere for σ 2 n =0.01(a) and σ 2 n = 0.1 (b). Points that lie outside the Poincaré sphere correspond to unphysical estimated states. The ratios of unphysical estimated states are 54% (a) and 57% (b).

Fig. 5.
Fig. 5.

Locations of the true polarization states (circles) and the estimated ones for σ 2 n = 0.01 (+ sign) and for σ 2 n = 0.1 (crosses).

Fig. 6.
Fig. 6.

Estimated noise variance vs. true noise variance. Circles represent the estimates that correspond to the largest class. Horizontal bars correspond to the variations of the estimates over the four classes.

Fig. 7.
Fig. 7.

Locations of the true polarization states (circles) and the estimated ones (crosses) where different noises reach each class.

Fig. 8.
Fig. 8.

Intensity images corresponding to four polarization probing states used to retrieve the Stokes image. Gray values are scaled for display purposes.

Fig. 9.
Fig. 9.

Stokes channels images issued from intensity images of Fig. 8 by min-norm solution. Gray values are scaled for display purposes.

Fig. 10.
Fig. 10.

The label map obtained with the proposed Bayesian approach. Different gray values correspond to different polarization signatures.

Fig. 11.
Fig. 11.

Locations of the estimated polarization states obtained by min-norm solution (a) and by our Bayesian approach (b).

Tables (1)

Tables Icon

Table 1. Estimated Stokes vectors of each class. The last two rows lists the RMSE and the ratio of misclassified pixels for each class.

Equations (31)

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I m j y j x = P ( η ) S in j y j x
I m = { I m j y j x j n ; j y 1 J y , j x [ 1 , J x ] , j n 0 N 1 }
S in = { S in j y j x j e ; j y 1 J y , j x 1 J x , j e 0,3 }
S ̂ in j y j x = P # I m j y j x
I m j y j x = P ( η ) S in j y j x + δ I j y j x
δ I i I i 1 n i ph
n i ph = Q E ( λ ) N i ph
δ I p j y j x = 1 Q E ( λ ) N ph I m j y j x = ρ I m j y j x = ρ PS in j y j x
I m j y j x = ( 1 + ρ ) P ( η ) S in j y j x + ε j y j x = I d j y j x + ε j y j x
I 0 = { I 0 j y j x ; I 0 j y j x [ 1 , K ] , j y 1 J y , j x 1 J x }
Φ = [ ϕ 0,1 ϕ 0,2 ϕ 0 K ϕ 3,1 ϕ 3,2 ϕ 3 K ]
{ p ( I 0 1,1 = k 1 ) = 1 K k 1 = 1 , . . . , K p ( I 0 1 j x = k 1 I 0 1 j x 1 = k 2 ) = { β 0 if k 1 = k 2 for j x > 1 1 β 0 K 1 if k 1 k 2 p ( I 0 j y 1 = k 1 I 0 ( j y 1,1 ) = k 2 ) = { β 0 if k 1 = k 2 for j y > 1 1 β 0 K 1 if k 1 k 2 p ( I 0 j y j x = k 4 I 0 j y 1 j x = k 1 I 0 j y j x 1 = k 2 I 0 j y 1 j x 1 = k 3 ) = { β 0 if k 1 = k 2 = k 3 = k 4 1 β 0 K 1 if k 1 = k 2 = k 3 k 4 for j x > 1 , j y > 1 1 K otherwise
ε j y j x j n I 0 j y j x = k N μ k ( j n ) σ k ( j n )
Θ ( j n ) = [ μ 1 ( j n ) μ 2 ( j n ) μ K ( j n ) σ 1 ( j n ) σ 2 ( j n ) σ K ( j n ) ]
π = p ( I m Θ , Φ , I 0 , ρ ) p ( Θ α 4 , K ) p ( α 4 ) p ( Φ K ) p ( I 0 β 0 , K ) p ( β 0 ) p ( ρ ) p ( K )
{ { p ( I m Θ , Φ , I 0 , ρ ) = j y , j x , j n p ( I m ( j y , j x , j n ) Θ , Φ , I 0 , ρ ) I m j y j x j n Θ , Φ , I 0 , ρ N ( I d j y j x j n + μ I 0 j y j x ( j n ) , σ I 0 j y j x ( j n ) ) ( see eqs . ( 8,12 ) ) { p ( Θ α 4 , K ) = j n , k p ( μ k ( j n ) ) p ( σ k ( j n ) α 4 ) μ k ( j n ) N α 1 α 2 ( σ k ( j n ) ) ‒2 Ga α 3 α 4 α 4 Ga α 41 α 42 { p ( Φ ) = k p ( ϕ ( : , k ) ) p ( ϕ ( : , k ) ) [ j e 1 2 π γ 2 exp ( 1 2 γ 2 ( ϕ j e k γ 1 ) 2 ) ] ( C k ) p ( I 0 β 0 , K ) = [ β 0 ] n 1 [ 1 β 0 K 1 ] n 2 [ 1 K ] n 3 β 0 Be β 1 β 2 ρ Be δ 1 δ 2 K U K min K max
U ( K max , K min ) : p ( z ) = 1 K max K min + 1 ( z K min K max )
Ga λ 1 λ 2 : p ( z ) = λ 2 Γ ( λ 1 ) z λ 1 1 exp ( λ 2 z ) ( z 0 )
Be λ 1 λ 2 ) p ( z ) = Γ ( λ 1 + λ 2 ) Γ ( λ 1 ) Γ ( λ 2 ) z λ 1 1 ( 1 z ) λ 2 1 ( z 0,1 )
π = p ( Θ , Φ , I 0 , ρ , α 4 , K , β 0 I m ) p ( I m )
s ̂ 1 = [ 1.0 0.039 0.03 0.003 ]
s ̂ 2 = [ 1.0 0.472 0.363 0.002 ]
s ̂ 3 = [ 1.0 0.782 0.007 0.025 ]
α 1 = ( max ( I m ) + min ( I m ) ) 2 , α 2 = ( max ( I m ) min ( I m ) ) 3 ,
α 3 = 1.2 ,
α 41 = 0.95 ,
α 42 = 2 ( max ( I m ) min ( I m ) ) 2 ,
β 1 = β 2 = 1 ,
γ 1 = ( max ( S ̂ LS ) + min ( S ̂ LS ) ) 2 , γ 2 = ( max ( S ̂ LS ) min ( S ̂ LS ) ) 3 ,
δ 1 = δ 2 = 1.01 ,
K min = 2 , K max = 10

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