Abstract

We address the optimization of Mueller polarimeters in the presence of additive Gaussian noise and signal-dependent shot noise, which are two dominant types of noise in most imaging systems. We propose polarimeter architectures in which the noise variances on each coefficient of the Mueller matrix are equalized and independent of the observed matrices.

© 2012 OSA

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References

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2010

P. A. Letnes, I. S. Nerbø, L. M. S. Aas, P. G. Ellingsen, and M. Kildemo, “Fast and optimal broad-band Stokes/Mueller polarimeter design by the use of a genetic algorithm,” Opt. Express18, 23,095–23,103 (2010).
[CrossRef]

2009

2008

2007

2006

J. Zallat, S. Ainouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A: Pure Appl. Opt.8, 807–814 (2006).
[CrossRef]

2004

A. N. Langvillea and W. J. Stewart, “The Kronecker product and stochastic automata networks,” J. Comp. Appl. Math.167, 429–447 (2004).
[CrossRef]

B. Laude-Boulesteix, A. De Martino, B. Drévillon, and L. Schwartz, “Mueller Polarimetric Imaging System with Liquid Crystals,” Appl. Opt.43(14), 2824–2832 (2004).
[CrossRef] [PubMed]

2003

2002

2000

1996

1995

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part II,” Opt. Eng.34, 1656–1658 (1995).
[CrossRef]

1988

Aas, L. M. S.

P. A. Letnes, I. S. Nerbø, L. M. S. Aas, P. G. Ellingsen, and M. Kildemo, “Fast and optimal broad-band Stokes/Mueller polarimeter design by the use of a genetic algorithm,” Opt. Express18, 23,095–23,103 (2010).
[CrossRef]

Ahmad, J. E.

Ainouz, S.

J. Zallat, S. Ainouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A: Pure Appl. Opt.8, 807–814 (2006).
[CrossRef]

Ambirajan, A.

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part II,” Opt. Eng.34, 1656–1658 (1995).
[CrossRef]

Azzam, R. M. A.

Chipman, R. A.

De Martino, A.

Drévillon, B.

Ellingsen, P. G.

P. A. Letnes, I. S. Nerbø, L. M. S. Aas, P. G. Ellingsen, and M. Kildemo, “Fast and optimal broad-band Stokes/Mueller polarimeter design by the use of a genetic algorithm,” Opt. Express18, 23,095–23,103 (2010).
[CrossRef]

Elminyawi, I. M.

El-Saba, A. M.

Garcia-Caurel, E.

Goudail, F.

Jeune, B. L.

Kildemo, M.

P. A. Letnes, I. S. Nerbø, L. M. S. Aas, P. G. Ellingsen, and M. Kildemo, “Fast and optimal broad-band Stokes/Mueller polarimeter design by the use of a genetic algorithm,” Opt. Express18, 23,095–23,103 (2010).
[CrossRef]

Kim, Y.-K.

Langvillea, A. N.

A. N. Langvillea and W. J. Stewart, “The Kronecker product and stochastic automata networks,” J. Comp. Appl. Math.167, 429–447 (2004).
[CrossRef]

Laude, B.

Laude-Boulesteix, B.

Lemaillet, P.

Letnes, P. A.

P. A. Letnes, I. S. Nerbø, L. M. S. Aas, P. G. Ellingsen, and M. Kildemo, “Fast and optimal broad-band Stokes/Mueller polarimeter design by the use of a genetic algorithm,” Opt. Express18, 23,095–23,103 (2010).
[CrossRef]

Look, D. C.

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part II,” Opt. Eng.34, 1656–1658 (1995).
[CrossRef]

Lu, S.

Martino, A. D.

Nerbø, I. S.

P. A. Letnes, I. S. Nerbø, L. M. S. Aas, P. G. Ellingsen, and M. Kildemo, “Fast and optimal broad-band Stokes/Mueller polarimeter design by the use of a genetic algorithm,” Opt. Express18, 23,095–23,103 (2010).
[CrossRef]

Rivet, S.

Schwartz, L.

Smith, M. H.

Stewart, W. J.

A. N. Langvillea and W. J. Stewart, “The Kronecker product and stochastic automata networks,” J. Comp. Appl. Math.167, 429–447 (2004).
[CrossRef]

Stoll, M. P.

J. Zallat, S. Ainouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A: Pure Appl. Opt.8, 807–814 (2006).
[CrossRef]

Takakura, Y.

Twietmeyer, K. M.

Tyo, J. S.

Zallat, J.

J. Zallat, S. Ainouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A: Pure Appl. Opt.8, 807–814 (2006).
[CrossRef]

Appl. Opt.

J. Comp. Appl. Math.

A. N. Langvillea and W. J. Stewart, “The Kronecker product and stochastic automata networks,” J. Comp. Appl. Math.167, 429–447 (2004).
[CrossRef]

J. Opt. A: Pure Appl. Opt.

J. Zallat, S. Ainouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A: Pure Appl. Opt.8, 807–814 (2006).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Eng.

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part II,” Opt. Eng.34, 1656–1658 (1995).
[CrossRef]

Opt. Express

K. M. Twietmeyer and R. A. Chipman, “Optimization of Mueller matrix polarimeters in the presence of error sources,” Opt. Express16, 11589–11603 (2008).
[CrossRef] [PubMed]

P. A. Letnes, I. S. Nerbø, L. M. S. Aas, P. G. Ellingsen, and M. Kildemo, “Fast and optimal broad-band Stokes/Mueller polarimeter design by the use of a genetic algorithm,” Opt. Express18, 23,095–23,103 (2010).
[CrossRef]

Opt. Lett.

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Figures (3)

Fig. 1
Fig. 1

tetrahedron obtained from the matrix A1 presented Eq. (29). (a) Top view. (b) Global view

Fig. 2
Fig. 2

Definition of the angles of rotation α and β to generate any regular tetrahedron on the Poincaré sphere from the optimal one presented in Eq. (29).

Fig. 3
Fig. 3

Evolution of the criterion in function of the angles α and β.

Tables (2)

Tables Icon

Table 1 Variance of each coefficient of the Mueller matrix and efficiency criterion values obtained by using different sets of polarization states. Min: Set of polarization states minimizing the criterion ��. Tetra: Set of polarization states forming a regular tetrahedron on the Poincaré sphere defined Eq. (33). TetraMin/max: Optimal set of polarization states forming a regular tetrahedron on the Poincaré sphere defined Eq. (29).

Tables Icon

Table 2 Variance of each coefficient of the Mueller matrix and efficiency criterion values obtained by using different sets of polarization states. Min: Set of polarization states minimizing the criterion �� for M1. Tetra: Set of polarization states forming a tetrahedron on the Poincaré sphere defined Eq. (33). TetraMin/max: Optimal set of polarization states forming a tetrahedron on the Poincaré sphere defined Eq. (29).

Equations (33)

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M = [ M 00 M 01 M 02 M 03 M 10 M 11 M 12 M 13 M 20 M 21 M 22 M 23 M 30 M 31 M 32 M 33 ]
U = 1 2 [ 1 1 1 1 s 1 U s 2 U s 3 U s 4 U ]
I = I 0 B T M A
V I = [ B A ] T V M
V ^ M = { [ B A ] T } 1 V I = [ ( B T ) 1 ( A T ) 1 ] V I
< V ^ M > = [ ( B T ) 1 ( A T ) 1 ] < V I > = V M
Γ V ^ M = [ B 1 A 1 ] T Γ V I [ B 1 A 1 ]
𝒞 ( A , B , V M ) = trace { Γ V ^ M }
𝒞 = trace { [ B 1 A 1 ] [ B 1 A 1 ] T Γ V I } = trace { [ Q B Q A ] T Γ V I }
𝒞 = σ 2 trace { [ Q B Q A ] T } = σ 2 trace { Q B } trace { Q A }
i [ 1 , 16 ] , σ i 2 = σ 2 [ [ B B T ] 1 [ A A T ] 1 ] i , i
var [ M ] = σ 2 [ 1 3 3 3 3 9 9 9 3 9 9 9 3 9 9 9 ]
𝒞 gauss = 10 2 σ 2
[ Γ V I ] i , i = I i = k = 1 16 [ B A ] i , k T [ V M ] k
𝒞 = i = 1 16 j = 1 16 [ Q B Q A ] i , j T [ Γ V I ] i , j
𝒞 = i = 1 16 [ Q B Q A ] i , j k = 1 16 [ B A ] i , k T [ V M ] k = k = 1 16 [ V M ] k [ i = 1 16 [ Q B Q A ] i , i T [ B A ] i , k T ] = V M T V ( A , B )
[ V ( A , B ) ] k = i = 1 16 [ Q B Q A ] i , i [ B A ] i , k T
𝒞 = [ V M ] 1 4 i = 1 16 [ Q B Q A ] i , i + V M T V ( A , B )
M = [ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ]
{ A , B } max M [ V M T V ( A , B ) ] 0
i [ 1 , 16 ] , [ Q B Q A ] i , i = ( 5 2 ) 2
k [ 2 , 16 ] , i = 1 16 [ B A ] i , k T = 0
𝒞 = 10 2 4 [ V M ] 1 + ( 5 2 ) 2 k = 2 16 [ V M ] k [ i = 1 16 [ B A ] i , k T ]
𝒞 min / max poi = 10 2 4 [ V M ] 1
σ i 2 = k = 1 16 [ V M ] k [ n = 1 16 ( [ B 1 A 1 ] n , i ) 2 [ B A ] n , k T ]
σ i 2 = [ V M ] 1 4 n = 1 16 ( [ B 1 A 1 ] n , i ) 2 + k = 2 16 [ V M ] k [ n = 1 16 ( [ B 1 A 1 ] n , i ) 2 [ B A ] n , k T ]
k = 2 16 [ V M ] k [ n = 1 16 ( [ B 1 A 1 ] n , i ) 2 [ B A ] n , k T ] = V M T W A , B i
W A , B k i = [ n = 1 16 ( [ B 1 A 1 ] n , i ) 2 [ B A ] n , k T ]
A 1 = B 1 = 1 2 [ 1 1 1 1 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 1 / 3 ]
= i = 2 16 [ k = 2 16 [ W A , B k i ] 2 ] 2
σ i 2 = [ V M ] 1 4 n = 1 16 ( [ B 1 A 1 ] n , i ) 2
var [ M ] = [ V M ] 1 4 [ 1 3 3 3 3 9 9 9 3 9 9 9 3 9 9 9 ]
A tetra M 1 = B tetra M 1 = [ 1.0000 1.0000 1.0000 1.0000 0.3166 0.1024 0.5438 0.9628 0.6290 0.9420 0.1319 0.1811 0.7100 0.3195 0.8288 0.2077 ]

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