Abstract

One can explicitly retrieve physically realizable Mueller matrices from quantified intensity data even in the presence of noise. This is done by integrating the physical realizability criterion obtained by Givens and Kostinski, [J. Mod. Opt. 40, 471 (1993)], as an active constraint in a global optimization process. Among different global optimization techniques, two of them have been tested and their robustness analyzed: a deterministic approach based on sequential quadratic programming and a stochastic approach based on constrained simulated annealing algorithms are implemented for this purpose. We illustrate the validity of both methods on experimental data and on the inadmissible Mueller matrix given by Howell, [Appl. Opt. 18, No. 6, 808-812 (1979)]. In comparison, the constrained simulated annealing method produced higher accuracy with similar computing time.

© 2008 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  31. S. R. Cloude, "Conditions for the realisability of matrix operators in polarimetry," Proc. SPIE 1166, 177-185 (1989).

2007

2006

2004

2002

2000

1998

P. Spellucci, "A SQP method for general nonlinear programs using only equality constrained subproblems," Math. Program 82, 413-448 (1998).
[CrossRef]

A. V. Gopala Rao,K. S. Mallesh, and Sudha, "On the algebraic characterization of a Mueller matrix in polarization optics," J. Mod. Opt. 45, 955-987 (1998).

E. Landi Degl�??Innocenti and J. C. del Toro Iniesta, "Physical significance of experimental Mueller matrices," J. Opt. Soc. Am. A 15, 533-537 (1998).
[CrossRef]

1996

S. R. Cloude and E. Pottier, "A Review of Target Decomposition Theorems in Radar Polarimetry," IEEE Trans. Geosci. Remote Sens. 34, 498-518 (1996).
[CrossRef]

F. Le Roy-Brehonnet, B. Le Jeune, P. Eliès, J. Cariou, and J. Lotrian, "Optical media characterization by Mueller matrix decomposition," J. Phys. D: Appl. Phys. 29, 34-38 (1996).
[CrossRef]

J. W. Hovenier and C. V. M. van der Mee, "Testing scattering matrices: A compendium of recipes," J. Quant. Spectrosc. Radiat. Transfer 55, 649-661 (1996).
[CrossRef]

1995

J. L. Pezzaniti and R. A. Chipman, "Mueller matrix imaging polarimetry," Opt. Eng. 34, 1558-1568 (1995).
[CrossRef]

1994

D. Anderson and R. Barakat, "Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix," J. Opt. Soc. Am. A. 11, 2305-2319 (1994).
[CrossRef]

1993

C. R. Givens and A. B. Kostinski, "A simple necessary and sufficient condition on physically realizable Mueller matrices," J. Mod. Opt. 40, 471-481 (1993).
[CrossRef]

1992

J. -F. Xing, "On the Deterministic and Non-deterministic Mueller Matrix," J. Mod. Opt. 39, 461-484 (1992).
[CrossRef]

1989

S. R. Cloude, "Conditions for the realisability of matrix operators in polarimetry," Proc. SPIE 1166, 177-185 (1989).

1986

J. W. Hovenier, H. C. van de Hulst, and C. V. M. van der Mee, "Conditions for the elements of the scattering matrix," Astron. Astrophys. 157, 301-310 (1986).

1981

R. Barakat, "Bilinear constraints between elements of the 4�?4 Mueller-Jones transfer matrix of polarization theory," Opt. Commun. 38, 159-161 (1981).
[CrossRef]

1979

B. J. Howell, " Measurements of the polarization effects of an instrument using partially polarized light," Appl. Opt. 18, 808-812 (1979).
[CrossRef]

1978

Aiello, A.

Anderson, D.

D. Anderson and R. Barakat, "Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix," J. Opt. Soc. Am. A. 11, 2305-2319 (1994).
[CrossRef]

Azzam, R. M.

Barakat, R.

D. Anderson and R. Barakat, "Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix," J. Opt. Soc. Am. A. 11, 2305-2319 (1994).
[CrossRef]

R. Barakat, "Bilinear constraints between elements of the 4�?4 Mueller-Jones transfer matrix of polarization theory," Opt. Commun. 38, 159-161 (1981).
[CrossRef]

Bueno, J. M.

Campbell, M. C. W.

Cariou, J.

F. Le Roy-Brehonnet, B. Le Jeune, P. Eliès, J. Cariou, and J. Lotrian, "Optical media characterization by Mueller matrix decomposition," J. Phys. D: Appl. Phys. 29, 34-38 (1996).
[CrossRef]

Chenault, D. B.

Cheney, M.

Chipman, R.

Chipman, R. A.

J. L. Pezzaniti and R. A. Chipman, "Mueller matrix imaging polarimetry," Opt. Eng. 34, 1558-1568 (1995).
[CrossRef]

Cloude, S. R.

S. R. Cloude and E. Pottier, "A Review of Target Decomposition Theorems in Radar Polarimetry," IEEE Trans. Geosci. Remote Sens. 34, 498-518 (1996).
[CrossRef]

S. R. Cloude, "Conditions for the realisability of matrix operators in polarimetry," Proc. SPIE 1166, 177-185 (1989).

DeBoo, B.

Dereniak, E.

Descour, M. R.

Eliès, P.

F. Le Roy-Brehonnet, B. Le Jeune, P. Eliès, J. Cariou, and J. Lotrian, "Optical media characterization by Mueller matrix decomposition," J. Phys. D: Appl. Phys. 29, 34-38 (1996).
[CrossRef]

Elsayed Ahmad, J.

Elsner, A.

Givens, C. R.

C. R. Givens and A. B. Kostinski, "A simple necessary and sufficient condition on physically realizable Mueller matrices," J. Mod. Opt. 40, 471-481 (1993).
[CrossRef]

Goldstein, D. L.

Gopala Rao,, A. V.

A. V. Gopala Rao,K. S. Mallesh, and Sudha, "On the algebraic characterization of a Mueller matrix in polarization optics," J. Mod. Opt. 45, 955-987 (1998).

Hovenier, J. W.

J. W. Hovenier and C. V. M. van der Mee, "Testing scattering matrices: A compendium of recipes," J. Quant. Spectrosc. Radiat. Transfer 55, 649-661 (1996).
[CrossRef]

J. W. Hovenier, H. C. van de Hulst, and C. V. M. van der Mee, "Conditions for the elements of the scattering matrix," Astron. Astrophys. 157, 301-310 (1986).

Howell, B. J.

B. J. Howell, " Measurements of the polarization effects of an instrument using partially polarized light," Appl. Opt. 18, 808-812 (1979).
[CrossRef]

Ikeuchi, K.

D. Miyazaki, K. Kagesawa, and K. Ikeuchi, "Transparent Surface Modeling from a Pair of Polarization Images," IEEE Trans. Pattern Anal. Mach. Intell. 26, 72-83 (2004).

Kagesawa, K.

D. Miyazaki, K. Kagesawa, and K. Ikeuchi, "Transparent Surface Modeling from a Pair of Polarization Images," IEEE Trans. Pattern Anal. Mach. Intell. 26, 72-83 (2004).

Kemme, S. A.

Kostinski, A. B.

C. R. Givens and A. B. Kostinski, "A simple necessary and sufficient condition on physically realizable Mueller matrices," J. Mod. Opt. 40, 471-481 (1993).
[CrossRef]

Landi, E.

Le Jeune, B.

F. Le Roy-Brehonnet, B. Le Jeune, P. Eliès, J. Cariou, and J. Lotrian, "Optical media characterization by Mueller matrix decomposition," J. Phys. D: Appl. Phys. 29, 34-38 (1996).
[CrossRef]

Le Roy-Brehonnet, F.

F. Le Roy-Brehonnet, B. Le Jeune, P. Eliès, J. Cariou, and J. Lotrian, "Optical media characterization by Mueller matrix decomposition," J. Phys. D: Appl. Phys. 29, 34-38 (1996).
[CrossRef]

Lotrian, J.

F. Le Roy-Brehonnet, B. Le Jeune, P. Eliès, J. Cariou, and J. Lotrian, "Optical media characterization by Mueller matrix decomposition," J. Phys. D: Appl. Phys. 29, 34-38 (1996).
[CrossRef]

Mallesh,, K. S.

A. V. Gopala Rao,K. S. Mallesh, and Sudha, "On the algebraic characterization of a Mueller matrix in polarization optics," J. Mod. Opt. 45, 955-987 (1998).

Miyazaki, D.

D. Miyazaki, K. Kagesawa, and K. Ikeuchi, "Transparent Surface Modeling from a Pair of Polarization Images," IEEE Trans. Pattern Anal. Mach. Intell. 26, 72-83 (2004).

Pezzaniti, J. L.

J. L. Pezzaniti and R. A. Chipman, "Mueller matrix imaging polarimetry," Opt. Eng. 34, 1558-1568 (1995).
[CrossRef]

Phipps, G. S.

Pottier, E.

S. R. Cloude and E. Pottier, "A Review of Target Decomposition Theorems in Radar Polarimetry," IEEE Trans. Geosci. Remote Sens. 34, 498-518 (1996).
[CrossRef]

Puentes, G.

Reimer, M.

Sabatke, D. S.

Sasian, J.

Shaw, J. A.

Smithwick, Q.

Spellucci, P.

P. Spellucci, "A SQP method for general nonlinear programs using only equality constrained subproblems," Math. Program 82, 413-448 (1998).
[CrossRef]

Sudha, K. S.

A. V. Gopala Rao,K. S. Mallesh, and Sudha, "On the algebraic characterization of a Mueller matrix in polarization optics," J. Mod. Opt. 45, 955-987 (1998).

Sweatt, W. C.

Takakura, Y.

Tyo, J. S.

van de Hulst, H. C.

J. W. Hovenier, H. C. van de Hulst, and C. V. M. van der Mee, "Conditions for the elements of the scattering matrix," Astron. Astrophys. 157, 301-310 (1986).

van der Mee, C. V. M.

J. W. Hovenier and C. V. M. van der Mee, "Testing scattering matrices: A compendium of recipes," J. Quant. Spectrosc. Radiat. Transfer 55, 649-661 (1996).
[CrossRef]

J. W. Hovenier, H. C. van de Hulst, and C. V. M. van der Mee, "Conditions for the elements of the scattering matrix," Astron. Astrophys. 157, 301-310 (1986).

Voigt, D.

Weber, A.

Woerdman, J. P.

Xing, J. -F.

J. -F. Xing, "On the Deterministic and Non-deterministic Mueller Matrix," J. Mod. Opt. 39, 461-484 (1992).
[CrossRef]

Yevick, D.

Appl. Opt.

Astron. Astrophys.

J. W. Hovenier, H. C. van de Hulst, and C. V. M. van der Mee, "Conditions for the elements of the scattering matrix," Astron. Astrophys. 157, 301-310 (1986).

IEEE Trans. Geosci. Remote Sens.

S. R. Cloude and E. Pottier, "A Review of Target Decomposition Theorems in Radar Polarimetry," IEEE Trans. Geosci. Remote Sens. 34, 498-518 (1996).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell.

D. Miyazaki, K. Kagesawa, and K. Ikeuchi, "Transparent Surface Modeling from a Pair of Polarization Images," IEEE Trans. Pattern Anal. Mach. Intell. 26, 72-83 (2004).

J. Mod. Opt.

J. -F. Xing, "On the Deterministic and Non-deterministic Mueller Matrix," J. Mod. Opt. 39, 461-484 (1992).
[CrossRef]

C. R. Givens and A. B. Kostinski, "A simple necessary and sufficient condition on physically realizable Mueller matrices," J. Mod. Opt. 40, 471-481 (1993).
[CrossRef]

A. V. Gopala Rao,K. S. Mallesh, and Sudha, "On the algebraic characterization of a Mueller matrix in polarization optics," J. Mod. Opt. 45, 955-987 (1998).

J. Opt. Soc. Am. A

J. Opt. Soc. Am. A.

D. Anderson and R. Barakat, "Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix," J. Opt. Soc. Am. A. 11, 2305-2319 (1994).
[CrossRef]

J. Phys. D: Appl. Phys.

F. Le Roy-Brehonnet, B. Le Jeune, P. Eliès, J. Cariou, and J. Lotrian, "Optical media characterization by Mueller matrix decomposition," J. Phys. D: Appl. Phys. 29, 34-38 (1996).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

J. W. Hovenier and C. V. M. van der Mee, "Testing scattering matrices: A compendium of recipes," J. Quant. Spectrosc. Radiat. Transfer 55, 649-661 (1996).
[CrossRef]

Math. Program

P. Spellucci, "A SQP method for general nonlinear programs using only equality constrained subproblems," Math. Program 82, 413-448 (1998).
[CrossRef]

Opt. Commun.

R. Barakat, "Bilinear constraints between elements of the 4�?4 Mueller-Jones transfer matrix of polarization theory," Opt. Commun. 38, 159-161 (1981).
[CrossRef]

Opt. Eng.

J. L. Pezzaniti and R. A. Chipman, "Mueller matrix imaging polarimetry," Opt. Eng. 34, 1558-1568 (1995).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. SPIE

S. R. Cloude, "Conditions for the realisability of matrix operators in polarimetry," Proc. SPIE 1166, 177-185 (1989).

Other

J. R. Huynen, Phenomenological Theory of Radar Targets, PhD. thesis, University of Technology, The Netherlands (1970).

R. Fletcher, Practical Methods of Optimization (Wiley, 1987).

B. W. Wah and T. Wang, in Principles and Practice of Constraint Programming, (Springer, Heidelberg, 1999) Vol. 461.

R. A. Chipman, Handbook of Optics, 2nd ed., M. Bass ed., (McGraw-Hill, 1995) Vol. II.

P. J. M. Laarhoven and E. H. L. Aarts, Simulated annealing: theory and applications (Kluwer Academic Publishers, 1987).

M. Lundy and A. Mees, in Mathematical Programming (Springer, 1986).

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge U. Press, 1985).

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Figures (7)

Fig. 1.
Fig. 1.

Classical imaging polarimeter [17]. LS, incoherent light source; F, filter; P H,V , horizontal and vertical linear polarizers; L1,2, rotating retardation plates; IF, interferential filter.

Fig. 2.
Fig. 2.

Modified Shepp-Logan phantom. Each color corresponds to different linear polarizer angle.

Fig. 3.
Fig. 3.

Mueller matrix of the modified Shepp-Logan phantom with additive Gaussian noise.

Fig. 4.
Fig. 4.

Top view of the Poincaré sphere. The four linear polarizers are represented by small colored spheres where their centers are located at an angle of 2ϕ from the S 1 axis. Left image: results without constrained optimization. Right image: results when employing CSA, all the points that were lying outside the Poincaré sphere are now translated into inside.

Fig. 5.
Fig. 5.

Two extreme cases of possible transitions. Point A will be transformed to point A′ which will induce error reduction. Point B will be transformed to B′ which will result an error increase.

Fig. 6.
Fig. 6.

Convergence test of the CSA algorithm performed on the modified Shepp-Logan phantom. (a) Geometrical definition of the distance δ: which is the minimum distance from the center of the Poincaré sphere to the CSA solution region. (b) Plot between the added noise variance σ 2 with respect to the distance δ; this graph shows that the CSA algorithm converges to an admissible solution (δ≈1) very near the theoretical solution for different levels of added noise.

Fig. 7.
Fig. 7.

Degree of polarization (DoP) map. (a) For Howell’s experiment: the DoP map shows that this matrix is physically inadmissible because its DoP surface protrudes the Poincaré sphere boundaries. (b) The estimated physical matrix retrieved by a CSA optimization. It is clear that the CSA algorithm has estimated an admissible Mueller matrix from Howell’s experiment without altering the overall matrix properties preserved in the shape and orientation of its DoP map.

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

I t AMP .
ε T ε = ( B m l I e ) T ( B m l I e )
= m l T B T B m l 2 m l T B T I e + I e T I e .
m l = ( B T B ) 1 B T I e = B + I e
s 0 0 , s 0 2 ( s 1 2 + s 2 2 + s 3 2 ) 0 .
min m l B m l I e 2
subject   to : { imag ( Σ k ) k = 1 . . . 4 = 0 ; S σ 1 T G S σ 1 0 ; s 0 , σ 1 0 .
( M , λ ) = Residual ( M ) + λ T h ( M ) + 1 2 h ( M ) 2
T ini = max ( ( M i , 1 ) ( M i , 1 ) , h ( M i ) ) .
P r ( X opt , X ) = { exp ( ( X ) ( x opt ) T ) if X = ( M , λ ) . exp ( ( X opt ) ( X ) T ) if X = ( M , λ )
M ϕ = [ 1 cos 2 ϕ sin 2 ϕ 0 cos 2 ϕ cos 2 2 ϕ cos 2 ϕ sin 2 ϕ 0 sin 2 ϕ cos 2 ϕ sin 2 ϕ sin 2 2 ϕ 0 0 0 0 0 ] .
M pr = 1 2 [ 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ]

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