## Estimation of physically realizable Mueller matrices from experiments using global constrained optimization

Optics Express, Vol. 16, Issue 18, pp. 14274-14287 (2008)

http://dx.doi.org/10.1364/OE.16.014274

Acrobat PDF (3297 KB)

### 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

## 1. Introduction

1. J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. **45**, 5453–5469 (2006). [CrossRef] [PubMed]

2. J. M. Bueno and M. C. W. Campbell, “Confocal scanning laser ophthalmoscopy improvement by use of Mueller-matrix polarimetry,” Opt. Lett. **27**, 830–832 (2002). [CrossRef]

3. A. Weber, M. Cheney, Q. Smithwick, and A. Elsner, “Polarimetric imaging and blood vessel quantification,” Opt. Express **12**, 5178–5190 (2004). [CrossRef] [PubMed]

5. J. L. Pezzaniti and R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. **34**, 1558–1568 (1995). [CrossRef]

*L*

^{1},

*L*

^{2},

*L*

^{∞}or Frobenius norm depending upon configuration. The major shortcoming within these conventional methods is their inadequacy to preserve the solution stability to perturbations and systematic errors. Therefore, in the presence of experimental noise, the situation regularly degenerates and reconstructed Mueller matrices often fail to be physically admissible [6

6. B. J. Howell, “ Measurements of the polarization effects of an instrument using partially polarized light,” Appl. Opt. **18**, 808–812 (1979). [CrossRef]

8. 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]

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

## 2. Experimental and theoretical considerations

### 2.1. Polarimetric system overview

**P**be the complete matrix that characterizes the Stokes parameters of the polarization state generator (PSG) and let

**A**be the complete matrix representing an elliptical diattenuator known as polarization state analyzer (PSA). The expected intensity matrix recorded by the observation system is given by:

**I**

*are obtained from averaging on pixels corresponding to the entrance pupil of the camera. Such values should be representative of the radiance received by the observation system. When the system is calibrated, i.e.,*

_{t}**A**and

**P**are known,

**M**can be easily determined from at least 16 measurements. Practically, it is recommended to have an over-determined system of measurements [16

16. M. Reimer and D. Yevick, “Least-squares analysis of the Mueller matrix,” Opt. Lett. **31**, 2399–2401 (2006). [CrossRef] [PubMed]

*I*∝

_{t}**B**

*m*, where

_{l}*m*is a reshape column vector of the Mueller matrix. The elements of matrix

_{l}**B**are obtained by identification with relation (1), they are directly connected to the Kronecker product of the analyzer + polarizer Stokes parameters,

**B**=

**P**

*⊗*

^{T}**A**.

*ε*=

*I*-

_{t}*I*=

_{e}**B**

*m*-

_{l}*I*. The minimization of the least-squares error is done by computing the solution vector

_{e}*m*that minimizes the error module norm ‖

_{l}*ε*‖

^{2}=

*ε*

^{T}

*ε*. Hence, we can write:

*m*we get

_{l}**B**

^{T}

**B**

*-*

_{ml}**B**

^{T}

*I*=0, thus the choice of the matrix that minimizes the least-squares error is:

_{e}**B**

^{+}is the pseudo-inverse of matrix

**B**. One clearly notices that the linear algorithm described above is devoted exclusively to calculate the solution that minimizes the residual error norm ‖

**B**

*m*-

_{l}*I*‖. This conventional approach cannot guarantee the admissibility of the solution because this specification was not integrated within optimization.

_{e}### 2.2. Physical realizability

*S*=[

*s*

_{ο},

*s*

_{1},

*s*

_{2},

*s*

_{3}]

^{T}represent an arbitrary Stokes vector. A given 4×4 real matrix is considered a physical Mueller matrix if for each possible incident physical Stokes vector

*S*the emergent Stokes vector

_{in}*S*=

_{e}**M**

*S*is also physical [19

_{in}19. 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]

*S*,∑

*k*]=eig(

**GM**

^{T}**GM**), where ∑

_{k=1...4}=diag(σ

_{1},σ

_{2},σ

_{3},σ

_{4}) has been arranged in the descending order of eigenvalues. The rigorous extraction of an admissible Mueller matrix can be realized by implementing a non-linear constrained optimization instead of simple matrix inversion.

*S*

_{σ1}corresponding to the largest eigenvalue is a physical Stokes vector. They are direct implementation of relation (4). In the remaining part of the paper we will denote the equality constraint by

*h*(

**M**) and the two inequality constraints by

*g*(

_{j}**M**) where

*j*={1,2}.

## 3. Global optimization setups

### 3.1. Initialization process

**H**corresponding to the Mueller matrix extracted from the PI method. This coherency matrix is then decomposed using target decomposition [21, 31] into

**H**=

**UDU**

^{†}, where

**D**is a diagonal matrix arranged in the descending order of positive eigenvalues,

**U**is a unitary matrix of corresponding eigenvectors and

**U**

^{†}is its transpose conjugate.

**D**. We suggest setting this value to zero and recalculating the matrix

**H**

*and to recalculate from this new coherency matrix, the Mueller matrix*

_{i}**M**

*which could be used as an initial estimate.*

_{i}### 3.2. Sequential quadratic programming

22. P. Spellucci, “ A SQP method for general nonlinear programs using only equality constrained subproblems,” Math. Prog. , 413–448 (Springer, 1998). [CrossRef]

### 3.3. Constrained simulated annealing

**M**and the Lagrangian

*λ*, satisfying both minimum objective function and constraints realization. The saddle point can be reached by carrying out probabilistic descents in the variable space (Mueller coefficients) and probabilistic local ascents in the Lagrange-multiplier space.

**Algorithm 1**solves only equality constraints, in the form of

*h*(

_{i}**M**)=0, with the following augmented Lagrangian function:

*g*(

_{j}**M**) ≤ 0 can always be transformed into equality constraints using a maximum function,

*g̃*(

_{j}**M**)=max(0,

*g*(

_{j}**M**))=0.

**Algorithm 1**represents the proposed global search for solving Eq. (5). The algorithm starts by regrouping all necessary parameters to be initiated. A physical starting point

*X*=(

_{i}**M**

*,*

_{i}*λ*) is selected after calculating

**M**

*from target decomposition and setting*

_{i}*λ*=0. At the beginning, optimum value

*X*is considered identical to

_{opt}*X*.

_{i}*T*is initialized to be large enough in order to allow several search directions to be accepted at the beginning. In this setup, initial temperature is obtained first by calculating

**M**

*and randomly generating 100 corresponding neighbors*

_{i}**M**′

*, then setting the initial temperature to take the maximum value between the Lagrangian and constraints functions:*

_{i}*α*after looping

*N*times for the same temperature

_{T}*T*. Theoretically, if

*T*is reduced very slowly CSA will converge to a constrained global minimum [26]. Unfortunately, a very slowly cooled temperature is time-consuming and thus duration for treating a large number of inadmissible pixels may become extensively long. It appears that selecting a polynomial cooling schedule is very consistent and reliable when dealing with Mueller 16-variables space [27].

*X*=(

_{opt}**M**,λ=0) and

*T*=

*T*

_{ini}*X*)

_{opt}**Algorithm 1**searches for a feasible neighbor around the optimal estimate

*X*. It searches first for a neighbor in the variable space by performing small perturbations to the Mueller matrix. If these perturbations were incapable of generating feasible points then it carries out probabilistic ascents of 𝓛(

_{opt}**M**,

*λ*) with respect to

*λ*for a fixed value of

**M**in order to increase the penalty of violated constraints and to force them into satisfaction. A local examination in the Lagrangian space can be viewed as a global search in variable space.

*X*′=(

**M**′,

*λ*) or

*X*′=(

**M**,

*λ*′). In the first case we set

**M**′=

**M**+

*α*ʘ

_{τ}*N*(0, σ

*), where*

_{i}*α*is a 4×4 scaling matrix generated from a seed distribution that decreases with time; 𝒩(0,

_{τ}*σ*) is a varying Gaussian distribution with a randomly generated variance, and ʘ is the Hadamard product. The scaling matrix

_{i}*α*provides us with an adaptive varying step algorithm based on the experimental observation that the CSA algorithm needs larger searching steps at the beginning and as time passes the algorithm will approach a region near the optimum value. At this point, for a proper convergence, smaller steps in the searching directions will be more appropriate.

_{τ}*X*′=(

**M**,

*λ*′) we apply the following:

*λ*′=

*λ*+

*βψ*, where

*β*is a random variable uniformly generated in the range of [-1

1. J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. **45**, 5453–5469 (2006). [CrossRef] [PubMed]

1. J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. **45**, 5453–5469 (2006). [CrossRef] [PubMed]

*ψ*is the maximum value of constraints violation. We set the ratio of generating (

**M**′,

*λ*) and (

**M**,λ

*′*) to be 20

*n*to

*m*, where

*n*is the number of variables and

*m*is the number of constraints [24], meaning that

**M**is updated more frequently than

*λ*.

*X*, it generates a physical trial candidate

_{opt}*X*′ in the vicinity of Xopt. If 𝓛(X′) < 𝓛(

*X*) then

_{opt}*X*′ is accepted with a probability of one as a starting point for the next iteration. Otherwise,

*X*′ is accepted with probability:

**M**or

*λ*is changed in

*X*′.

## 4. Simulated case study

### 4.1. Modified Shepp-Logan phantom

*φ*with respect to the vertical optical axis, Fig. 2.

### 4.2. Physical interpretation

*SNR*≈30dB, Frobenius norm of the error between theoretical and synthesized data is about 8.30%. The CSA algorithm reduces such an error to 7.78%: an improvement can be noticed.

*A*lying inside the outer hemisphere and having high error percentage (Euclidean distance

*d*1) compared to the theoretical point

*C*could be transformed to a point

*A*′ inside the physical hemisphere. This operation will result in an error decrease between the admissible and the inadmissible Mueller matrices. On the contrary, CSA could transform an inadmissible point

*B*with modest percentage error to a new admissible point

*B*′ with larger percentage error.

*SNR*of 30 dB.

### 4.3. CSA convergence check

*σ*

^{2}. The test was carried out as follows: First, the modified Shepp-Logan phantom is used to generate the intensity data. Second, Gaussian noise with controlled variance is added to these intensity images. Third, we retrieve the noisy Mueller image from the noisy intensity measurements. Finally, we run the CSA algorithm for each Mueller matrix associated to each pixel within the Mueller image.

*δ*, it is the nearest distance from each class to the center of the Poincaré sphere. This distance can be directly interpreted as the worst convergent point of the algorithm. If this distance is very far from the center of the sphere,

*δ*≈1, then the algorithm has attained the convergence. Otherwise the algorithm is not converging and thus the estimated Mueller matrix is quite wrong.

*σ*

^{2}=2×10

^{-3}the algorithm converges to

*δ*=0.97. Even for large noise values in the order of

*σ*

^{2}=3×10

^{-2}the CSA algorithm is converging near the theoretical (optimum) solution,

*δ*=0.92.

## 5. Experimental results

### 5.1. Mueller matrix of the air

**M**is then obtained by inverting relation (1),

**M**=

**A**#

**I**

_{t}**P**#. Where # represents the matrix inverse when

**A**and

**P**are square matrices, or it represents the matrix Pseudo-Inverse when

**A**and

**P**are rectangular matrices.

29. Y. Takakura and J. Elsayed Ahmad, “Noise distribution of Mueller matrices retrieved with active rotating polarimeters,” Appl. Opt. **46**, 7354–7364 (2007). [CrossRef] [PubMed]

**M**

*, is not admissible: ∑={1.021,0.997,0.985,0.963} and*

_{PI}**M**

*-*

_{PI}**M**

*‖/‖*

_{opt}**M**

*+*

_{PI}**M**

*‖ between the physical matrix and the inadmissible one is 0.62%. The error in the physical matrix is slightly larger than the inadmissible matrix. This result highlights the fact that there exists an error increase penalty when using global optimization. In fact, the experimental polarimeter that we dispose of is very accurate, in addition we have estimated the Mueller matrix based on 8×8 measurements which is more precise than methods based on 16 acquisitions. The contribution of this method in noise reduction is more appreciated if the signal-to-noise ratio is lower than the threshold of 30 dB highlighted in the previous section.*

_{opt}### 5.2. Howell’s matrix

**A**and

**P**to become invertible, i.e.,

**B**becomes invertible [30

30. D. S. Sabatke, M. R. Descour, E. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, “ Optimization of retardance for complete Stokes polarimeter,” Opt. Lett. **25**, 802–804 (2000). [CrossRef]

_{1}and L

_{2}that minimizes the equal weighted variance, EWV=Trace [(

**B**

^{+})

^{T}**B**

^{+}]. A suitable choice of retardation orientation angles based on minimizing the EWV criterion will eliminate numerical errors that may arise from ill-conditioned matrix inversion.

*I*=

_{e}**B**

*m*, is simulated. After that, minimization algorithm SQP or CSA are conducted searching for the optimum vector

_{h}*m*that minimizes the residual error ‖

_{l}**B**

*m*-

_{l}*I*‖ provided that the solution must not violate the admissibility conditions explicated in Eq. (6). The admissible estimate given to initiate the process is extracted from target decomposition in the same way as explained earlier in the paper.

_{e}**M**

*and*

_{H}**M**

*is 5.2%.*

_{SQP}**M**

*matrix is 3.3%. It is believed that by using the CSA technique we have attained a physical saddle point closer to the global minimum of the objective function than the SQP method.*

_{H}**M**:

_{pr}## 6. Conclusion

*SNR*of 30 dB.

## References and links

1. | J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. |

2. | J. M. Bueno and M. C. W. Campbell, “Confocal scanning laser ophthalmoscopy improvement by use of Mueller-matrix polarimetry,” Opt. Lett. |

3. | A. Weber, M. Cheney, Q. Smithwick, and A. Elsner, “Polarimetric imaging and blood vessel quantification,” Opt. Express |

4. | D. Miyazaki, K. Kagesawa, and K. Ikeuchi, “Transparent Surface Modeling from a Pair of Polarization Images,” |

5. | J. L. Pezzaniti and R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. |

6. | B. J. Howell, “ Measurements of the polarization effects of an instrument using partially polarized light,” Appl. Opt. |

7. | 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. |

8. | J. W. Hovenier and C. V. M. van der Mee, “Testing scattering matrices: A compendium of recipes,” J. Quant. Spectrosc. Radiat. Transfer |

9. | J. -F. Xing, “ On the Deterministic and Non-deterministic Mueller Matrix,” J. Mod. Opt. |

10. | E. Landi Degl’Innocenti and J. C. del Toro Iniesta, “Physical significance of experimental Mueller matrices,” J. Opt. Soc. Am. A |

11. | C. R. Givens and A. B. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt. |

12. | S. R. Cloude and E. Pottier, “A Review of Target Decomposition Theorems in Radar Polarimetry,” |

13. | 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. |

14. | A. Aiello, G. Puentes, D. Voigt, and J. P. Woerdman, “ Maximum-likelihood estimation of Mueller matrices,” Opt. Lett. |

15. | A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics, ” J. Mod. Opt. |

16. | M. Reimer and D. Yevick, “Least-squares analysis of the Mueller matrix,” Opt. Lett. |

17. | R. M. Azzam, “ Photopolarimetric measurement of the Mueller matrix by Fourier analysis of a single detected signal,” Opt. Lett. |

18. | R. A. Horn and C. R. Johnson, |

19. | 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. |

20. | R. Barakat, “Bilinear constraints between elements of the 4×4 Mueller-Jones transfer matrix of polarization theory,” Opt. Commun. |

21. | J. R. Huynen, |

22. | P. Spellucci, “ A SQP method for general nonlinear programs using only equality constrained subproblems,” Math. Prog. , 413–448 (Springer, 1998). [CrossRef] |

23. | R. Fletcher, |

24. | B. W. Wah and T. Wang, in |

25. | R. A. Chipman, |

26. | P. J. M. Laarhoven and E. H. L. Aarts, |

27. | M. Lundy and A. Mees, in |

28. | B. DeBoo, J. Sasian, and R. Chipman, “Degree of polarization surfaces and maps for analysis of depolarization,” Opt. Express , |

29. | Y. Takakura and J. Elsayed Ahmad, “Noise distribution of Mueller matrices retrieved with active rotating polarimeters,” Appl. Opt. |

30. | D. S. Sabatke, M. R. Descour, E. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, “ Optimization of retardance for complete Stokes polarimeter,” Opt. Lett. |

31. | S. R. Cloude, “ Conditions for the realisability of matrix operators in polarimetry,” in Polarization Considerations for Optical Systems II, Proc. SPIE1166, 177–185 (1989). |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(120.4820) Instrumentation, measurement, and metrology : Optical systems

(120.5410) Instrumentation, measurement, and metrology : Polarimetry

(230.5440) Optical devices : Polarization-selective devices

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: July 9, 2008

Revised Manuscript: August 15, 2008

Manuscript Accepted: August 16, 2008

Published: August 28, 2008

**Citation**

Jawad Elsayed Ahmad and Yoshitate Takakura, "Estimation of physically realizable
Mueller matrices from experiments
using global constrained optimization," Opt. Express **16**, 14274-14287 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-18-14274

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### References

- J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, "Review of passive imaging polarimetry for remote sensing applications," Appl. Opt. 45, 5453-5469 (2006). [CrossRef] [PubMed]
- J. M. Bueno and M. C. W. Campbell, "Confocal scanning laser ophthalmoscopy improvement by use of Muellermatrix polarimetry," Opt. Lett. 27, 830-832 (2002). [CrossRef]
- A. Weber, M. Cheney, Q. Smithwick, and A. Elsner, "Polarimetric imaging and blood vessel quantification," Opt. Express 12, 5178-5190 (2004). [CrossRef] [PubMed]
- 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. L. Pezzaniti and R. A. Chipman, "Mueller matrix imaging polarimetry," Opt. Eng. 34, 1558-1568 (1995). [CrossRef]
- B. J. Howell, " Measurements of the polarization effects of an instrument using partially polarized light," Appl. Opt. 18, 808-812 (1979). [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).
- 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. -F. Xing, "On the Deterministic and Non-deterministic Mueller Matrix," J. Mod. Opt. 39, 461-484 (1992). [CrossRef]
- 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]
- 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]
- 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]
- A. Aiello, G. Puentes, D. Voigt, and J. P. Woerdman, "Maximum-likelihood estimation of Mueller matrices," Opt. Lett. 31, 817-819 (2006). [CrossRef] [PubMed]
- 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).
- M. Reimer and D. Yevick, "Least-squares analysis of the Mueller matrix," Opt. Lett. 31, 2399-2401 (2006). [CrossRef] [PubMed]
- R. M. Azzam, "Photopolarimetric measurement of the Mueller matrix by Fourier analysis of a single detected signal," Opt. Lett. 2, 148-150 (1978). [CrossRef] [PubMed]
- R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge U. Press, 1985).
- 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]
- J. R. Huynen, Phenomenological Theory of Radar Targets, PhD. thesis, University of Technology, The Netherlands (1970).
- P. Spellucci, "A SQP method for general nonlinear programs using only equality constrained subproblems," Math. Program 82, 413-448 (1998). [CrossRef]
- 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).
- B. DeBoo, J. Sasian, and R. Chipman, "Degree of polarization surfaces and maps for analysis of depolarization," Opt. Express 12, 4941-4958 (2004). [CrossRef] [PubMed]
- Y. Takakura and J. Elsayed Ahmad, "Noise distribution of Mueller matrices retrieved with active rotating polarimeters," Appl. Opt. 46, 7354-7364 (2007). [CrossRef] [PubMed]
- D. S. Sabatke, M. R. Descour, E. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, " Optimization of retardance for complete Stokes polarimeter," Opt. Lett. 25, 802-804 (2000). [CrossRef]
- S. R. Cloude, "Conditions for the realisability of matrix operators in polarimetry," Proc. SPIE 1166, 177-185 (1989).

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