## Mueller matrix roots algorithm and computational considerations |

Optics Express, Vol. 20, Issue 1, pp. 17-31 (2012)

http://dx.doi.org/10.1364/OE.20.000017

Acrobat PDF (1331 KB)

### Abstract

Recently, an order-independent Mueller matrix decomposition was proposed in an effort to elucidate the nine depolarization degrees of freedom [*Handbook of Optics*, Vol. 1 of *Mueller Matrices* (2009)]. This paper addresses the critical computational issues involved in applying this Mueller matrix roots decomposition, along with a review of the principal matrix root and common methods for its calculation. The calculation of the *p*th matrix root is optimized around *p* = 10^{5} for a 53 digit binary double precision calculation. A matrix roots algorithm is provided which incorporates these computational results. It is applied to a statistically significant number of randomly generated physical Mueller matrices in order to gain insight on the typical ranges of the depolarizing Matrix roots parameters. Computational techniques are proposed which allow singular Mueller matrices and Mueller matrices with a half-wave of retardance to be evaluated with the matrix roots decomposition.

© 2011 OSA

## 1. Introduction

1. S.-Y. Lu and R. A. Chipman, “Interpretation of mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A **13**, 1106–1113 (1996). [CrossRef]

2. R. Ossikovski, “Analysis of depolarizing mueller matrices through a symmetric decomposition,” J. Opt. Soc. Am. A **26**, 1109–1118 (2009). [CrossRef]

*N*-matrices [5

5. R. C. Jones, “A new calculus for the treatment of optical systems. vii. properties of the n-matrices,” J. Opt. Soc. Am. **38**, 671–683 (1948). [CrossRef]

**M**into

*p*infinitesimal slices: as illustrated in Fig. 1, where

**N**is the Mueller matrix for one small slice. When

*p*becomes very large, the polarization properties of

*p*approaches infinity, the principal

*p*th root of a large class of Mueller matrices approaches the identity matrix

**I**, In this study, the class of Mueller matrices that obey Eq. (2) are called

*uniform*Mueller matrices. The Mueller matrix roots decomposition from [4] is applicable to this subset of Mueller matrices, with the exception of the non-uniform special cases highlighted in section 4.

*p*th root of square matrices is an extensively studied subject [6

6. N. J. Higham, *Functions of Matrices: Theory and Computation* (Society for Industrial and Applied Mathematics, 2008). [CrossRef]

8. C.-H. Guo, “On newton’s method and halley’s method for the principal pth root of a matrix,” Linear Algebra Appl. **432**, 1905–1922 (2010). [CrossRef]

*p*th root must be optimized with all of these topics in mind. This paper reviews and updates the Mueller matrix roots decomposition from [4], addresses several computational issues, and highlights several common non-uniform special cases that arise. Considerations for these computational issues and special cases are implemented in a matrix roots decomposition algorithm, which is applied in a statistical analysis of a large quantity of randomly generated physical Mueller matrices.

## 2. Matrix roots decomposition

**M**, a Mueller matrix

**N**with infinitesimal polarization properties is first calculated from the

*p*th principal root of

**M**, where

*p*is some large integer (typically 10

^{5}), The appropriate choice of

*p*is discussed in detail in section 5. If

*p*is too small, the Mueller matrix roots decomposition fails to be order-independent.

*d*

_{0}through

*d*

_{15}are defined from the symmetric and antisymmetric parts of N: where

*d*

_{13},

*d*

_{14}, and

*d*

_{15}are solved from the first-order generator products in terms of the parameters

*f*

_{13},

*f*

_{14}, and

*f*

_{15}: The infinitesimal polarization parameters

*d*

_{0}through

*d*

_{15}are rescaled by

*p*to produce the matrix roots parameters

*D*

_{0}through

*D*

_{15}:

*D*

_{0}through

*D*

_{15}parameterize the sixteen degrees of freedom of

**M**.

*D*

_{1},

*D*

_{2},

*D*

_{3}, and three matrix roots parameters for retardance,

*D*

_{4},

*D*

_{5},

*D*

_{6}. The three degrees of freedom for each property correspond to the axes in the Stokes/Mueller formalism (horizontal/vertical, 45°/135°, right/left circular).

## 3. *p*th root

*p*th matrix root along with relevant examples and common methods of calculating the

*p*th matrix root. This topic is of critical importance for calculating the Matrix roots, since matrices have multiple roots.

**A**∈ ℂ

*(in complex space) with*

^{n×n}*s*distinct eigenvalues, there are precisely

*p*

^{s}*p*th roots [6

6. N. J. Higham, *Functions of Matrices: Theory and Computation* (Society for Industrial and Applied Mathematics, 2008). [CrossRef]

**A**∈ ℂ

*has no negative, real eigenvalues, there is a unique*

^{n×n}*p*th root of

**A**whose eigenvalues’ arguments lie between −

*π*/

*p*and

*π*/

*p*, and that unique root is defined as the principal root of

**A**[6

6. N. J. Higham, *Functions of Matrices: Theory and Computation* (Society for Industrial and Applied Mathematics, 2008). [CrossRef]

**A**is real, then its principal root

**A**

^{1/p}is real.

*π*retardance are discussed in section 4.2.

### 3.1. Principal matrix root algorithms

**A**and compute a

*p*th root of the resulting upper triangular factor using various (stable) recursive formulae [13

13. N. J. Higham, “Stable iterations for the matrix square root,” Num. Algor. **15**, 227–242 (1997). [CrossRef]

*p*th root of

**A**using an iterative approach [6

*Functions of Matrices: Theory and Computation* (Society for Industrial and Applied Mathematics, 2008). [CrossRef]

14. M. H. Smith, “Optimization of a dual-rotating-retarder mueller matrix imaging polarimeter,” Appl. Opt. **41**, 2488–2493 (2002). [CrossRef] [PubMed]

14. M. H. Smith, “Optimization of a dual-rotating-retarder mueller matrix imaging polarimeter,” Appl. Opt. **41**, 2488–2493 (2002). [CrossRef] [PubMed]

7. D. A. Bini, N. J. Higham, and B. Meini, “Algorithms for the matrix pth root,” Num. Algor. **39**, 349–378 (2005). [CrossRef]

*Functions of Matrices: Theory and Computation* (Society for Industrial and Applied Mathematics, 2008). [CrossRef]

7. D. A. Bini, N. J. Higham, and B. Meini, “Algorithms for the matrix pth root,” Num. Algor. **39**, 349–378 (2005). [CrossRef]

**A**into the product of three matrices, where

*λ*are the eigenvalues of

_{i}**A**, the columns of

**Z**are its eigenvectors, and diag(

*λ*) is a diagonal matrix with its

_{i}*i*th diagonal element equal to

*λ*. Then the matrix root of

_{i}**A**is calculated as This method often (but not always) yields the principal root of a diagonalizable Mueller matrix, so long as its principal root exists. However if

**A**is real and has some complex eigenvalues, then the computed

**A**

^{1/p}, which should be real, may acquire a tiny imaginary part due to computational rounding errors. This imaginary part should be discarded. However, numerical instability can produce a large spurious imaginary part, so diagonalization-based computations of matrices with any imaginary eigenvalues should be treated with care [6

*Functions of Matrices: Theory and Computation* (Society for Industrial and Applied Mathematics, 2008). [CrossRef]

## 4. Special cases

### 4.1. Polarizers

15. S.-Y. Lu and R. A. Chipman, “Homogeneous and inhomogeneous jones matrices,” J. Opt. Soc. Am. A **11**, 766–773 (1994). [CrossRef]

*p*th root is calculated. For example, the formula for a homogeneous linear polarizer as a function of orientation

*θ*is Note that and thus The linear polarizer Mueller matrix is its own root to all orders, and the roots do not approach the identity matrix.

*T*, 0), where

_{max}*T*is the maximum transmission, and the minimum transmission

_{max}*T*is 0. The polarizer can be replaced with a nearby uniform Mueller matrix by adjusting the maximum and minimum transmission by a small number

_{min}*ɛ*. The partial polarizer matrix is now uniform, since the partial polarizer is a diattenuator with the same orthogonal eigenpolarizations, but the eigenvalues associated with its physical eigenpolarizations are (

*T*–

_{max}*ɛ*,

*ɛ*).

*η*and orientation

*θ*of the state of maximum transmission on the Poincare sphere from the polarizer’s three dimensional diattenuation vector {

*d*,

_{H}*d*

_{45},

*d*} [1

_{R}1. S.-Y. Lu and R. A. Chipman, “Interpretation of mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A **13**, 1106–1113 (1996). [CrossRef]

*η*and

*θ*are then plugged in to the general formula for an elliptical diattenuator [4, 16

16. D. Goldstein, *Polarized Light*, 2nd ed. (Marcel Dekker, 2003). [CrossRef]

*T*= 1 and

_{max}*T*= 0. The

_{min}*p*th root of this partial polarizer approaches the identity matrix for large

*p*- after replacing the polarizer with the nearby partial polarizer, the Mueller matrix roots decomposition algorithm can proceed as in the general case.

### 4.2. Half wave retarders

*p*th root of an ideal homogeneous linear retarder

**LR**(

*δ*,

*θ*) with retardance

*δ*at orientation

*θ*is A retarder with a half-wave of retardance has negative, real eigenvalues, and therefore no principal root, so the half-wave retarder must be treated as a special case.

*λ*= {−1,−1,1,1}), so the half wave retarder has no principal

*p*th root. For any value of

*p*, two eigenvalues of

**HWR**

^{1/p}are (−1)

*with arguments of −*

^{p}*π*/

*p*and

*π*/

*p*, which lie on the edge of the principal root segment defined in section 3. While the desired solution in this case is not a principal root, it is the solution which approaches the identity matrix. The half wave retarder’s uniform

*p*th root can be calculated analogously to the polarizer case, by means of a small perturbation.

*θ*and latitude

*η*can be calculated from the three-dimensional retardance vector {

*δ*,

_{H}*δ*

_{45},

*δ*} [1

_{R}1. S.-Y. Lu and R. A. Chipman, “Interpretation of mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A **13**, 1106–1113 (1996). [CrossRef]

*δ*, orientation

*θ*, and latitude

*η*can be written in terms of three linear retarders as where the general form for the linear retarder

**LR**(

*δ*,

*θ*) with retardance

*δ*and an orientation of

*θ*is [4]

*θ*and

*η*are calculated from the retardance vector, the half-wave retarder is perturbed by some small number (

*ɛ*≈ 10

^{−7}) to a nearby elliptical retarder

**ER**′ of retardance (

*π*–

*ɛ*) by using equation 21: This is a unitary transformation - therefore any higher order root of a half wave retarder can be calculated as follows: After following this procedure, the matrix roots of the perturbed half-wave retarders can be calculated without issue.

### 4.3. Depolarizing non-uniform Mueller matrices

*a*,

*b*, and

*c*has eigenvalues

*λ*= {1,

*a*,−

*b*, −

*c*}, and therefore no principal root. For cases where

*b*=

*c*, the perturbation approach used in section 4.2 can be modified to yield a uniform Mueller matrix. When

*b*≠

*c*, the perturbation approach fails to yield non-negative eigenvalues, and therefore the matrix will remain non-uniform.

**13**, 1106–1113 (1996). [CrossRef]

**M**

_{Δ}is a depolarizing Mueller matrix,

**M**

*is a pure retarder, and*

_{R}**M**

*is a pure diattenuator. The half-wave retarder found in the*

_{D}**M**

*term is perturbed to the nearby retarder*

_{R}**M**′

*with (*

_{R}*π*–

*ɛ*) retardance as shown in equation 24. Then the terms are recombined to form

**M**′, where The eigenvalues of

**M**′ are no longer negative, since

**M**

_{Δ}and

**M**

*always have positive eigenvalues [1*

_{D}**13**, 1106–1113 (1996). [CrossRef]

**M**′

*are positive. Therefore the uniform*

_{R}*p*th root of the perturbed depolarizing half-wave retarder from Eq. (28) can be calculated from

**M**′, and its Mueller matrix roots decomposition parameters can be found.

*D*of magnitude

*D*= 1 and a depolarization index less than 1.

**13**, 1106–1113 (1996). [CrossRef]

*ɛ*to the nearby uniform diattenuator (or partial polarizer) according to the procedure in section 4.1. The modified

**M**′

*is substituted into Eq. (29) in place of*

_{D}**M**

*, and the three matrices are recombined to form a modified, non-singular matrix*

_{D}**M**′: Then the

*p*th root of

**M**′ can be calculated, and its Mueller matrix roots decomposition parameters can be found.

## 5. Numerical accuracy and root order

### 5.1. Choice of p

*p*is an important consideration when calculating the matrix roots parameters.

*p*must be large enough so that the Mueller matrix elements become sufficiently small for the polarization properties to separate and achieve and order-independent decomposition. However, the magnitude of

*p*should also be as small as possible so as to minimize unnecessary loss of numerical precision from the calculation.

#### 5.1.1. Accuracy

*p*th root of a retarder Mueller matrix with retardance

*δ*results in a

*p*th principal matrix root with retardance

*δ*/

*p*. Therefore a Mueller matrix of an ideal elliptical retarder with retarder vector {

*δ*,

_{H}*δ*

_{45},

*δ*} should have matrix roots parameters of

_{R}*D*

_{4}=

*δ*,

_{H}*D*

_{5}=

*δ*

_{45}, and

*D*

_{6}=

*δ*. In order to evaluate the correct choice of

_{R}*p*in consideration of this criteria, the matrix roots retardance parameters (

*D*

_{4}through

*D*

_{6}) were calculated (using Mathematica’s default double-precision machine arithmetic) for a pure elliptical retarder with retardance parameters {

*δ*,

_{H}*δ*

_{45},

*δ*} = {0.294,0.302,0.997} for different values of root order

_{R}*p*. The relative error Δ

*was then calculated according to the following expression,*

_{x}*δ*,

_{H}*δ*

_{45},

*δ*} and the corresponding matrix roots retardance vector

_{R}*D*

_{4}through

*D*

_{6}for different choices of

*p*for the

*p*th root. In Fig. 2, the relative retardance error converges to a minimum value just beyond the 10

^{5}th root. For large

*p*, the relative error increases, due to numerical rounding and the loss of precision associated with machine arithmetic.

*p*is sufficiently large, the

*D*-parameters converge to values that are independent of

*p*. In order to demonstrate this convergence, a more complex Mueller matrix was generated by multiplying an elliptical retarder of randomly generated input retardance vector by a partial depolarizer

**PD**of the form The randomly generated elliptical retarder with retardance vector {

*δ*,

_{H}*δ*

_{45},

*δ*} = {0.210,0.033,1.003} was multiplied by

_{R}**PD**(0.1,0.2,0.3), generating the Mueller matrix

*p*. Figure 4 shows the convergence of the norm of the matrix roots retardance parameters (

*D*

_{4},

*D*

_{5}, and

*D*

_{6}). The norm of the matrix roots retardance parameters converges to a steady value of approximately 1.07 following the 10

^{4}th root. The convergence of the norm of the matrix roots diagonal depolarization parameters (

*D*

_{13},

*D*

_{14}, and

*D*

_{15}) behaves in a strikingly similar manner, as shown in Fig. 3.

#### 5.1.2. Numerical precision

**M**

*as a function of root order*

_{R}*p*= 10

*, for*

^{k}*k*between 1 and 14. This relative error was calculated as follows, The relative error increases linearly with each operation, so it is recommended to balance this with the optimization of the convergence properties discussed in section 5.1.1. The linear increase in error with each numerical operation is independent of the choice of Mueller matrix.

*p*is on the order of 10

^{5}. This choice balances the relative error generated from the root calculation while achieving convergence of its matrix root polarization properties. Relative error of 5 · 10

^{−12}is achieved with

*p*= 10

^{5}for the randomly generated Mueller matrix

**M**

*.*

_{R}## 6. Algorithm and flow chart

16. D. Goldstein, *Polarized Light*, 2nd ed. (Marcel Dekker, 2003). [CrossRef]

17. R. Barakat, “Conditions for the physical realizability of polarization matrices characterizing passive systems,” J. Mod. Opt. **34**, 1535–1544 (1987). [CrossRef]

18. K. M. Twietmeyer, R. A. Chipman, A. E. Elsner, Y. Zhao, and D. VanNasdale, “Mueller matrix retinal imager with optimized polarization conditions,” Opt. Express **16**, 21339–21354 (2008). [CrossRef] [PubMed]

^{5}th matrix root of the Mueller matrix. (This choice of

*p*= 10

^{5}was discussed in section 5.) The matrix root can be calculated using a builtin matrix root algorithm (such as Mathematica’s matrix power function), or with any principal matrix root algorithm such as those discussed in section 3.1. The principal root algorithms discussed in section 3.1 can fail to converge to a solution, particularly for such a high root order. As discussed in section 3.1, for any matrix with imaginary eigenvalues, computational rounding errors can lead to spurious imaginary parts. These may be discarded so long as they are small enough not to affect the accuracy of the calculation. Step 6 discards these spurious imaginary parts. In step 7, the parameters

*d*

_{0}through

*d*

_{15}are determined from the principal root according to Eq. (4). Step 8 rescales the infinitesimal roots parameters by

*p*= 10

^{5}, resulting in the desired matrix roots parameters

*D*

_{0}through

*D*

_{15}.

## 7. Statistical algorithm implementation

*m*

_{0,0}element to a value of one, and the other fifteen matrix elements are uniformly randomly distributed between negative one and one. If the matrix is nonphysical or has negative real eigenvalues, it is discarded. The Mueller matrix roots parameters were calculated for all of the remaining matrices. 76,336 physical, non-singular Mueller matrices with no real negative eigenvalues were found from the set of 10

^{9}randomly generated matrices.

*D*

_{7},

*D*

_{8}, and

*D*

_{9}) are entirely overlapping and largely lie within the range of −1 to 1, with a full-width half maximum (FWHM) of 0.6. The distributions for the phase depolarization parameters (

*D*

_{10},

*D*

_{11}, and

*D*

_{12}) also overlap entirely and range mostly between −2 and 2, with a FWHM of 0.8. The distributions of the diagonal depolarization parameters

*D*

_{13}and

*D*

_{14}overlap and have a very similar distribution to the phase depolarization parameters, with a range primarily between −2 and 2 and FWHM of 0.8.

*D*

_{15}has a distinct distribution. It has a hard limit at zero, as it cannot have a negative value, since

*f*

_{13},

*f*

_{14}, and

*f*

_{15}are always positive. Its distribution cuts off near 4, with a FWHM of 1.1. Out of all the depolarizing matrix roots parameters, it has the only non-symmetric distribution.

## 8. Conclusion

*p*th matrix root is reviewed, along with a brief discussion of the most common methods of calculating the

*p*th principal matrix root. Our study indicates that the decomposition is optimized around

*p*= 10

^{5}for a 53 digit binary double precision calculation, in consideration of numerical accuracy and noise as well as parameter convergence. Practical values for the roots of singular Mueller matrices can be obtained through perturbing them to nearby diattenuating matrices. Similarly, Mueller matrices with a half wave of retardance can be evaluated by perturbing their retardance from a half wave, without changing the retarder form. An algorithm is provided which incorporates the computational considerations involved in calculating the matrix roots decomposition. Finally, the algorithm is implemented to perform a statistical analysis on a large set of randomly generated Mueller matrices in order to yield insight on the typical ranges of the matrix roots parameters for physical Mueller matrices.

## References and links

1. | S.-Y. Lu and R. A. Chipman, “Interpretation of mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A |

2. | R. Ossikovski, “Analysis of depolarizing mueller matrices through a symmetric decomposition,” J. Opt. Soc. Am. A |

3. | S. R. Cloude, “Conditions for the physical realisability of matrix operators in polarimetry,” Proc. SPIE |

4. | R. A. Chipman, |

5. | R. C. Jones, “A new calculus for the treatment of optical systems. vii. properties of the n-matrices,” J. Opt. Soc. Am. |

6. | N. J. Higham, |

7. | D. A. Bini, N. J. Higham, and B. Meini, “Algorithms for the matrix pth root,” Num. Algor. |

8. | C.-H. Guo, “On newton’s method and halley’s method for the principal pth root of a matrix,” Linear Algebra Appl. |

9. | R. D. Skeel and J. B. Keiper, |

10. | R. A. Chipman, “Depolarization index and the average degree of polarization,” Appl. Opt. |

11. | J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta |

12. | H. D. Noble, “Mueller matrix roots,” Doctoral dissertation (2011). |

13. | N. J. Higham, “Stable iterations for the matrix square root,” Num. Algor. |

14. | M. H. Smith, “Optimization of a dual-rotating-retarder mueller matrix imaging polarimeter,” Appl. Opt. |

15. | S.-Y. Lu and R. A. Chipman, “Homogeneous and inhomogeneous jones matrices,” J. Opt. Soc. Am. A |

16. | D. Goldstein, |

17. | R. Barakat, “Conditions for the physical realizability of polarization matrices characterizing passive systems,” J. Mod. Opt. |

18. | K. M. Twietmeyer, R. A. Chipman, A. E. Elsner, Y. Zhao, and D. VanNasdale, “Mueller matrix retinal imager with optimized polarization conditions,” Opt. Express |

**OCIS Codes**

(120.5410) Instrumentation, measurement, and metrology : Polarimetry

(260.5430) Physical optics : Polarization

(290.5855) Scattering : Scattering, polarization

(240.2130) Optics at surfaces : Ellipsometry and polarimetry

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: August 2, 2011

Revised Manuscript: October 13, 2011

Manuscript Accepted: October 21, 2011

Published: December 19, 2011

**Citation**

H. D. Noble and R. A. Chipman, "Mueller matrix roots algorithm and computational considerations," Opt. Express **20**, 17-31 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-1-17

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### References

- S.-Y. Lu and R. A. Chipman, “Interpretation of mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A13, 1106–1113 (1996). [CrossRef]
- R. Ossikovski, “Analysis of depolarizing mueller matrices through a symmetric decomposition,” J. Opt. Soc. Am. A26, 1109–1118 (2009). [CrossRef]
- S. R. Cloude, “Conditions for the physical realisability of matrix operators in polarimetry,” Proc. SPIE1166, 177–185 (1989).
- R. A. Chipman, Handbook of Optics, Vol. 1 of Mueller Matrices (McGraw Hill, 2009), 3rd ed.
- R. C. Jones, “A new calculus for the treatment of optical systems. vii. properties of the n-matrices,” J. Opt. Soc. Am.38, 671–683 (1948). [CrossRef]
- N. J. Higham, Functions of Matrices: Theory and Computation (Society for Industrial and Applied Mathematics, 2008). [CrossRef]
- D. A. Bini, N. J. Higham, and B. Meini, “Algorithms for the matrix pth root,” Num. Algor.39, 349–378 (2005). [CrossRef]
- C.-H. Guo, “On newton’s method and halley’s method for the principal pth root of a matrix,” Linear Algebra Appl.432, 1905–1922 (2010). [CrossRef]
- R. D. Skeel and J. B. Keiper, Elementary Numerical Computing with Mathematica, Chapter 2 (McGraw Hill, 1993).
- R. A. Chipman, “Depolarization index and the average degree of polarization,” Appl. Opt.44, 2490–2495 (2005). [CrossRef] [PubMed]
- J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta33, 185–189 (1986). [CrossRef]
- H. D. Noble, “Mueller matrix roots,” Doctoral dissertation (2011).
- N. J. Higham, “Stable iterations for the matrix square root,” Num. Algor.15, 227–242 (1997). [CrossRef]
- M. H. Smith, “Optimization of a dual-rotating-retarder mueller matrix imaging polarimeter,” Appl. Opt.41, 2488–2493 (2002). [CrossRef] [PubMed]
- S.-Y. Lu and R. A. Chipman, “Homogeneous and inhomogeneous jones matrices,” J. Opt. Soc. Am. A11, 766–773 (1994). [CrossRef]
- D. Goldstein, Polarized Light, 2nd ed. (Marcel Dekker, 2003). [CrossRef]
- R. Barakat, “Conditions for the physical realizability of polarization matrices characterizing passive systems,” J. Mod. Opt.34, 1535–1544 (1987). [CrossRef]
- K. M. Twietmeyer, R. A. Chipman, A. E. Elsner, Y. Zhao, and D. VanNasdale, “Mueller matrix retinal imager with optimized polarization conditions,” Opt. Express16, 21339–21354 (2008). [CrossRef] [PubMed]

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