## Optimal Mueller matrix estimation in the presence of Poisson shot noise |

Optics Express, Vol. 20, Issue 19, pp. 21331-21340 (2012)

http://dx.doi.org/10.1364/OE.20.021331

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### 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

## 1. Introduction

1. 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]

2. K. M. Twietmeyer and R. A. Chipman, “Optimization of Mueller matrix polarimeters in the presence of error sources,” Opt. Express **16**, 11589–11603 (2008). [CrossRef] [PubMed]

9. 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. Express **18**, 23,095–23,103 (2010). [CrossRef]

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

11. F. Goudail, “Noise minimization and equalization for Stokes polarimeters in the presence of signal-dependent Poisson shot noise,” Opt. Lett. **34**, 647–649 (2009). [CrossRef] [PubMed]

## 2. Performance criterion for a Mueller polarimeter

*N*= 16 intensity measurements to estimate the Mueller matrix of a material. Let us denote the 16×16 Mueller matrix to estimate. The measurement system is composed of a unpolarized light source of intensity

*I*

_{0}, a polarization state generator with matrix of states

*A*, and a polarization state analyzer with matrix

*B*. The matrices

*A*and

*B*contain sets of 4 Stokes vectors used respectively in illumination and analysis to acquire the Mueller matrix: where

*U*= {

*A*,

*B*} and

*denotes the transpose of the matrix,*

^{T}*I*

_{0}is the intensity coming from the light source,

*I*is a 4 × 4 matrix containing the intensities obtained from the 16 measurements using the polarization states defined in the

*A*and

*B*matrices. In the following, to simplify equations, we will consider that we are estimating the Mueller matrix

*I*

_{0}

*M*. Eq. (3) can be thus rewritten as follows: where ⊗ denotes the Kronecker product [14

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

**V**

*and*

_{M}**V**

*are 16 dimensional vectors obtained by reading respectively the matrices*

_{I}*I*

_{0}

*M*and

*I*in the lexicographic order.

*σ*

_{2}while the Poisson noise has intrinsically the interesting property that its variance is equal to its mean. The variance of the noise disturbing the acquisition will thus be equal to the mean of the intensity measured.

**V**

*) from the noisy intensity measurements stacked in the vector*

_{M}**V**

*, we use the following estimator, which consists in inverting Eq. (4):*

_{I}**V**̂

*is an unbiased estimator, since where < . > denotes ensemble averaging. Its covariance matrix has the following expression [10*

_{M}10. Y. Takakura and J. E. Ahmad, “Noise distribution of Mueller matrices retrieved with active rotating polarimeters,” Appl. Opt. **46**, 7354–7364 (2007). [CrossRef] [PubMed]

_{VI}is the covariance matrix of

**V**

*. Finally, a standard performance criterion for a Mueller polarimeter is the sum of the variances of all the elements of the Mueller matrix, which is the trace of Γ*

_{I}_{V̂M}: This criterion can be rewritten in a simpler form obtained by using some properties of the Kronecker product and trace functions: with

*Q*= (

_{U}*U*)

^{T}U^{−1}.

**V**

*is a random vector such that each of its elements [*

_{I}**V**

*]*

_{I}*,*

_{i}*i*∈ [1, 16] is a Gaussian random variable of mean value <

*I*> and variance

_{i}*σ*

^{2}. We assume that the fluctuations are statistically independent from one intensity measurement to the other. The covariance matrix Γ

_{VI}of

**V**

*is thus a diagonal matrix with diagonal elements equal to*

_{I}*σ*

^{2}. In this case, the expression of the criterion

*𝒞*can be simplified as follows: It has been shown that trace{

*Q*} is minimized if the 4 vectors

_{U}*i*∈ [1

1. 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]

4. J. S. Tyo, “Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error,” Appl. Opt. **41**, 619–630 (2002). [CrossRef] [PubMed]

4. J. S. Tyo, “Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error,” Appl. Opt. **41**, 619–630 (2002). [CrossRef] [PubMed]

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

13. J. S. Tyo, “Noise equalization in Stokes parameter images obtained by use of variable-retardance polarimeters,” Opt. Lett. **25**, 1198–1200 (2000). [CrossRef]

*𝒞*, the matrices

*A*and

*B*must be of this form. It can be noticed that they may not be identical.

*A*and

*B*are two sets of polarization states forming a regular tetrahedron on the Poincaré sphere, the variances associated with each coefficient are given by: where

*var*[

*M*] is a matrix containing the estimation variance of each coefficients of the Mueller matrix. The minimal value of

*𝒞*is thus equal to: The variance on each coefficient does not depend on the observed Mueller matrix, which is normal in the presence of additive noise. In the next section, we will show that specific polarimeter configurations enable us to obtain similar properties in the presence of Poisson distributed shot noise.

## 3. Optimal Mueller matrix estimation in the presence of Poisson shot noise

**V**

*is a random vector such that each of its elements [*

_{I}**V**

*]*

_{I}*,*

_{i}*i*∈ [1, 16] is a Poisson random variable of mean value <

*I*> and variance <

_{i}*I*>. From the properties of Poisson shot noise, the fluctuations are statistically independent from one intensity measurement to the other. The covariance matrix Γ

_{i}_{VI}is thus diagonal of the form: Using the properties of the trace of a matrix, it is possible to rewrite the

*𝒞*criterion from Eq. (9) as: and substituting the expression of [Γ

_{VI}]

*in Eq. (14), we obtain: with the vector: It is interesting to notice that, as the first row of matrices*

_{i,j}*A*and

*B*only consists of 1/2, the first column of the matrix [

*B*⊗

*A*]

*is thus equal to 1/4 and thus the criterion*

^{T}*𝒞*can be separated in two terms: where

**V**′

*is a 15 dimensional vector containing the coefficients of*

_{U}**V**

*from 2 to 16. It has to be noted that, contrary to the case where the noise is additive Gaussian, the criterion*

_{U}*𝒞*depends on both the measurement matrices (

*A*,

*B*) and the observed Mueller matrix. It is thus possible to find the best couple of matrices (

*A*,

*B*) that minimize this criterion when observing a particular Mueller matrix

*M*. However, as it will be shown in the last section of this paper, this couple of measurement matrices is optimal only for one observed Mueller matrix, and can lead to high values of the criterion when used to estimate other matrices, which means high variances of some coefficients of the Mueller matrix. It is thus interesting to find the best couple of matrices (

*A*,

*B*) that allows us minimizing the criterion

*𝒞*whatever the observed Mueller matrix. For that purpose, we will use a

*min/max*approach.

*M*, it is always possible to find a couple of matrices (

*A*,

*B*) leading to a negative value of the product

*V*′

*′*

^{T}_{M}V_{(A,B)}. However, if we consider the physical Mueller matrix associated with a perfect depolarizer: the vector

**V**′

*is null whatever the matrices (*

_{M}*A*,

*B*). We can thus say that, if we consider all the possible physical Mueller matrices, the maximal value of the product

**V**′

_{M}^{T}**V**′

_{(A,B)}is larger or equal to zero. This is true for all measurement matrices (

*A*,

*B*). We thus have the relation: Let us now consider two sets of 4 Stokes vectors spread over the Poincaré sphere and forming a regular tetrahedron. They are gathered in the two matrices of illumination and analysis

*A*and

*B*as presented in Eq. (2). This type of matrices has two interesting properties: By substituting Eq. (21) in Eq. (18), the criterion

*𝒞*is rewritten as: Finally, using the property in Eq. (22), we obtain that the product

**V**′

_{M}^{T}**V**′

_{(}

_{A,B}_{)}is equal to 0, which is the minimal value that can be reached if we want to minimize the criterion

*𝒞*considering all the possible vectors

**V**′

*(see Eq. (20)). The conclusion is thus that, using Stokes vectors forming a regular tetrahedron on the Poincaré sphere, it is possible to minimize the maximal variance over all observed Mueller matrices, and the obtained value of the criterion*

_{M}*𝒞*is then equal to: The min/max value of

*𝒞*has the same expression as in the case of additive Gaussian noise, with

*σ*

^{2}replaced by [

**V**

*]*

_{M}_{1}, which corresponds to a variance since we are in the presence of Poisson shot noise.

*ℱ*will be equal to 0 if and only if ∀

*i*,

*k*

*α*from −90° to 90° and

*β*from −180° to 180°, it is possible to generate all the possible regular tetrahedra, and compute for each of them the criterion

*ℱ*. It has to be noted that for

*α*= 0 and

*β*= 0, the generated tetrahedron is the optimal one. The obtained results are presented in Fig. 3.

*α*and

*β*only equal to −90°, 0° and 90° and it is easily observed that all these combinations always lead to the two optimal tetrahedra defined in Eq. (29). It is interesting to notice that, by using the couple of matrices (

*A*

_{1},

*B*

_{2}) and (

*A*

_{2},

*B*

_{1}), we obtain also a value of

*ℱ*equal to 0 and the conclusions are the same as those we present when using couples (

*A*

_{1},

*B*

_{1}) and (

*A*

_{2},

*B*

_{2}) to estimate the Mueller matrix.

*A*) and analysis (

*B*), the estimation variance on each coefficient of the Mueller matrix will be independent of observed matrix and the variance of each coefficient is given by: These variance are gathered in the matrix

*var*[

*M*] that is equal to: We can observe that we obtain a result similar to the one obtained in the case of intensity disturbed by additive Gaussian noise (see Eq. (12)). The only difference is that the variance is replaced by the coefficient [

**V**

*]*

_{M}_{1}, which also represents a variance in the presence of Poisson noise. However, in the case of Poisson noise, these properties are not obtained for all polarimeter structures based on regular tetrahedra, but only in the case of the measurement matrices in Eq. (29).

## 4. Examples & discussion

*D*= 0.5 and axis

**D**given by

**D**

*= [0.8, 0.6, 0] [15*

^{T}15. S. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A **13**, 1106–1113 (1996). [CrossRef]

*A*=

*B*).

*Min*, consists in using the set of polarization states minimizing the criterion

*𝒞*presented Eq. (23) for this matrix. The second, that we call

*Tetra*, consists in using a set of polarization states forming an arbitrary regular tetrahedron on the Poincaré sphere. The associated matrix

*A*is given by:

_{tetra}*Tetra*, consists in using the set of polarization states forming the optimal regular tetrahedron on the Poincaré sphere defined Eq. (29). For these 3 configurations, we compute the criterion

^{Min/max}*𝒞*(see Eq. (18)) and the variance matrix

*Var*[

*M*] by using the analytical form of the matrix in Eq. (25). We have checked the validity of this expression with Monte Carlo simulations: when a sufficient numbers of realizations is used, one obtains a very good agreement with the theoretical values for all the Mueller matrices we have tested. The results are gathered in Table 1.

*𝒞*is, as expected, minimal in the configuration

*Min*because the set of polarization states have been adapted to the measured matrix. It has to be noticed, that, in this configuration, the polarization states are not forming a regular tetrahedron on the Poincaré sphere. Considering the two other configurations

*Tetra*and

*Tetra*, the criterion

^{Min/max}*𝒞*is equal to (5/2)

^{2}= 6.25, as found previously in Eq. (24). Let us now look at the variances of the different coefficients of the Mueller matrix. We can notice that some coefficients have a lower variance than the one obtained by using the optimized regular tetrahedron presented Eq. (29). However, others have a higher variance. It means that, even if the global estimation of the Mueller matrix seems to be more efficient by using the set of polarization states minimizing

*𝒞*, some coefficients have a worse estimation precision than when using the optimized regular tetrahedron (like, for example, the coefficient

*M*

_{33}that has a variance 13% larger). The same observation can be done considering the arbitrary regular tetrahedron. Even if the use of this latter leads to the same value of

*𝒞*as with the optimal regular tetrahedron, some coefficients have a bad estimation precision compared to the optimal case. For example, the coefficient

*M*

_{11}that has a 64% larger variance.

*Min*has been optimized for one particular matrix. What are the consequences of the use of this set to estimate another Mueller matrix? Let us consider that we observe another diattenuator matrix of diattenuation

*D*= 0.42 with

**D**

*= [0.24, 0.24, 0.94]. The sets of polarization states used to estimate the Mueller matrix are kept the same and the results are presented in the table 2.*

^{T}*𝒞*in the configuration

*Min*is now larger than the one obtained with regular tetrahedron. Indeed, the set of polarization states used is absolutely not optimized for this matrix, that is why the variance increases. As expected, the value of the criterion does not change using the tetrahedron. Considering now the variance of each coefficient, we can see that, in the configuration

*Min*, some of them have a variance that is now 90% larger than the one obtained with the optimal tetrahedron, such as the coefficient

*M*

_{11}. The same observation can be done with the configuration

*Tetra*where the variance of the coefficient

*M*

_{22}is now 74% larger than in the optimal configuration.

## 5. Conclusion

## Acknowledgments

## References and links

1. | B. Laude-Boulesteix, A. De Martino, B. Drévillon, and L. Schwartz, “Mueller Polarimetric Imaging System with Liquid Crystals,” Appl. Opt. |

2. | K. M. Twietmeyer and R. A. Chipman, “Optimization of Mueller matrix polarimeters in the presence of error sources,” Opt. Express |

3. | R. M. A. Azzam, I. M. Elminyawi, and A. M. El-Saba, “General analysis and optimization of the four-detector photopolarimeter,” J. Opt. Soc. Am. A |

4. | J. S. Tyo, “Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error,” Appl. Opt. |

5. | 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. |

6. | M. H. Smith, “Optimization of a dual-rotating-retarder Mueller matrix polarimeter,” Appl. Opt. |

7. | A. D. Martino, Y.-K. Kim, E. Garcia-Caurel, B. Laude, and B. Drévillon, “Optimized Mueller polarimeter with liquid crystals,” Opt. Lett. |

8. | P. Lemaillet, S. Rivet, and B. L. Jeune, “Optimization of a snapshot Mueller matrix polarimeter,” Opt. Lett. |

9. | 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. Express |

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

11. | F. Goudail, “Noise minimization and equalization for Stokes polarimeters in the presence of signal-dependent Poisson shot noise,” Opt. Lett. |

12. | A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part II,” Opt. Eng. |

13. | J. S. Tyo, “Noise equalization in Stokes parameter images obtained by use of variable-retardance polarimeters,” Opt. Lett. |

14. | A. N. Langvillea and W. J. Stewart, “The Kronecker product and stochastic automata networks,” J. Comp. Appl. Math. |

15. | S. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A |

**OCIS Codes**

(100.0100) Image processing : Image processing

(110.5405) Imaging systems : Polarimetric imaging

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: June 18, 2012

Revised Manuscript: July 27, 2012

Manuscript Accepted: July 27, 2012

Published: September 4, 2012

**Citation**

Guillaume Anna and François Goudail, "Optimal Mueller matrix estimation in the presence of Poisson shot noise," Opt. Express **20**, 21331-21340 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-19-21331

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### References

- 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]
- 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]
- R. M. A. Azzam, I. M. Elminyawi, and A. M. El-Saba, “General analysis and optimization of the four-detector photopolarimeter,” J. Opt. Soc. Am. A5, 681–689 (1988). [CrossRef]
- J. S. Tyo, “Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error,” Appl. Opt.41, 619–630 (2002). [CrossRef] [PubMed]
- 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]
- M. H. Smith, “Optimization of a dual-rotating-retarder Mueller matrix polarimeter,” Appl. Opt.41, 2488–2493 (2002). [CrossRef] [PubMed]
- A. D. Martino, Y.-K. Kim, E. Garcia-Caurel, B. Laude, and B. Drévillon, “Optimized Mueller polarimeter with liquid crystals,” Opt. Lett.28, 616–618 (2003). [CrossRef] [PubMed]
- P. Lemaillet, S. Rivet, and B. L. Jeune, “Optimization of a snapshot Mueller matrix polarimeter,” Opt. Lett.33, 144–146 (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]
- Y. Takakura and J. E. Ahmad, “Noise distribution of Mueller matrices retrieved with active rotating polarimeters,” Appl. Opt.46, 7354–7364 (2007). [CrossRef] [PubMed]
- F. Goudail, “Noise minimization and equalization for Stokes polarimeters in the presence of signal-dependent Poisson shot noise,” Opt. Lett.34, 647–649 (2009). [CrossRef] [PubMed]
- A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part II,” Opt. Eng.34, 1656–1658 (1995). [CrossRef]
- J. S. Tyo, “Noise equalization in Stokes parameter images obtained by use of variable-retardance polarimeters,” Opt. Lett.25, 1198–1200 (2000). [CrossRef]
- A. N. Langvillea and W. J. Stewart, “The Kronecker product and stochastic automata networks,” J. Comp. Appl. Math.167, 429–447 (2004). [CrossRef]
- S. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A13, 1106–1113 (1996). [CrossRef]

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