## Estimation of Mueller matrices using non-local means filtering |

Optics Express, Vol. 21, Issue 4, pp. 4424-4438 (2013)

http://dx.doi.org/10.1364/OE.21.004424

Acrobat PDF (8908 KB)

### Abstract

This article addresses the estimation of polarization signatures in the Mueller imaging framework by non-local means filtering. This is an extension of previous work dealing with Stokes signatures. The extension is not straightforward because of the gap in complexity between the Mueller framework and the Stokes framework. The estimation procedure relies on the Cholesky decomposition of the coherency matrix, thereby ensuring the physical admissibility of the estimate. We propose an original parameterization of the boundary of the set of Mueller matrices, which makes our approach possible. The proposed method is fully unsupervised. It allows noise removal and the preservation of edges. Applications to synthetic as well as real data are presented.

© 2013 OSA

## 1. Introduction

1. J. Zallat, S. Anouz, 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]

2. J. Zallat, Ch. Collet, and Y. Takakura, “Clustering of polarization-encoded images,” Appl. Opt. **43**, 283–292 (2004). [CrossRef] [PubMed]

3. Ch. Collet, J. Zallat, and Y. Takakura, “Clustering of Mueller matrix images for skeletonized structure detection,” Opt. Express **12**, 1271–1280 (2004). [CrossRef] [PubMed]

4. J. Zallat, Ch. Heinrich, and M. Petremand, “A Bayesian approach for polarimetric data reduction: the Mueller imaging case,” Opt. Express **16**, 7119–7133 (2008). [CrossRef] [PubMed]

5. S. Faisan, Ch. Heinrich, F. Rousseau, A. Lallement, and J. Zallat, “Joint filtering-estimation of Stokes vector images based on a non-local means approach,” J. Opt. Soc. Am. A **29**, 2028–2037 (2012). [CrossRef]

6. A. Buades, B. Coll, and J. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Simul. **4**, 490–530 (2005). [CrossRef]

## 2. Related work

5. S. Faisan, Ch. Heinrich, F. Rousseau, A. Lallement, and J. Zallat, “Joint filtering-estimation of Stokes vector images based on a non-local means approach,” J. Opt. Soc. Am. A **29**, 2028–2037 (2012). [CrossRef]

### 2.1. Physical admissibility of Mueller matrices

### 2.2. Joint filtering-estimation of Stokes vectors

*I*, the estimate of the denoised image

*I*at pixel

_{nlm}**x**is a weighted average of all pixels in the image: where Ω is the support of the digital image (Ω ⊂

^{2}), and where

*w*(

**x**,

**y**) represents the similarity between pixels

**x**and

**y**with 0 ≤

*w*(

**x**,

**y**) ≤ 1, and ∑

_{y}*w*(

**x**,

**y**) = 1. The similarity between two pixels derives from Euclidean distance between patches, a patch being a square window centered on

**x**or

**y**. More precisely, the similarity

*w*(

**x**,

**y**) is proportional to

*P*(resp.

_{x}*P*) is the vectorized set of gray level intensities of the

_{y}**x**-patch (resp.

**y**-patch).

**x**is thereby estimated using the whole set of pixels in the image, in a filtering-estimation procedure. Since the gradient of Eq. (3) is equal to the gradient of ‖

**I**

*(*

_{nlm}**x**) −

**P**·

**S**‖

^{2}(see [5

5. S. Faisan, Ch. Heinrich, F. Rousseau, A. Lallement, and J. Zallat, “Joint filtering-estimation of Stokes vector images based on a non-local means approach,” J. Opt. Soc. Am. A **29**, 2028–2037 (2012). [CrossRef]

**V̂**(

**x**) = (

**P**

*·*

^{t}**P**)

^{−1}·

**P**·

**I**

*(*

_{nlm}**x**). However, this solution may not verify the admissibility constraints:

**V̂**(

**x**) may not be a Stokes vector. The following constrained criterion has been proposed: where

**B**is the set of admissible Stokes vectors. Note that optimizing the criterion of Eq. (5) is equivalent to optimizing the one of Eq. (3) under the physical admissibility constraint since both criteria are equal up to a constant.

**B**is a convex set, the criterion of Eq. (5) has a unique minimum. Instead of using a constrained optimization procedure, we use a much simpler procedure, which was defined in [5

**29**, 2028–2037 (2012). [CrossRef]

*∂*

**B**of

**B**can be easily parameterized.

## 3. Joint filtering-estimation of Mueller matrices

### 3.1. Definition of the criterion

**D**(

_{x}**y**) is a

*pq*×

*pq*diagonal matrix, ∑

**(**

_{y}D_{x}**y**) being the identity matrix. The diagonal element

*d*(

_{ii}*i*= 1 ...

*pq*) is the similarity (or weight) between pixel

**x**and pixel

**y**for the

*i*channel. As in the Stokes vector case, the estimate of Eq. (8) is equivalent to: where

^{th}**I**

*is defined as: Finally, incorporating the physical admissibility constraints in Eq. (8) (or equivalently in Eq. (9)) leads to: where*

_{nlm}**B**denotes in this part the set of admissible Mueller matrices.

### 3.2. Properties of the Mueller matrix set

**M**and the system coherency matrix

**H**, which reads: where

**T**is a 16 × 16 invertible complex transformation matrix whose expression can be found for example in [4

4. J. Zallat, Ch. Heinrich, and M. Petremand, “A Bayesian approach for polarimetric data reduction: the Mueller imaging case,” Opt. Express **16**, 7119–7133 (2008). [CrossRef] [PubMed]

**M**is defined here as a matrix that verifies the Jones criterion (see Sec. 2.1),

**M**is physically acceptable

*iff*

**H**is positive semidefinite (

12. S. Cloude and E. Pottier, “Concept of polarization entropy in optical scattering: polarization analysis and measurement,” Opt. Eng. **34**, 1599–1610 (1995). [CrossRef]

**B**is a convex set.

*∂*

**B**of

**B**. As

*∂*

**B**is the image of the boundary

**T**, the key point is to define a parameterization for

**Property 1:**

*iff*

**H**is not invertible.

**Λ**

*is the conjugate transpose of*

^{t}**Λ**. Matrix

**Λ**is a lower triangular matrix composed of 16 real parameters {

*λ*}, Property 1 and the fact that

_{i}*det*(

**Λ**·

**Λ**

*) = (*

^{t}*λ*

_{1}

*λ*

_{2}

*λ*

_{3}

*λ*

_{4})

^{2}lead to property 2.

**Property 2:**

*iff*at least one value amongst

*λ*(

_{i}*i*= 1 ... 4) is null.

**Property 3:**if

**Λ**with

*λ*

_{4}= 0 such that

**H**=

**Λ**·

**Λ**

*.*

^{t}### 3.3. Optimization algorithm

**B**is convex, Eq. (11) has a unique minimum denoted

**M**

^{★}(

**x**).

**V̂**(

**x**) ∈

**B**, then

**M**

^{★}(

**x**) =

**V̂**(

**x**). Otherwise,

**M**

^{★}(

**x**) belongs to

*∂*

**B**and can be estimated by using these two strategies iteratively:

*∂*

**B**, with the orthogonal projection of

**V̂**(

**x**) onto

**B**, and strategy (

*i*) is considered. When a minimum is obtained, computing the gradient of criterion of Eq. (11) enables to determine if this minimum is the global one or a local one: the gradient gives the direction of the interior of

**B**

*iff*the minimum is the global one. If this minimum is a local one, the criterion can be decreased by entering into

**B**. The descent is then continued with strategy (

*ii*). In this strategy, the boundary is ensured to be met since there is no local minimum in

**B̆**. Then, strategy (

*i*) is used again. This procedure is repeated until the global minimum is reached with strategy (

*i*).

*i*) is a subspace trust region method that is based on the interior-reflective Newton method described in [17

17. T. Coleman and Y. Li, “On the convergence of reflective Newton methods for large-scale nonlinear minimization subject to bounds,” Math. Program. **67**, 189–224 (1994). [CrossRef]

18. T. Coleman and Y. Li, “An interior, trust region approach for nonlinear minimization subject to bounds,” SIAM J. Optim. **6**, 418–445 (1996). [CrossRef]

*ii*), a simple gradient approach is used: the maximal admissible step constraining

**M̂**(

**x**) to be a Mueller matrix is computed at each iteration. The proposed algorithm is finally much simpler than a standard constrained optimization procedure, and it is ensured to converge to the unique minimum.

*i*) is used starting from a Mueller matrix

**M**that belongs to

*∂*

**B**. This matrix has been obtained either from the orthogonal projection of

**V̂**(

**x**) onto

**B**, or from strategy (

*ii*) that is ensured to converge to a Mueller matrix of

*∂*

**B**. In both cases, starting from

**M**∈

*∂*

**B**, initialization of strategy (

*i*) requires the determination of

**Λ**such that

*λ*

_{4}= 0 and

**M**=

**T**·

**Λ**·

**Λ**

*. Due to numerical problems, this computation has to be done carefully (see Appendix B).*

^{t}**M̂**denotes the estimated Mueller matrix image, the image

**Î**defined as

**Î**(

**x**) =

**P**·

**M̂**(

**x**) for each

**x**, can be considered as the denoised version of

**I**. For

*pq*= 16, the images

**Î**and

**I**

*differ only for the pixels for which the pseudo-inverse solution does not satisfy the admissibility constraints.*

_{nlm}## 4. Results

### 4.1. Results on synthetic data

**M**

*composed of two distinct regions: (i) a background with a uniform polarization signature*

^{gt}**M**, and (ii) a 100 pixel radius circle with a smoothly varying polarized Mueller signature placed in the center of the image (see Fig. 1). The Mueller matrices of the synthetic target lie on

*∂*

**B**and even a small noise level may lead to non physical solutions if one uses the pseudo-inverse approach.

*λ*parameterization with

*λ*

_{4}= 0 as follows:

- The Mueller matrix associated to the background has been defined by drawing each component of
**Λ**(except*λ*_{4}that is set to 0) according to a normal distribution of standard deviation 1. - For the circle, 15 images of size 256×256 (an image for each
*λ*,_{i}*i*≠ 4) are computed by drawing the value at each pixel according to a normal distribution of standard deviation 1. Each image is then filtered using an isotropic Gaussian filter of standard deviation of 5 pixels, thereby creating 15 smooth images. A constant is then added to each image (each constant is drawn according to a normal distribution of standard deviation 1).

**M**

*can then be computed. A standard observation model was finally used to generate intensity images*

^{gt}*i*= 1...16) that were degraded by adding white Gaussian noise of variance

*σ*

^{2}.

**M̂**and the associated intensity values

**Î**) with the original ones (

**M**

*and*

^{gt}**I**

*). The method is first evaluated by comparing the original image*

^{gt}**I**

*(noise-free intensity image) with its estimation*

^{gt}**Î**using the Peak Signal-to-Noise Ratio (PSNR): where

*α*is computed so that the dynamic of

_{j}*d*(for example 255), and where

*P*is the number of pixels. Complementary to the PSNR, a Mueller matrix dedicated measurement is also used to evaluate the estimation accuracy:

*M*,

_{PI}*M*,

_{PIortho}*M*, and

_{PIproj}*P*are evaluated:

_{M}- for
*M*:_{PI}**M̂**(**x**) is the pseudo-inverse solution; - for
*M*:_{PIortho}**M̂**(**x**) is the pseudo-inverse solution, further orthogonally projected onto the Mueller matrix set if it does not verify the admissibility constraints; - for
*M*:_{PIproj}**M̂**(**x**) is the pseudo-inverse solution, further projected onto the Mueller matrix set if it does not verify the admissibility constraints (the projection is performed with the proposed optimization algorithm so as to reduce at most the error reconstruction between the observed measurements and the predicted ones); *M*is the proposed approach._{P}

*M*,

_{PI}*M*,

_{PIortho}*M*do not use any spatial filtering.

_{PIproj}*M*,

_{PI}*M*,

_{PIortho}*M*and

_{PIproj}*M*are given in Tab. 1 for different values of

_{P}*σ*(from

*σ*= 0.05 to 1).

*σ*= 0.1) with the NLM approach (a), and with

*M*(b). Since results obtained with the methods

_{PI}*M*and

_{PIproj}*M*are visually very similar to Fig. 2(b), they are not presented here. The channels associated to the third line of the Mueller matrix are for example noisy with

_{PIortho}*M*but not with the NLM approach. Besides performing noise reduction, we can observe that edges have been preserved. This nice property is inherited from NLM filtering: NLM filtering preserves sharp edges and fine texture details [6

_{PI}6. A. Buades, B. Coll, and J. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Simul. **4**, 490–530 (2005). [CrossRef]

*M*is the one providing the best results. This shows that the traditional way of projecting onto the Mueller matrix set (method

_{PIproj}*M*, orthogonal projection) is not an efficient approach, compared to the proposed method which reduces the most the error reconstruction between the observed measurements and the predicted ones, while constraining the solution to be physically acceptable. Note also that

_{PIortho}*M*provides better results than

_{PIortho}*M*in terms of accuracy of the Mueller matrix estimation, but less satisfactory results in terms of PSNR. This highlights the fact that the choice of orthogonal projection is arbitrary.

_{PI}### 4.2. Results on real data

1. J. Zallat, S. Anouz, 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]

*i*) the first scene (Fig. 3) corresponds to two shapes made from two different transparent thin layers (cellophane for wrapping food) sandwiched between two glass sections. This object is expected to show class-wise constant polarization responses; (

*ii*) the second one consists of an ensemble of objects (Fig. 4) leading to an image with various polarization responses and complex geometrical properties. Observations were carried out through a narrow band interferential filter to ensure that the PMM is known with high accuracy. The Mueller image was reconstructed from the raw intensity images using

*M*(the proposed approach) and

_{P}*M*(the classical pseudo-inverse approach except that the solution is further projected orthogonally onto the Mueller matrix set if it does not verify the admissibility constraints).

_{PIortho}*m*

_{11},

*m*

_{12},

*m*

_{21}, and

*m*

_{22}element images). In particular, a lot of fine details are definitely lost (channels

*m*

_{12}and

*m*

_{21}). All channels estimated with

*M*are particularly noisy for the second scene which is not the case with the proposed approach. For the sake of conciseness, only channels

_{PIortho}*m*

_{22}(Fig. 4(a)) and

*m*

_{44}(Fig. 4(c)) are presented. The

*M*approach propagates intensity noises to the Mueller channels leading to less workable Mueller images. Results obtained with the second scene illustrate also the efficiency of the NLM filtering approach to preserve edges and thin structures (see Fig. 4(b) and (d)).

_{PIortho}*M*and the

_{P}*M*solutions for the first scene. The poor quality of the result obtained with the

_{PIortho}*M*approach can be explained by the fact that the upper left 2 × 2 block of the Mueller matrix is in this case particularly noisy (see Fig. 3(b)).

_{PIortho}## 5. Conclusion

## A. Characterization of the boundary
∂ ℋ + 4 of the set of H-type matrices

**Λ**with

*λ*

_{4}= 0 such that

**H**=

**Λ**·

**Λ**

*.*

^{t}*λ*’s that has at least one solution if

**H**is invertible, then the signs of

*λ*

_{1},

*λ*

_{2},

*λ*

_{3}, and

*λ*

_{4}can be set arbitrarily leading to 2

^{4}different solutions. In the general case where

*λ*’s writes:

**H**is invertible (

*λ*> 0 for

_{i}*i*= 1...4), Eq. (20) can be either obtained by solving the system of equations in

*λ*’s directly or by using the algorithm of [19, p. 145]. In [19], the algorithm is described for symmetric positive definite (real) matrices but the extension to (complex) Hermitian positive definite matrices is straightforward.

**H**is not invertible (i.e. ∃

*i*∈ [1

1. J. Zallat, S. Anouz, 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]

4. J. Zallat, Ch. Heinrich, and M. Petremand, “A Bayesian approach for polarimetric data reduction: the Mueller imaging case,” Opt. Express **16**, 7119–7133 (2008). [CrossRef] [PubMed]

*λ*= 0), the

_{i}*i*-th column of

**Λ**is set to 0 (see Eq. (20)). This is justified in [19, p. 148] for real matrices, but the same reasoning applies to complex ones.

*λ*

_{1}= 0, or

*λ*

_{2}= 0, or

*λ*

_{3}= 0, or

*λ*

_{4}= 0. In the following,

**Λ**denotes the lower triangular matrix whose elements are defined in Eq. (20). By construction,

**Λ**verifies

**H**=

**Λ**·

**Λ**

*but*

^{t}*λ*

_{4}may be greater than 0. To obtain a solution with

*λ*

_{4}= 0, an equivalence class is defined on the set of the lower triangular matrices (see Eq. (14)):

**Λ**

*and*

_{a}**Λ**

*are equivalent if there exists an orthonormal matrix*

_{b}**R**such that

**Λ**

*=*

_{a}**Λ**

*·*

_{b}**R**. Matrices

**Λ**

*and*

_{a}**Λ**

*correspond then to the same underlying coherency matrix*

_{b}**Λ**

*·*

_{a}**Λ**

*=*

_{a}^{t}**Λ**

*·*

_{b}**Λ**

*, and consequently to the same Mueller matrix. If*

_{b}^{t}**Λ**a matrix with

*λ*

_{4}= 0. Three cases have to be considered, depending on which

*λ*vanishes:

_{i}- if
**H**is such that*λ*_{1}= 0 then from Eq. (20), we have: with**H**=**Λ**·**Λ**. A matrix equivalent to^{t}**Λ**can be derived from**Λ**by swapping its first and last columns. This can be formally defined as follows:**Λ**is equivalent to**Λ**·**R**, with - In the class of
**Λ**, we thereby have a matrix (**Λ**·**R**) with*λ*_{4}= 0. This means that**Λ**·**R**is a lower triangular matrix with*λ*_{4}= 0 that verifies**H**= (**Λ**·**R**) · (**Λ**·**R**);^{t} - if
**H**is such that*λ*_{2}= 0 in Eq. (20), then the second column of**Λ**vanishes and the reasoning above applies with - if
**H**is such that*λ*_{3}= 0 in Eq. (20), then the third column of**Λ**vanishes and the reasoning above applies with

**H**=

**Λ**·

**Λ**

*, with*

^{t}*λ*

_{4}= 0.

*λ*= 0 (

_{i}*i*= 1 ... 3), the elements of the

*i*-th column cannot be uniquely determined. In this case, these elements have been set to 0 (see Eq. (20)). This has provided us a way to easily find a matrix equivalent to

**Λ**that verifies

*λ*

_{4}= 0. If the undetermined elements had not been set to 0, it would not have been possible to use such a simple procedure, because swapping columns would have led to a non-triangular matrix.

## B. Algorithm for the estimation of Λ from M ∈ *∂*B

**M**be a Mueller matrix that belongs to the boundary

*∂*

**B**of the Mueller matrix set. Let

**H**be the associated coherency matrix (

**Λ**with

*λ*

_{4}= 0 such that

**H**=

**Λ**·

**Λ**

*. The goal of this Appendix is to explain how to determine*

^{t}**Λ**. At first sight, an algorithm can be easily derived from Appendix A:

- Determine
**Λ**using Eq. (20). - Let
*i*_{0}be an integer in [1**8**, 807–814 (2006). [CrossRef]**16**, 7119–7133 (2008). [CrossRef] [PubMed]*λ*_{i0}= 0. Such an integer exists since at least one value of*λ*is null (_{i}*i*= 1 ... 4). - Swap the
*i*_{0}-th column of**Λ**with its last one to obtain the desired result.

**Λ**. Indeed, due to numerical problems, it may happen that

*λ*

_{1},

*λ*

_{2},

*λ*

_{3}and

*λ*

_{4}are all strictly positive. We could then define a threshold beyond which the values of

*λ*would be set to 0 (

_{i}*i*= 1 ... 4). However, there is in this case a risk that

*λ*is set to 0 while it should not be, thereby leading to a matrix

_{i}**Λ**with

**Λ**·

**Λ**

*highly different from*

^{t}**H**.

**Λ**are computed by setting some values of

*λ*to 0 (

_{i}*i*= 1...4). This can be formally described as follows. Let 𝒜 denote a combination of

*p*elements of {

*λ*

_{1},

*λ*

_{2},

*λ*

_{3},

*λ*

_{4}} (1 ≤

*p*≤ 4). Each

*λ*of 𝒜 is set to 0, and the other values of

_{i}**Λ**are computed from Eq. (20). All combinations are tested, leading to 15 different matrices

**Λ**. The matrix

**Λ**that minimizes the quadratic error between

**H**and

**Λ**·

**Λ**

*is chosen. The equivalence class defined in Apprendix A enables finally to derive a lower triangular matrix*

^{t}**Λ**

_{0}from

**Λ**, with

*λ*

_{4}= 0 that verifies

**H**≃

**Λ**

_{0}·

**Λ**

_{0}

*.*

^{t}## Acknowledgment

## References and links

1. | J. Zallat, S. Anouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A: Pure Appl. Opt. |

2. | J. Zallat, Ch. Collet, and Y. Takakura, “Clustering of polarization-encoded images,” Appl. Opt. |

3. | Ch. Collet, J. Zallat, and Y. Takakura, “Clustering of Mueller matrix images for skeletonized structure detection,” Opt. Express |

4. | J. Zallat, Ch. Heinrich, and M. Petremand, “A Bayesian approach for polarimetric data reduction: the Mueller imaging case,” Opt. Express |

5. | S. Faisan, Ch. Heinrich, F. Rousseau, A. Lallement, and J. Zallat, “Joint filtering-estimation of Stokes vector images based on a non-local means approach,” J. Opt. Soc. Am. A |

6. | A. Buades, B. Coll, and J. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Simul. |

7. | Z. Xing, “On the deterministic and nondeterministic Mueller matrix,” J. Mod. Opt. |

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

9. | C. Vandermee, “An eigenvalue criterion for matrices transforming Stokes parameters,” J. Math. Phys. |

10. | 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. | A. Gopala Rao, K. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics – I. Identifying a Mueller matrix from its N matrix,” J. Mod. Opt. |

12. | S. Cloude and E. Pottier, “Concept of polarization entropy in optical scattering: polarization analysis and measurement,” Opt. Eng. |

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

14. | C. Deledalle, L. Denis, and F. Tupin, “Iterative weighted maximum likelihood denoising with probabilistic patch-based weights,” IEEE Trans. Image Process. |

15. | S. Boyd and L. Vandenberghe, |

16. | G. Stewart, |

17. | T. Coleman and Y. Li, “On the convergence of reflective Newton methods for large-scale nonlinear minimization subject to bounds,” Math. Program. |

18. | T. Coleman and Y. Li, “An interior, trust region approach for nonlinear minimization subject to bounds,” SIAM J. Optim. |

19. | G. H. Golub and C. F. Van Loan, |

**OCIS Codes**

(100.3020) Image processing : Image reconstruction-restoration

(100.3190) Image processing : Inverse problems

(120.5410) Instrumentation, measurement, and metrology : Polarimetry

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: November 2, 2012

Revised Manuscript: December 7, 2012

Manuscript Accepted: December 14, 2012

Published: February 13, 2013

**Citation**

Sylvain Faisan, Christian Heinrich, Giorgos Sfikas, and Jihad Zallat, "Estimation of Mueller matrices using non-local means filtering," Opt. Express **21**, 4424-4438 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-4-4424

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### References

- J. Zallat, S. Anouz, 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. Zallat, Ch. Collet, and Y. Takakura, “Clustering of polarization-encoded images,” Appl. Opt.43, 283–292 (2004). [CrossRef] [PubMed]
- Ch. Collet, J. Zallat, and Y. Takakura, “Clustering of Mueller matrix images for skeletonized structure detection,” Opt. Express12, 1271–1280 (2004). [CrossRef] [PubMed]
- J. Zallat, Ch. Heinrich, and M. Petremand, “A Bayesian approach for polarimetric data reduction: the Mueller imaging case,” Opt. Express16, 7119–7133 (2008). [CrossRef] [PubMed]
- S. Faisan, Ch. Heinrich, F. Rousseau, A. Lallement, and J. Zallat, “Joint filtering-estimation of Stokes vector images based on a non-local means approach,” J. Opt. Soc. Am. A29, 2028–2037 (2012). [CrossRef]
- A. Buades, B. Coll, and J. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Simul.4, 490–530 (2005). [CrossRef]
- Z. Xing, “On the deterministic and nondeterministic Mueller matrix,” J. Mod. Opt.39, 461–484 (1992). [CrossRef]
- C. Givens and A. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt.40, 471–481 (1993). [CrossRef]
- C. Vandermee, “An eigenvalue criterion for matrices transforming Stokes parameters,” J. Math. Phys.34, 5072–5088 (1993). [CrossRef]
- D. Anderson and R. Barakat, “Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix,” J. Opt. Soc. Am. A11, 2305–2319 (1994). [CrossRef]
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