## Characterization of a medium by estimating the constituent components of its coherency matrix in polarization optics |

Optics Express, Vol. 19, Issue 22, pp. 21665-21672 (2011)

http://dx.doi.org/10.1364/OE.19.021665

Acrobat PDF (716 KB)

### Abstract

In this work, an alternative route to analyze a set of coherency matrices associated to a medium is addressed by means of the Independent Component Analysis (*ICA*) technique. We highlight the possibility of extracting an underlying structure of the medium in relation to a model of constituent components. The medium is considered as a mixture of unknown constituent components weighted by unknown but statistically independent random coefficients of *thickness.* The *ICA* technique can determine the number of components necessary to characterize a set of sample of the medium. An estimate of the value of these components and their respective weights is also determined. Analysis of random matrices generated by multiplying random diattenuators and depolarizers is presented to illustrate the proposed approach and demonstrate its capabilities.

© 2011 OSA

## 1 - Introduction

*incoherent*decomposition (the emerging light is a sum of several incoherent contributions) of coherency matrices is not the only one like it. Other parallel decompositions may be found in the review paper of Gil [7

7. J. J. Gil, “Polarimetric characterization of light and media,” Eur. Phys. J. Appl. Phys. **40**(1), 1–47 (2007), doi:. [CrossRef]

**N**and

**m**differential matrices for non depolarizing media. However, the formal relation between these both matrices was formulated by Barakat [10

10. R. Barakat, “Exponential versions of the Jones and Mueller-Jones polarization matrices,” J. Opt. Soc. Am. A **13**(1), 158–163 (1996). [CrossRef]

*LE*process, we derive the hypothesis of our approach and describe the

*ICA*processing over the space of coherency matrices. The case of random matrices generated by multiplying random diattenuators and depolarizers is eventually presented for illustrating the proposed approach.

## 2 – Coherency matrix space and LE metrics

### 2.1. Mueller and coherency matrix definition

### 2.2. Log-Euclidean distance on coherency matrix space

*HPD*(4) matrices as a smooth manifold, it is also possible to define a Riemannian framework on this set of matrices. This is the way used by Arsigny [14

14. V. Arsigny, P. Fillard, X. Pennec, and N. Ayache, “Geometric means in a novel vector space structure on symmetric positive-definite matrices,” SIAM J. Matrix Anal. Appl. **29**(1), 328–347 (2007). [CrossRef]

*LE*distance on Symmetric Positive Definite matrices. Proving that this distance can easily be extended to

*HPD*(4) matrices and is well adapted to coherency matrix interpolation was done in [11

11. V. Devlaminck, “Mueller matrix interpolation in polarization optics,” J. Opt. Soc. Am. A **27**(7), 1529–1534 (2010). [CrossRef] [PubMed]

*LE*distance, interpolating from

**H**

_{1}=

**H**(0) to

**H**

_{2}=

**H**(1) consists in changing the proportion of both the elements

**D**

_{1}=log(

**H**

_{1}) and

**D**

_{2}=log(

**H**

_{2}) constituting the thin laminae (see [11

11. V. Devlaminck, “Mueller matrix interpolation in polarization optics,” J. Opt. Soc. Am. A **27**(7), 1529–1534 (2010). [CrossRef] [PubMed]

**D**= (1-

*t*).

**D**

_{1}+

*t*.

**D**

_{2}. This is a sandwich of both the previous media with proportions (1-

*t*) and

*t*respectively. It is worth noticing that this result is independent of the order in which both the sections are placed since matrix summation is commutative. Thus constituent components are not localised into the medium but may be regarded as distributed contributions along the light path.

*logarithmic scalar multiplication*of a

*HPD*matrix

**H**by a scalar α∈Ρ in the same way as Arsigny proposed for symmetric matrix [14

14. V. Arsigny, P. Fillard, X. Pennec, and N. Ayache, “Geometric means in a novel vector space structure on symmetric positive-definite matrices,” SIAM J. Matrix Anal. Appl. **29**(1), 328–347 (2007). [CrossRef]

**V**= (

*HPD*(4), ⊕, •) can be viewed as a vector space when a

*HDP*matrix is identified with its logarithm. The reader is referred to [14

14. V. Arsigny, P. Fillard, X. Pennec, and N. Ayache, “Geometric means in a novel vector space structure on symmetric positive-definite matrices,” SIAM J. Matrix Anal. Appl. **29**(1), 328–347 (2007). [CrossRef]

**H**can be written as a

*linear mixtures*of vectors {

**G**

_{1},

**G**, …,

_{2}**G**

_{N}} of this space

**V**.

*LE*interpolation process,

**H**may be considered a sandwich of constituent components

**D**

_{i}= log(

**G**

_{i}) with the corresponding thickness α

_{i}. Decomposing

**H**over a set of basis vectors of

**V**could be a first available solution. It could be also possible to choose the constituent components

**D**

_{i}in such a way that the corresponding coefficients α

_{i}are homogeneous to optical quantities like for

**N**matrix defined by Jones.

## 3 – Analysis of a medium by a constituent components model

**H**has 16 degrees of freedom.

*thickness*α

_{i}, the

*Independent Component Analysis*(

*ICA*) technique can be used to estimate the

**D**

_{i}.

### 3.1. ICA model used for coherency matrix of a medium

15. A. Hyvärinen and E. Oja, “Independent component analysis: algorithms and applications,” Neural Netw. **13**(4-5), 411–430 (2000). [CrossRef] [PubMed]

*ICA*technique and explain how this approach can be used for our problem

**H**and we can write:

**H**a Coherency matrix,

**LH**the logarithm of this matrix or

**M**the associated Mueller matrix. There is no ambiguity between the three notations.

_{j}as the realization of a random variable. The random thickness α

_{i}are the latent variables of our problem since they cannot be directly observed and the

**D**

_{j}may be considered as mixing elements. It is possible to derive a more explicit expression of these mixtures. Since

**H**∈

*HPD*(4),

**D**

_{j}is necessarily a Hermitian matrix and we have only 16 unknown entries for each

**D**

_{j}matrix:

**X**> represents the mean value of

**LH**matrices and corresponds exactly to the mean value of coherency matrices according to the

*LE*distance:

11. V. Devlaminck, “Mueller matrix interpolation in polarization optics,” J. Opt. Soc. Am. A **27**(7), 1529–1534 (2010). [CrossRef] [PubMed]

**29**(1), 328–347 (2007). [CrossRef]

**A**matrix of Eq. (8) and the corresponding constituent components

**D**

_{i}may be viewed as a characterization of fluctuations around this mean value. Thus we are exhibiting polarimetric structure of the medium directly from the data itself in which case the feature extraction process can be regarded as well suited to data which is being processed.

**S**and

**A**being unknown, any scalar multiplier in one of the thickness α

_{i}could be cancelled by dividing the corresponding column of

**A**by the same scalar. This is a well-known ambiguity of the

*ICA*model. Normalizing the

**D**

_{j}matrices of constituent components, is the way we choose to fix the magnitudes of the thickness α

_{j}. After this step of normalization, Tr(

**D**

_{j}

**D**

_{j}

^{†}) = 1 where † stands for a Hermitian conjugate and Tr(.) denotes the trace of the matrix. Nevertheless, there is always the ambiguity of the sign. Since α

_{i}are presumed to be homogeneous to thickness, we can always multiply the α

_{j}and the corresponding

**D**

_{j}matrix by −1 without affecting the model in order to have positive quantities for the thickness.

_{j}. This is not an so unrealistic assumption in many cases. Considering a Mueller matrix as a mixing of independent physical quantities (birefringence or dichroïsm for instance) is one example. Since these quantities can have different values from one moment to another or from one location to another, a set of matrices with independent latent variables can be measured and analyzed according to the proposed approach. Illustrating this issue is exactly what we will do with the example below.

### 3.2. The example of matrices with random diattenuation and depolarization values

**M**. According to Lu and Chipman [1

1. S. Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A **13**(5), 1106–1113 (1996). [CrossRef]

**M**can be decomposed as

**M**where

_{R}M_{D}M_{DEP}**M**stands for a retardance matrix,

_{R}**M**stands for the diattenuation one and

_{D}**M**for the depolarizing one. For the sake of simplicity we will suppose

_{DEP}**M**to be the identity matrix in order to limit the number of independent components, but this is absolutely not a limitation of the approach.

_{R}**M**matrix can be expressed as:

_{DEP}*a*,

*b*,

*p*uniformly distributed random variables with <

*a*> = 0.3, <

*b*> = 0.4 and <

*p*> = 0.1.

*c*,

*q*and

*r*are fixed at 0.12, 0 and 0 respectively. Similarly, a

**M**matrix is completely defined by its diattenuation vector

_{D}**DV**=[

*f g h*].

*f*and

*g*are fixed to 0.2 and −0.14 respectively and

*h*is assumed uniformly distributed random variable with <

*h*> = 0.12. All these values are arbitrarily chosen but in order to have physical Mueller matrices. A set of random Mueller matrices is then generated by multiplying these random diattenuators and depolarizers

**M**.

*ICA*method. The algorithm estimates a number of independent components equal to 4. The mean value of

**LH**matrices and the four components are also estimated. The Mueller matrix

**M**

_{mean}corresponding to this mean value of

**LH**is listed in Tab. 2 with its polar decomposition. This is exactly the Mueller matrix with the mean values of the random variables used to generate the set of input data. It could be argued that if a Lu and Chipman decomposition was first applied on each random matrix and the classical Euclidean mean computed on the set of

**M**

_{D}and

**M**

_{DEP}respectively, the good mean value is then obtained by the multiplication of both these mean matrices. This is true but is only possible because the order of the decomposition is a priori known which is not generally the case in experimental measurements.

**M**

_{c}(i), associated to the estimated constituent components when all the α

_{j}except one are set to 0. These matrices are thus computed from the N coherency matrices given for i∈{1;N} by:

## 4 - Conclusion

## Appendix A

*ICA*technique, we have an estimation of

**A**and

**W**in such a way that

**WA**=

**Id**but it is worth noticing that

**AW**is not equal to the identity matrix. The algorithm assumes that the data is preprocessed by centering so we actually estimate:

**W:**

**S**=

**W X**and then:

## 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. | J. Morio and F. Goudail, “Influence of the order of diattenuator, retarder, and polarizer in polar decomposition of Mueller matrices,” Opt. Lett. |

3. | R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett. |

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

5. | S. R. Cloude, “Group theory and polarisation algebra,” Optik (Stuttg.) |

6. | A. Aiello and J. P. Woerdman, arXiv.org e-print archive math-ph/0412061 (2004) |

7. | J. J. Gil, “Polarimetric characterization of light and media,” Eur. Phys. J. Appl. Phys. |

8. | R. C. Jones, “A new calculus for the treatement of optical systems. VII properties of the N-matrices,” J. Opt. Soc. Am. |

9. | R. M. A. Azzam, “Propagation of partially polarized light through anisotropic media with or without depolarization: A differential 4x4 matrix calculus,” J. Opt. Soc. Am. |

10. | R. Barakat, “Exponential versions of the Jones and Mueller-Jones polarization matrices,” J. Opt. Soc. Am. A |

11. | V. Devlaminck, “Mueller matrix interpolation in polarization optics,” J. Opt. Soc. Am. A |

12. | K. Kim, L. Mandel, and E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A |

13. | B. N. Simon, S. Simon, N. Mukunda, F. Gori, M. Santarsiero, R. Borghi, and R. Simon, “A complete characterization of pre-Mueller and Mueller matrices in polarization optics,” J. Opt. Soc. Am. A |

14. | V. Arsigny, P. Fillard, X. Pennec, and N. Ayache, “Geometric means in a novel vector space structure on symmetric positive-definite matrices,” SIAM J. Matrix Anal. Appl. |

15. | A. Hyvärinen and E. Oja, “Independent component analysis: algorithms and applications,” Neural Netw. |

16. | A. Hyvärinen, “Fast and robust fixed-point algorithms for independent component analysis,” IEEE Trans. Neural Netw. |

**OCIS Codes**

(110.2960) Imaging systems : Image analysis

(120.5410) Instrumentation, measurement, and metrology : Polarimetry

(100.4995) Image processing : Pattern recognition, metrics

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: May 27, 2011

Revised Manuscript: July 22, 2011

Manuscript Accepted: August 4, 2011

Published: October 19, 2011

**Citation**

V. Devlaminck, P. Terrier, and J. M. Charbois, "Characterization of a medium by estimating the constituent components of its coherency matrix in polarization optics," Opt. Express **19**, 21665-21672 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-22-21665

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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(5), 1106–1113 (1996). [CrossRef]
- J. Morio and F. Goudail, “Influence of the order of diattenuator, retarder, and polarizer in polar decomposition of Mueller matrices,” Opt. Lett.29(19), 2234–2236 (2004). [CrossRef] [PubMed]
- R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett.32(6), 689–691 (2007). [CrossRef] [PubMed]
- R. Ossikovski, “Analysis of depolarizing Mueller matrices through a symmetric decomposition,” J. Opt. Soc. Am. A26(5), 1109–1118 (2009). [CrossRef] [PubMed]
- S. R. Cloude, “Group theory and polarisation algebra,” Optik (Stuttg.)75, 26–36 (1986).
- A. Aiello and J. P. Woerdman, arXiv.org e-print archive math-ph/0412061 (2004)
- J. J. Gil, “Polarimetric characterization of light and media,” Eur. Phys. J. Appl. Phys.40(1), 1–47 (2007), doi:. [CrossRef]
- R. C. Jones, “A new calculus for the treatement of optical systems. VII properties of the N-matrices,” J. Opt. Soc. Am.38(8), 671–685 (1948). [CrossRef]
- R. M. A. Azzam, “Propagation of partially polarized light through anisotropic media with or without depolarization: A differential 4x4 matrix calculus,” J. Opt. Soc. Am.68(12), 1756–1767 (1978). [CrossRef]
- R. Barakat, “Exponential versions of the Jones and Mueller-Jones polarization matrices,” J. Opt. Soc. Am. A13(1), 158–163 (1996). [CrossRef]
- V. Devlaminck, “Mueller matrix interpolation in polarization optics,” J. Opt. Soc. Am. A27(7), 1529–1534 (2010). [CrossRef] [PubMed]
- K. Kim, L. Mandel, and E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A4(3), 433–437 (1987). [CrossRef]
- B. N. Simon, S. Simon, N. Mukunda, F. Gori, M. Santarsiero, R. Borghi, and R. Simon, “A complete characterization of pre-Mueller and Mueller matrices in polarization optics,” J. Opt. Soc. Am. A27(2), 188–199 (2010). [CrossRef] [PubMed]
- V. Arsigny, P. Fillard, X. Pennec, and N. Ayache, “Geometric means in a novel vector space structure on symmetric positive-definite matrices,” SIAM J. Matrix Anal. Appl.29(1), 328–347 (2007). [CrossRef]
- A. Hyvärinen and E. Oja, “Independent component analysis: algorithms and applications,” Neural Netw.13(4-5), 411–430 (2000). [CrossRef] [PubMed]
- A. Hyvärinen, “Fast and robust fixed-point algorithms for independent component analysis,” IEEE Trans. Neural Netw.10(3), 626–634 (1999). [CrossRef] [PubMed]

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