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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 22 — Oct. 24, 2011
  • pp: 21665–21672
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Characterization of a medium by estimating the constituent components of its coherency matrix in polarization optics

V. Devlaminck, P. Terrier, and J. M. Charbois  »View Author Affiliations


Optics Express, Vol. 19, Issue 22, pp. 21665-21672 (2011)
http://dx.doi.org/10.1364/OE.19.021665


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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

H=ijmij(σiσj*)
(1)

Coherency matrix is thus another possibility of representation of the polarimetric properties of a material medium. But this parallel 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]

] for instance. Once again, this tool does not lead to a straightforward characterisation of optical properties of the medium since we only access to an equivalent parallel model.

dM(z)dz=mM(z)
(2)

Azzam also derived the relations between the entries of 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]

] from the concept of exponential versions of the Mueller-Jones matrices.

The paper is thus organized as follows. After a reminder of definitions and relations associated with coherency matrix and 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

When considering 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]

] to define the 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]

].

This approach can be extended when defining the 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]

]. This multiplication is defined by:

(αH)=exp[αLog(H)]
(4)

With both the operations defined by Eq. (3) and (4), 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]

] for the demonstration of this property with Symmetric Positive Definite matrices.

A straightforward consequence is that H can be written as a linear mixtures of vectors { G1, G2, …, GN} of this space V.

H=(α1G1)(α2G2)(αNGN)=exp[i=1NαiLog(Gi)]
(5)

Following the same approach used to exhibit the physical meaning of LE interpolation process, H may be considered a sandwich of constituent components Di = log(Gi) with the corresponding thickness αi. 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 Di in such a way that the corresponding coefficients αi are homogeneous to optical quantities like for N matrix defined by Jones.

3 – Analysis of a medium by a constituent components model

Without any supplementary information, we are just able to tell that N = 16 components are obviously needed to represent the most general medium since H has 16 degrees of freedom.

Nevertheless, one approach to solving this problem could be to assume some statistical properties of the quantities to estimate. If the medium may be considered as a mixture of unknown constituent components weighted by unknown but statistically independent random coefficients of thickness αi, the Independent Component Analysis (ICA) technique can be used to estimate the Di.

3.1. ICA model used for coherency matrix of a medium

It is beyond the scope of this paper to develop a general presentation of this method and the reader is referred to the literature on this topic, see [15

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

] for instance. We just develop the classical formalism of ICA technique and explain how this approach can be used for our problem

From Eq. (5) at any point in the space or any instant of time, we have a coherency matrix H and we can write:

LH=log(H)=j=1NαjDj
(6)

It is worth noticing that it is equivalent to speak about H a Coherency matrix, LH the logarithm of this matrix or M the associated Mueller matrix. There is no ambiguity between the three notations.

Equation (6) defined αj as the realization of a random variable. The random thickness αi are the latent variables of our problem since they cannot be directly observed and the Dj may be considered as mixing elements. It is possible to derive a more explicit expression of these mixtures. Since HHPD(4), Dj is necessarily a Hermitian matrix and we have only 16 unknown entries for each Dj matrix:

Dj=[d1jd5j+id6jd7j+id8jd9j+id10j.d2jd11j+id12jd13j+id14j..d3jd15j+id16j...d4j]
(7)

X=A(SS)+X
(8)

(See Appendix A for the proof) where <.> denotes the ensemble average.

It is worth noticing that <X> represents the mean value of LH matrices and corresponds exactly to the mean value of coherency matrices according to the LE distance:

HLE=exp(1Kj=1Klog(Hj))
(9)

See [11

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

] [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]

] for more details on this definition. The A matrix of Eq. (8) and the corresponding constituent components Di 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.

Both S and A being unknown, any scalar multiplier in one of the thickness αi 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 Dj matrices of constituent components, is the way we choose to fix the magnitudes of the thickness αj. After this step of normalization, Tr(Dj Dj) = 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 αi are presumed to be homogeneous to thickness, we can always multiply the αj and the corresponding Dj matrix by −1 without affecting the model in order to have positive quantities for the thickness.

Obviously, the major assumption of this approach is on the actual existence of statistically independent coefficients αj. 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

We consider the example of a medium characterized by its Mueller matrix 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 MRMDMDEP where MR stands for a retardance matrix, MD stands for the diattenuation one and MDEP for the depolarizing one. For the sake of simplicity we will suppose MR to be the identity matrix in order to limit the number of independent components, but this is absolutely not a limitation of the approach.

The MDEP matrix can be expressed as:

MDEP=[1pqr0a0000b0000c]
(10)

We assume 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 MD matrix is completely defined by its diattenuation vector 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

Two examples of the diattenuation and depolarization matrices are listed in Tab. 1

Table 1. Examples of Generated Mueller Matrices. First Column: MD Values, Second Column: MDEP Values, Third Column: M= MDMDEP

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with the corresponding generated matrices M.

All these 4 random variables are supposed to be independent. This is not an unrealistic assumption since diattenuation and depolarization can be regarded as derived from different physical properties and their components are specifically introduced as degrees of freedom of Mueller matrices in the Lu and Chipman decomposition.

A set of 1000 matrices is built following the procedure previously described. The corresponding coherency matrices are computed and this set of matrices is the input data analyzed by 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 Mmean corresponding to this mean value of LH is listed in Tab. 2

Table 2. Estimated Mean Mueller Matrix. First Column: Mmean, Second Column: MD Values, Third Column: MDEP Values

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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 MD and MDEP 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.

In order to analyze the constituent components estimated by the algorithm, we compute the Mueller matrices denoted Mc(i), associated to the estimated constituent components when all the αj except one are set to 0. These matrices are thus computed from the N coherency matrices given for i∈{1;N} by:

Hi=exp[αiDi+log(HLE)]
(11)

Tab. 3

Table 3. Estimated Mc Mueller Matrix Associated to Each Component. First Column: Mc(i), Second Column: MD(i) Values, Third Column: MDEP(i) Values

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shows an example of these N=4 Mueller matrices computed when the αj are the estimated values corresponding to the Mueller matrix at the first line of Tab. 1. For each of these Mc(i) matrices, the polar decomposition is computed and corresponding MD and MDEP matrices are shown.

4 - Conclusion

The case of random matrices generated by multiplying random diattenuators and depolarizers was presented to illustrate the proposed approach and demonstrate its capabilities. Applications of this method to the case of coherency matrix from experimental data are being analyzed.

Appendix A

From the 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:

X-X=AS1
(A1)

then

S=S1+WXorS1=SWX
(A2)

But from (A1), we have by a left multiplication by W:

S1=WX-WX
(A3)

from (A3) and (A2) we have: S = W X and then:

S=WX
(A4)

Thus from (A1), (A2) and (A4) we eventually have X - <X> = A (S - <S>) that gives Eq. (8).

References and Links

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]

2.

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]

3.

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]

4.

R. Ossikovski, “Analysis of depolarizing Mueller matrices through a symmetric decomposition,” J. Opt. Soc. Am. A 26(5), 1109–1118 (2009). [CrossRef] [PubMed]

5.

S. R. Cloude, “Group theory and polarisation algebra,” Optik (Stuttg.) 75, 26–36 (1986).

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. 40(1), 1–47 (2007), doi:. [CrossRef]

8.

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]

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. 68(12), 1756–1767 (1978). [CrossRef]

10.

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

11.

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

12.

K. Kim, L. Mandel, and E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A 4(3), 433–437 (1987). [CrossRef]

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 27(2), 188–199 (2010). [CrossRef] [PubMed]

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]

15.

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

16.

A. Hyvärinen, “Fast and robust fixed-point algorithms for independent component analysis,” IEEE Trans. Neural Netw. 10(3), 626–634 (1999). [CrossRef] [PubMed]

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

  1. 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]
  2. 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]
  3. 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]
  4. R. Ossikovski, “Analysis of depolarizing Mueller matrices through a symmetric decomposition,” J. Opt. Soc. Am. A26(5), 1109–1118 (2009). [CrossRef] [PubMed]
  5. S. R. Cloude, “Group theory and polarisation algebra,” Optik (Stuttg.)75, 26–36 (1986).
  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.40(1), 1–47 (2007), doi:. [CrossRef]
  8. 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]
  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.68(12), 1756–1767 (1978). [CrossRef]
  10. R. Barakat, “Exponential versions of the Jones and Mueller-Jones polarization matrices,” J. Opt. Soc. Am. A13(1), 158–163 (1996). [CrossRef]
  11. V. Devlaminck, “Mueller matrix interpolation in polarization optics,” J. Opt. Soc. Am. A27(7), 1529–1534 (2010). [CrossRef] [PubMed]
  12. 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]
  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. A27(2), 188–199 (2010). [CrossRef] [PubMed]
  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]
  15. A. Hyvärinen and E. Oja, “Independent component analysis: algorithms and applications,” Neural Netw.13(4-5), 411–430 (2000). [CrossRef] [PubMed]
  16. 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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