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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 25 — Dec. 5, 2011
  • pp: 25188–25198
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Full analytical solution of adapted polarisation state contrast imaging

Debajyoti Upadhyay, Sugata Mondal, Eric Lacot, and Xavier Orlik  »View Author Affiliations


Optics Express, Vol. 19, Issue 25, pp. 25188-25198 (2011)
http://dx.doi.org/10.1364/OE.19.025188


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Abstract

We have earlier proposed a 2-channel imaging technique: Adapted Polarisation State Contrast Imaging (APSCI), which noticeably enhances the polarimetric contrast between an object and its background using fully polarised incident state adapted to the scene, such that the polarimetric responses of those regions are located as far as possible on the Poincaré sphere. We address here the full analytical and graphical analysis of the ensemble of solutions of specific incident states, by introducing 3-Distance Eigen Space and explain the underlying physical structure of APSCI and the effect of noise over the measurements.

© 2011 OSA

1. Introduction

Taking into account the vectorial nature of the electromagnetic fields, polarimetric imaging contains obviously additional informations compared to classical imaging. The Mueller formalism describes the full polarimetric response of an object and thus is able to describe completely its properties of birefringence, dichroism, depolarization and polarisance [1

1. J. L. Pezzaniti and R. A. Chipman, “Mueller matrix imaging polarimetry (journal paper),” Opt. Eng. 34(06), 1558–1568 (1995). [CrossRef]

4

4. R. Ossikovski, M. Anastasiadoua, and A. De Martino, “Product decompositions of depolarizing Mueller matrices with negative determinants,” Opt. Commun. 281(9), 2406–2410 (2008). [CrossRef]

]. However, these above mentioned properties exhibit respectively 3, 4 and 9 degrees of freedom and only a few of them remain pertinent for a given polarimetric scene. Moreover, there could be some situations when the object and the background would differ very weakly from 2 or more polarimetric degrees of freedom and that they could become distinguishable only by cumulating the global polarimetric information. Thus, we have proposed in earlier works [5

5. M. Richert, X. Orlik, and A. De Martino, “Optimized orthogonal state contrast image,” Proceedings of the 21th Congress of the Int. Comm. for Opt. (2008), Vol. 139.

, 6

6. M. Richert, X. Orlik, and A. De Martino, “Adapted polarization state contrast image,” Opt. Express 17, 14199–14210 (2009). [CrossRef] [PubMed]

] a specific method to maximize the polarimetric contrast between an object and its background that takes into account the total polarimetric information of the scene [7

7. F. Goudail, “Optimization of the contrast in active Stokes images,” Opt. Lett. 34, 121–123 (2009). [CrossRef] [PubMed]

9

9. L. Meng and J. P. Kerekes, “Adaptive target detection with a polarization-sensitive optical system,” Appl. Opt. 50, 1925–1932 (2011). [CrossRef] [PubMed]

]. This method consists of finding the fully polarised incident states that excite the object and background in such a way that they scatter light with polarimetric states located as far as possible on the Poincaré sphere [10

10. H. Poincaré, Théorie mathématique de la lumière (GABAY, 1892). [PubMed]

]. Also optimized with adapted detection states, this method has provided in the presence of shot noise, gains in contrasts quantified by Bhattacharyya distances [11

11. A. Bhattacharyya, “On a measure of divergence between two statistical populations defined by probability distributions,” Bull. Calcutta Math. Soc. 35, 99–109 (1943).

, 12

12. F. Goudail, P. Réfrégier, and G. Delyon, “Bhattacharyya distance as a contrast parameter for statistical processing of noisy optical images,” J. Opt. Soc. Am. A 21, 1231–1240 (2004). [CrossRef]

] that could reach up to some factors of the order of 10 in some situations [6

6. M. Richert, X. Orlik, and A. De Martino, “Adapted polarization state contrast image,” Opt. Express 17, 14199–14210 (2009). [CrossRef] [PubMed]

]. It has also been shown that the performances were not noticeably degraded by additional high contrast circular Gaussian speckle noise [13

13. D. Upadhyay, M. Richert, E. Lacot, A. De Martino, and X. Orlik, “Effect of speckle on APSCI method and Mueller imaging,” Opt. Express 19(5), 4553–4559 (2010). [CrossRef]

]. However, the ensemble of solutions describing the incident adapted polarimetric states were determined numerically by an iterative optimization process and no explanations were given concerning the nature of these solutions (circle or point on the Poincaré sphere) in function of the scene under investigation.

In this paper, we provide the full analytical solution of the adapted polarimetric states and explain the underlying physics through introducing the concept of 3-Distance Eigen Space. By mapping the mathematical theory with physical imaging technique we address the following issues :
  1. What are the ensembles of solutions for the specific excitations states depending on the polarimetric scenes considered.
  2. What is the effect of the shot noise on these solutions and
  3. for a given polarimetric scene, what is the theoretical maximal distance in the Poincaré sphere between the two Stokes vectors describing the field scattered by the object and background region.

2. The 3-Distance Eigen Space and Adapted Polarisation State Contrast Imaging

2.1. The Mathematical Modeling of the Problem

We consider a scene with a polarimetrically homogeneous circular object surrounded by a homogeneous background. The Mueller matrices MO and M B fully describes the polarimetric properties of the two disjoint regions 𝒪 and respectively for the object and the background. Here MO ij and MB ij belongs to R ∀ (i, j) ∈ [0, 3]. Considering experimental situation, the scene is considered to be a priori unknown and hence we need an initial estimation of the Mueller matrices of the object O and of the background B by Classical Mueller Imaging (CMI) before using APSCI method. Here similarly, from the physicality criterion O and B belongs to R 4XR 4. During CMI, we consider that each pixel of the detector indexed by (u, v) receive an intensity I(u, v) perturbed by a shot noise of Poisson distribution. Then from noisy intensity matrices Ĩ(u, v), the Mueller matrices (u, v) are retrieved for each pixel. The noise at any two pixels of the detector are also considered to be mutually independent.

Let us consider a Stokes vector S used to illuminate the scene after the initial evaluation of Mueller matrices for 𝒪 and . Here SB 4. B 4 is the closed ball in 4 dimensions. Which implies, SR 4 : ||S||2 ≤ 2. The estimations of the Stokes vectors of the field scattered by the object O and background B can be expressed as :
S˜O=[SO0,SO1,SO2,SO3]T=M˜OS,S˜B=[SB0,SB1,SB2,SB3]T=M˜BS
(1)

In APSCI, we only use fully polarised illumination states and hence, ||S||2 = 2 i.e. S𝒫, where 𝒫 is the set of fully polarised Stokes vectors corresponding to all the possible states on the surface of Poincaré sphere.

We define the 3-Distance between O and B in the Poincaré sphere by the Euclidean distance D considering their last three parameters in following way:
D(S)=k=13(SOkSBk)2
(2)

The goal is to find the specific Stokes vector Smax for which,
D(Smax)=maxS𝒫(D(S))
(3)

2.2. The Solution

Let us consider the following equations,
ΔM=M˜OM˜B=[Δm00Δm01Δm02Δm03Δm10Δm11Δm12Δm13Δm20Δm21Δm22Δm23Δm30Δm31Δm32Δm33]
(4)

Then, under illumination of the scene with the Stokes vector S = (s 0, s 1, s 2, s 3)T, the difference vector is
ΔM.S=(M˜OM˜B).S=[Δm00s0+Δm01s1+Δm02s2+Δm03s3Δm10s0+Δm11s1+Δm12s2+Δm13s3Δm20s0+Δm21s1+Δm22s2+Δm23s3Δm30s0+Δm31s1+Δm32s2+Δm33s3]
(5)

Hence, from Eq. (3), we have to maximise the sum of the squares of the last three components of the right hand side of Eq. (5). Now this term is the 3-Distance quadratic function of si’s, ∀i ∈ 0, 1, 2, 3. Let us denote this quadratic function by 𝒬 (s 0, s 1, s 2, s 3). Then we get,
𝒬(s0,s1,s2,s3)=j=13(i=03Δmjisi)2=i,j=03aijsisj
(6)

Here, we want to specify that we would like to have a symmetric solution for aij and which is unique. It can be calculated from Eq. (6) that
aij=k=13ΔmkiΔmkj,i,j0,1,2,3
(7)

From Eq. (7), we can easily verify that aij’s ∀i, j ∈ 0, 1, 2, 3 are symmetric. Hence,
A(i,j)=aij=aji,i,j0,1,2,3
(8)

Here A is a real symmetric 4X4 matrix, which implies that all its eigen values and eigen vectors are real and can be diagonalised to Ad by an orthogonal matrix transformation of form OT AO, where O is orthogonal matrix (columns of which are eigen vectors of A) and Ad is real and diagonal (having the corresponding eigenvalues of A on the diagonal).

Thus,
max(𝒬(s0,s1,s2,s3))=max(i,j=03aijsisj)=max(STAS)
(9)

From linear algebra we know that 𝒬 will be maximum for Sin = Smax, where Smax is the eigen vector of A corresponding to maximum eigen value λmax. Hence we get,
max(𝒬)=SmaxTASmax=λmaxSmax2
(10)

2.3. 3-Distance Eigen Space

An orthogonal transformation in 2D is similar to an operation of rotation. Thus, the transformation of S 0, S 1, S 2, S 3 space to the eigen space of matrix corresponding to 3-Distance Quadratic Function is equivalent to a rotation in 4D space. Moreover, from a set of orthogonal basis (here it is S 0, S 1, S 2, S 3) if we move to the eigen bases of a symmetric matrix (in this case it is A), the new bases will also remain orthogonal. The nomenclature for this new rotated space is from now on will be 3-Distance Eigen Space (3DES). 3DES has two intrinsic properties. Firstly, it depends on the quadratic form we work with, which can be very general and irrespective of the imaging problem we are addressing here (in our case the Cartesian distance function on Poincaré space of Stokes vectors) Eq. (6) and can be broadened further and secondly it depends on the exact Mueller matrices of the scene we work with and this is what makes the 3DES adaptive to our imaging. Now if we concentrate on the trace of A, we can see that:
Tr(A)=i=03aii=i=03k=13ΔmkiΔmki=i,k=03(Δmki)2j=03(Δm0j)2=i=03λi
(11)

Here λi’s (∀i ∈ (0, 1, 2, 3)) are the eigen values of A. As we know under similarity transformation the trace is conserved and hence Eq. (11). In the result section we analyse some properties of 3DES and it becomes immensely helpful to understand underlying physics of APSCI.

3. The Illumination Technique of APSCI

Here we finally determine the exact polarimetric illumination state of APSCI by taking into account the above analytical solution along with physical realisation criterions. Firstly, as we have seen in the last section λmax and Smax are real, hence we can reach a physically realisable solution. Though due to the solution space (i.e. ||S||2 ≤ 2) we work with, we may come up with solutions where input Stokes vectors are not fully polarised. For the sake of APSCI, only fully polarised solutions are selected. As we can readily see from Eq. (10), the maximum 3-Distance is proportional to Smax2, which is maximum and equal to 2 for only fully polarised illumination. However, the upper limit of the 3-Distance is 2λmax, but it might not be achievable in most of the imaging situations. In fact here, we are obtaining the direction of the illumination state using mathematical algorithm and then we are stretching out Stokes vector norm to unity in order to achieve specific fully polarised incident light. So, the algorithm we use to get Smaxnew after initial evaluation of Smax is as follow :
  1. Normalisation by the last 3 parameters:
    Smaxnew=Smaxk=13(Smaxk)2
    (12)
  2. Selection of the Physical solution by multiplying by −1 the entire Stokes vector in case its component So describing the intenty is negative.

Concerning step 2 as we can see from Eq. (10), Smax and –Smax both will yield the maximum of 3-Distance Quadratic Function, and hence we choose the physically possible solution so that S max0 is positive.

We would also like to mention here that though the input Stokes vector are fully polarised, the output Stokes vectors O and B are not normalized and hence they can exhibit different rates of depolarization. As a consequence, the maximization of the 3-Distance mentioned above takes into account both the physical entities: the polarization state and the intensity of the polarized part of the scattered field.

4. The Detection Technique of APSCI

In the detection part we project the scattered fields SoutO and SoutB respectively from 𝒪 and into some state Sd on the surface of Poincaré sphere. Then to maximise the imaging contrast we have to maximise the difference of projection. Hence we have to maximize the following function F(Sd)
F(Sd)=Sd.SoutOSd.SoutB
(13)
where ||Sd||2 = 2

In order to optimize F(Sd) we differentiate both side of Eq. (13) and set it equal to 0 and get,
dF(Sd)=dSd.(SoutOSoutB)=0
(14)

Hence we can say from Eq. (14), that dSd is orthogonal to (SoutOSoutB). Again differentiating ||Sd||2 = 2 we get,
2dSd.Sd=0
(15)

So by comparing Eqs. (14) and (15), |F| will be maximum for Sd parallel and antiparallel to (SoutOSoutB). Now, we use a Two Channel Imaging (TCI) system that projects SoutO and SoutB, into the following 2 orthogonal polarisation states of analyzer:
Sout1=[1,ΔST/ΔS]T,Sout2=[1,ΔST/ΔS]T
(16)

Where ΔSi=SoutiOSoutiB, ∀i ∈ (1, 2, 3). Thus, Sout 1 and Sout 2 will increase the detected intensities corresponding respectively to the object and background pixels while imaging.

Now we define the APSCI parameter for each pixel of the detector indexed by corresponding coordinates (u, v) as :
APSCI(u,v)=I1(u,v)I2(u,v)I1(u,v)+I2(u,v),
(17)

Where I 1(u, v) and I 2(u, v) are the detected intensities after projection respectively on the 2 states of polarization Sout 1 and Sout 2.

Finally, in order to quantify the imaging contrast of the scene under investigation, we calculate the Bhattacharyya Distance, defined by the measurement of the amount of overlap between the two statistical populations that are the APSCI parameter over the background and object region.

5. Results and Analysis

We can see in Fig. 1(a), that under several evaluations of Smax for different realizations of shot noise, the ensemble of solutions is on and around a great circle over Poincaré sphere. From theoretical solutions, we can see in this case if we consider pure Mueller matrices (not affected by shot noise), then the eigen space corresponding to λmax is degenerate. In this kind of cases, two eigen vectors, let’s say Smax1, Smax2, both corresponds to the maximum eigen value. Hence we get the solution ensemble as the plane defined by Smax1, Smax2. In presence of shot, we consider the evaluated Mueller matrices and we see that the degeneracy in 3DES corresponding to maximal eigen value breaks and hence one of the eigen directions becomes preferred over the others. In this figure we have plotted the degenerate eigen vectors ([1, 0, 1, 0] using the red arrow and [1, 0, 0,1] using the pink arrow, both correspond to same eigen value λmax = 0.0979) considering pure Mueller matrices and modified eigen directions ([1, 0.0184, 0.2636, −0.9644] using the dark green arrow corresponds to eigen value = 0.0887 and [1, 0.0107, 0.9645, 0.2639] using the light green arrow corresponds to eigen value = 0.0991) considering Mueller matrices perturbed by a single instance of shot noise. We can easily verify from the Fig. 1(a) that the two eigen planes have a small angle between them in (S 1, S 2, S 3) co-ordinate system and this statistics of the angular shift between the eigen planes in 3DES can provide the effect of noise during measurements. To provide an intuitive understanding we investigate the case graphically in Fig. 1(b). In case of birefringence, the direction of the birefringence vector in Poincaré sphere is defined by corresponding azimuth and ellipticity, while the scalar birefringence is defined by the rotation angle around it. Now as in our scene the object and the background only differ in scalar birefringence, hence the angles of rotation (for the object denoted by dark blue, for the background denoted by bluish green) of a polarimetric state around the same birefringence axis (denoted by the red arrow) are different. The distance between the final states (heads of arrows) will in turn denote the 3-Distance (denoted by the pink arrow) in the Poincaré sphere. For any point Sin not lying in the equatorial plane perpendicular to the birefringence vector, the same angular rotation will always yield lesser length of the pink arrow compared to any point Smaxnew chosen over the great circle lying on this plane and for all the points Smaxnew’s on that great circle we expect same length of the pink arrow. From this graphical explanation, then we can further confirm the reason of circular cluster for ensemble of solutions.

Fig. 1 (a) Ensemble of solutions of the incident adapted states for a scene with difference in 10% in scalar birefringence. The arrows defines the eign vectors using pure Mueller matrices(red and pink) and using Mueller matrices pertubed by a generic shot noise(light and dark green) (b) Graphical representation of the birefringence scene and at upper right corner: the ensemble of solutions over Poincaré sphere for different realisations of shot noise. (c) Ensemble of solutions for a scene with difference in 10% in azimuth of dichroic vector. The arrows show the average direction of incident adapted states from iterative simplex search method (blue) and from analytical method (green) (d) Graphical representation of the dichroic scene and at the upper right corner: the ensemble of solutions over Poincaré sphere for different realisations of shot noise. The colour mapping on the spheres at the upper right corners for (b) and (d) maps linearly the 3-Distance for all the polarimetric states on Poincaré sphere considering corresponding pure Mueller matrices respectively.

In the upper right corner image of Fig. 1(b), we show the distribution of evaluated states on Poincaré sphere. The colourmap of the surface of sphere maps the 3-Distance considering pure Mueller matrices. The black lines defines the boundary of the region where 3-Distance is 95% of maximum or above.

From Fig. 1(d), using graphical analysis we explain intuitively the ensemble of solutions in this case. For a dichroic object we define the direction dichroic vector by its azimuth and ellipticity. Under some polarimetric illumination of a dichroic object the Stokes vector turns towards the dichroic vector and the decrement of its magnitude is dictated by the scalar dichroism.

In the case concerned, our object and background differ only in the azimuth of their dichroic vectors. Object dichroic vector is denoted by D⃗O using the blue arrow, while the background dichroic vector is by D⃗B using the greenish blue arrow in Fig. 1(d). Any incident Stokes vector Sin on Poincaré sphere will provide more or less similar 3-Distance. Only the Stokes vector, that is bisecting the angle between D⃗B and D⃗O, will provoke a scattering in almost opposite direction for object and background and hence will yield the maximum 3-Distance(denoted by the pink arrow) and that’s the reason why we get the ensemble of solutions as a cluster of points.

From Fig. 1(d), using graphical analysis we explain intuitively the ensemble of solutions in this case. For a dichroic object we define the direction dichroic vector by its azimuth and ellipticity. Under some polarimetric illumination of a dichroic object the Stokes vector turns towards the dichroic vector and the decrement of its magnitude is dictated by the scalar dichroism. In the case concerned, our object and background differ only in the azimuth of their dichroic vectors. Object dichroic vector is denoted by D⃗O using the blue arrow, while the background dichroic vector is by D⃗B using the greenish blue arrow in Fig. 1(d). Any incident Stokes vector Sin on Poincaré sphere will provide more or less similar 3-Distance. Only the Stokes vector, that is bisecting the angle between D⃗B and D⃗O, will provoke a scattering in almost opposite direction for object and background and hence will yield the maximum 3-Distance(denoted by the pink arrow) and that’s the reason why we get the ensemble of solutions as a cluster of points.

In Fig. 2 we analyze a complex scene with a difference of 10% between object and background in scalar birefringence, scalar dichroism and scalar depolarisation simultaneously. In this case the real Mueller matrices of the object and background regions are the following,
MO=[0.5000.300000.1800.30000000.1120.212000.1760.094]andMB=[0.5000.250000.1250.25000000.0370.213000.2130.037]
(18)

Fig. 2 (a) Comparisons of contrast from different pertinent parameters by Bhattacharyya distance vs. SNR curves. (b) at SNR = 3.4, (i.) cluster of solutions from analytic (green points) and simplex search (blue points) method on Poincaré sphere, and Visual comparison of the complex scene obtained from (ii.) using best parameter of Mueller Matrix, (iii.) using APSCI by simplex search method and (iv.) using APSCI by analytical method.

Under the effect of the shot noise the average of the retrieved Mueller matrices using CMI for the object and the background are as follows,
M˜O=[0.5010.2960.0010.0020.1820.2970.0020.0030.0010.0010.1120.2090.0020.0010.1710.091]andM˜B=[0.4990.2490.0010.0060.1230.2450.0020.0000.0000.0020.0400.2130.0000.0080.2100.042]
(19)

6. Conclusion

The APSCI method takes into account the global cumulative polarimetric differences between the object and background and such technique is expected to increase noticeably our ability to detect and characterize objects. In this work, the underlying physics driving the APSCI method has been explained, using linear algebraic structure of the problem. In the general case, we have provided the full analytical solution of the ensemble of adapted illumination polarimetric states needed to optimize the contrast between an object and its background in function of their respective polarimetric properties. Given a scene with polarimetric object and background, we have provided the maximum value of contrast achievable using the APSCI method. We have also shown that the presence of shot noise broke the degeneracy in the ensemble of adapted states. Moreover, at low SNR condition, the analytical method has significantly improved the value and reliability of contrast in imaging, which is an upshot from experimental point of view. Such specific polarimetric imaging technique opens the vast field to promising applications concerning biological tissues that exhibit often several polarimetric properties simultaneously.

Acknowledgments

We would like to thank the Région Midi-Pyrénées for providing us the financial support for this work.

References and links

1.

J. L. Pezzaniti and R. A. Chipman, “Mueller matrix imaging polarimetry (journal paper),” Opt. Eng. 34(06), 1558–1568 (1995). [CrossRef]

2.

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]

3.

J. E. Wolfe and R. A. Chipman, “Polarimetric characterization of liquid-crystal-on-silicon panels,” Appl. Opt. 45, 1688–1703 (2006). [CrossRef] [PubMed]

4.

R. Ossikovski, M. Anastasiadoua, and A. De Martino, “Product decompositions of depolarizing Mueller matrices with negative determinants,” Opt. Commun. 281(9), 2406–2410 (2008). [CrossRef]

5.

M. Richert, X. Orlik, and A. De Martino, “Optimized orthogonal state contrast image,” Proceedings of the 21th Congress of the Int. Comm. for Opt. (2008), Vol. 139.

6.

M. Richert, X. Orlik, and A. De Martino, “Adapted polarization state contrast image,” Opt. Express 17, 14199–14210 (2009). [CrossRef] [PubMed]

7.

F. Goudail, “Optimization of the contrast in active Stokes images,” Opt. Lett. 34, 121–123 (2009). [CrossRef] [PubMed]

8.

F. Goudail and A. Béniére, “Optimization of the contrast in polarimetric scalar images,” Opt. Lett. 34, 1471–1473 (2009). [CrossRef] [PubMed]

9.

L. Meng and J. P. Kerekes, “Adaptive target detection with a polarization-sensitive optical system,” Appl. Opt. 50, 1925–1932 (2011). [CrossRef] [PubMed]

10.

H. Poincaré, Théorie mathématique de la lumière (GABAY, 1892). [PubMed]

11.

A. Bhattacharyya, “On a measure of divergence between two statistical populations defined by probability distributions,” Bull. Calcutta Math. Soc. 35, 99–109 (1943).

12.

F. Goudail, P. Réfrégier, and G. Delyon, “Bhattacharyya distance as a contrast parameter for statistical processing of noisy optical images,” J. Opt. Soc. Am. A 21, 1231–1240 (2004). [CrossRef]

13.

D. Upadhyay, M. Richert, E. Lacot, A. De Martino, and X. Orlik, “Effect of speckle on APSCI method and Mueller imaging,” Opt. Express 19(5), 4553–4559 (2010). [CrossRef]

14.

R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett. 32, 689–691 (2007). [CrossRef] [PubMed]

OCIS Codes
(110.2970) Imaging systems : Image detection systems
(120.5410) Instrumentation, measurement, and metrology : Polarimetry
(260.5430) Physical optics : Polarization
(110.5405) Imaging systems : Polarimetric imaging

ToC Category:
Imaging Systems

History
Original Manuscript: May 17, 2011
Revised Manuscript: June 24, 2011
Manuscript Accepted: July 5, 2011
Published: November 23, 2011

Citation
Debajyoti Upadhyay, Sugata Mondal, Eric Lacot, and Xavier Orlik, "Full analytical solution of adapted polarisation state contrast imaging," Opt. Express 19, 25188-25198 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-25-25188


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References

  1. J. L. Pezzaniti and R. A. Chipman, “Mueller matrix imaging polarimetry (journal paper),” Opt. Eng. 34(06), 1558–1568 (1995). [CrossRef]
  2. 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]
  3. J. E. Wolfe and R. A. Chipman, “Polarimetric characterization of liquid-crystal-on-silicon panels,” Appl. Opt. 45, 1688–1703 (2006). [CrossRef] [PubMed]
  4. R. Ossikovski, M. Anastasiadoua, and A. De Martino, “Product decompositions of depolarizing Mueller matrices with negative determinants,” Opt. Commun. 281(9), 2406–2410 (2008). [CrossRef]
  5. M. Richert, X. Orlik, and A. De Martino, “Optimized orthogonal state contrast image,” Proceedings of the 21th Congress of the Int. Comm. for Opt. (2008), Vol. 139.
  6. M. Richert, X. Orlik, and A. De Martino, “Adapted polarization state contrast image,” Opt. Express 17, 14199–14210 (2009). [CrossRef] [PubMed]
  7. F. Goudail, “Optimization of the contrast in active Stokes images,” Opt. Lett. 34, 121–123 (2009). [CrossRef] [PubMed]
  8. F. Goudail and A. Béniére, “Optimization of the contrast in polarimetric scalar images,” Opt. Lett. 34, 1471–1473 (2009). [CrossRef] [PubMed]
  9. L. Meng and J. P. Kerekes, “Adaptive target detection with a polarization-sensitive optical system,” Appl. Opt. 50, 1925–1932 (2011). [CrossRef] [PubMed]
  10. H. Poincaré, Théorie mathématique de la lumière (GABAY, 1892). [PubMed]
  11. A. Bhattacharyya, “On a measure of divergence between two statistical populations defined by probability distributions,” Bull. Calcutta Math. Soc. 35, 99–109 (1943).
  12. F. Goudail, P. Réfrégier, and G. Delyon, “Bhattacharyya distance as a contrast parameter for statistical processing of noisy optical images,” J. Opt. Soc. Am. A 21, 1231–1240 (2004). [CrossRef]
  13. D. Upadhyay, M. Richert, E. Lacot, A. De Martino, and X. Orlik, “Effect of speckle on APSCI method and Mueller imaging,” Opt. Express 19(5), 4553–4559 (2010). [CrossRef]
  14. R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett. 32, 689–691 (2007). [CrossRef] [PubMed]

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