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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 1 — Jan. 8, 2007
  • pp: 83–96
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Polarimetric data reduction: a Bayesian approach

Jihad Zallat and Christian Heinrich  »View Author Affiliations


Optics Express, Vol. 15, Issue 1, pp. 83-96 (2007)
http://dx.doi.org/10.1364/OE.15.000083


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Abstract

In this paper, we introduce a general Bayesian approach to estimate polarization parameters in the Stokes imaging framework. We demonstrate that this new approach yields a neat solution to the polarimetric data reduction problem that preserves the physical admissibility constraints and provides a robust clustering of Stokes images in regard to image noises. The proposed approach is extensively evaluated by using synthetic simulated data and applied to cluster and retrieves the Stokes image issuing from a set of real measurements.

© 2007 Optical Society of America

1. Introduction

The main interest of the Stokes-Mueller formalism in optical imaging is mainly due to the definition of light polarization parameters in terms of real quadratic observables (intensities) which are directly sensed by CCD detectors. This allows extending classical intensity-wise imaging systems to acquire Stokes images through the use of Polarization State Analyzers (PSA) in front of the camera. Many such systems that use different polarization modulation techniques, have been built in recent years for many application areas ranging from metrology [1

1. D. Miyazaki, M. Saito, Y. Sato, and K. Ikeuchi, “Determining surface orientations of transparent objects based on polarization degrees in visible and infrared wavelengths,” J. Opt. Soc. Am. A 19,687–694 (2002). [CrossRef]

, 2

2. D. Miyazaki, M. Kagesawa, and K. Ikeuchi, “Transparent surface modeling from a pair of polarization images,” IEEE Trans. PAMI 26,920–932 (2004). [CrossRef]

] to biomedical imaging [3

3. J. M. Bueno and P. Artal, “Double-pass imaging polarimetry in the human eye,” Opt. Letters. 2464–66 (1999) [CrossRef]

, 4

4. S. D. Giattina, et al., “Assessment of coronary plaque collagen with polarization sensitive optical coherence tomography (PS-OCT),” Int. J. Cardiol. 107,400–409 (2006). [CrossRef] [PubMed]

], and remote sensing [5

5. D. H. Goldstein and D. B. Chenault, and Society of Photo-optical Instrumentation Engineers, Polarization: measurement, analysis, and remote sensing II, 19–21 July, 1999, Denver, Colorado. 1999, Bellingham, Washington: SPIE. ix, 426 p.

].

Whatever the modulation principle used in the PSA configuration, the measurement procedure remains the same. Indeed, the PSA analyses the incoming Stokes vector by measuring its projections onto N ( N ≥ 4 ) independent basis states. The complete set of the N measurements yields a matrix equation, which relates the out-coming Stokes vector S from the sample to the measured raw intensity data vector I for each pixel. This matrix relation has to be inverted properly. This implies that an efficient calibration procedure of polarimetric imaging systems must be employed in order to extract the desired polarization-encoded images effectively. Many interesting studies in regard to this problem have been published in the recent literature, see for example [6

6. M. H. Smith, “Optimizing a dual-rotating-retarder Mueller matrix polarimeter,” in Polarization Analysis and Measurements IV, SPIE (2001).

9

9. S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng. 41,965–972 (2002). [CrossRef]

]. All of these papers focused on the calibration strategies to adopt and on the suited optical configurations in order to obtain the best conditioning of the aforementioned matrix equation to yield the polarization-encoded images.

In this paper, we reformulate the polarization data reduction problem in the general framework of Bayesian inference theory and demonstrate that this new approach allows a neat solution to the polarimetric data reduction problem that preserves the physical admissibility constraints and provides a robust clustering of Stokes images in regard to image noises.

2. Problem statement

2.1 Observation model in the Stokes imaging polarimetry framework

In such a system, the polarization state of the light coming from the scene is acquired. This can be done by inserting a complete state analyzer in front of the camera. At least, four independent states of the analyzer are needed to acquire all elements of the Stokes vector for each pixel location (jy, jx). This can be summarized by the following equation:

Imjyjx=P(η)Sinjyjx
(1)

where I m is the acquired image, P is the “polarization measurement matrix” (PMM) that depends on a parameter vector (η), while Sin is the Stokes vector to be estimated.

We note further that I m(jy, jx) will denote the intensity vector measured at the (jy, jx) pixel location while S in(jy, jx) represents the corresponding Stokes vector.

In the following, the image I m, and the Stokes image will be arranged in data cube structures such that:

Im={Imjyjxjn;jy1Jy,jx[1,Jx],jn0N1}
Sin={Sinjyjxje;jy1Jy,jx1Jx,je0,3}
(2)

Classical data reduction employs the pseudoinverse P # of the PMM to obtain a minimum-norm, least-square estimate Ŝin for the Stokes vector at each pixel location:

Ŝinjyjx=P#Imjyjx
(3)

If one uses four probing states to measure the Stokes vector, P # is identical to the inverse of P provided that η is chosen such that P is invertible. The reason for using N intensity measurements where N > 4 that yield an over-determined system for Eq. (1) is to reduce the system sensitivity to systematic errors as well as image noises. The drawback of such a strategy is an increase in acquisition, and data processing and handling time.

Systematic errors and the use of CCD detectors lead to noises that contaminate the measured raw data. This includes misalignment errors of optical elements, readout noise, thermal noise, dark current noise and photon noise among others. These perturbations can be accounted for by replacing Eq. (1) by:

Imjyjx=P(η)Sinjyjx+δIjyjx
(4)

where the term δI incorporates all sources of possible noises.

We write now δI as the sum of two components, i.e. δII p + ε. The first noise term is proportional to the measured intensities while the second one includes other independent additive noises. ε is assumed to follow a Gaussian distribution and δI p is attributed to photon noise that can be estimated by observing that, for each intensity measurement Ii, δI i/Iiis proportional to the inverse of the square root of the number of detected photons (nphi):

δIiIi1niph
(5)

We note further that the number of detected photons depends mainly on the quantum efficiency (QE ) of the used photosensitive device (e.g. CCD camera) which in turn depends on the wavelength. This dependency can be expressed formally as:

niph=QE(λ)Niph
(6)

where Nphi is the number of photons impinging onto the detector and λ is the considered wavelength.

If now we consider the case of N intensity measurements where each one contains nearly the same amount of statistical noise, we can assume that the same number of photons nph(λ) contributes to each one. Finally, δI p can be written as:

δIpjyjx=1QE(λ)NphImjyjx=ρImjyjx=ρPSinjyjx
(7)

We note further that in the case of coherent illumination, the ρ factor includes also the contribution of the speckle noise.

The final observation model can be written down:

Imjyjx=(1+ρ)P(η)Sinjyjx+εjyjx=Idjyjx+εjyjx
(8)

where I d = (1+ρ)PS in.

The Stokes images estimation problem will be tackled in a Bayesian framework [10

10. J. Bernardo and A. Smith, Bayesian Theory, (Wiley, 2000).

, 11

11. A. Gelman, J. Carlin, H. Stern, and D. Rubin, Bayesian data analysis, Second ed., (CRC Press, 2003).

], where all variables involved will be considered as random variables.

2.2 Underlying segmentation model

The main feature of the estimation procedure is to explicitly model and estimate the unknown underlying segmentation image and to account for the physical admissibility constraints. The segmentation image will be denoted

I0={I0jyjx;I0jyjx[1,K],jy1Jy,jx1Jx}
(9)

where K is the number of classes.

Considering such segmentation is justified since each Stokes vector is taken from a set of reduced cardinality: a Stokes vector is assumed to be invariant for a given type of class and the cardinality of the set is equal to the number of classes types in the analyzed sample. Even though the model can handle any number of probing states, without loss of generality, we consider here the case of a four probing states polarimeter, i.e., four intensity images are acquired to estimate the Stokes image. Let Φ being the matrix defined as:

Φ=[ϕ0,1ϕ0,2ϕ0Kϕ3,1ϕ3,2ϕ3K]
(10)

where Sin(jy, jx, je)=ϕ(je, I 0(jy, jx)) and consequently, the kth column of the matrix Φ represents the Stokes vector that corresponds to the kth class in the image. This hypothesis amounts to considering each Stokes parameter as being constant inside each class.

Fig. 1. The directed acyclic graph (DAG) that represents the dependence relations between the variables. According to usual conventions, square boxes represent fixed or observed quantities whereas circles represent quantities to be estimated.

Since there is a high probability that two neighboring pixels in a Stokes channel image have the same ϕ value, 0 I will be modeled as a Markov random field (MRF) [12

12. S. Z. Li, Markov random field modeling in image analysis, Second ed., (Springer, 2001).

]. Having a closed form partition function is a desirable feature to ease estimation of the MRF hyperparameter. We consider here the Markov mesh model studied by Gray et al. [13

13. A. Gray, J. Kay, and D. Titterington, “An empirical study of the simulation of various models used for images,” IEEE Trans. PAMI ,16,507–513 (1994). [CrossRef]

] and by Dunmur and Titterington [14

14. A. Dunmur and D. Titterington, “Computational Bayesian analysis of hidden Markov mesh models,” IEEE PAMI. 19,1296–1300 (1997). [CrossRef]

]. Only the main lines of the model are summarized here, the interested reader may refer to the cited references for more details. The MRF is governed by the hyperparameter β 0, which controls the size of the clusters: values of β 0 close to 1 yield monochrome images, whereas β 0 = 1/K corresponds to an independent pixel wise prior. The proposed probabilistic model reads:

{p(I01,1=k1)=1Kk1=1,...,Kp(I01jx=k1I01jx1=k2)={β0ifk1=k2forjx>11β0K1ifk1k2p(I0jy1=k1I0(jy1,1)=k2)={β0ifk1=k2forjy>11β0K1ifk1k2p(I0jyjx=k4I0jy1jx=k1I0jyjx1=k2I0jy1jx1=k3)={β0ifk1=k2=k3=k41β0K1ifk1=k2=k3k4forjx>1,jy>11Kotherwise
(11)

where ki ∈[1, K] are the four neighbors labels values. This third-order Markov mesh model, where a given pixel admits three causal neighbors, is equivalent to a second order MRF where a pixel admits eight neighbors.

This model yields p(I0|β0,K)=[β0]n1[1β0K1]n2[1K]n3, where the integers ni depend only on I 0.

2.3 Noise model

The noise will be supposed to be independent and Gaussian distributed according to

εjyjxjnI0jyjx=kNμk(jn)σk(jn)
(12)

where (μ(jn), σ(jn)) are the noise parameters (mean and standard deviation) that affect the kth class in the jn channel image.

Let us define Θ as the set Θ={Θ(jn) ;jn ∈ [0, N−1]}, where

Θ(jn)=[μ1(jn)μ2(jn)μK(jn)σ1(jn)σ2(jn)σK(jn)]
(13)

All these parameters are unknown and will have to be estimated.

2.4 General probabilistic model

The overall probabilistic model may be depicted using a directed acyclic graph (DAG) (see Fig. 1). All variables in square boxes but I m represent fixed hyperparameters. Hyperparameters values are set so that the corresponding priors are weakly informative in the region of interest (see for example p. 735 of Ref. [15

15. S. Richardson and P. Green, “On Bayesian analysis of mixtures with an unknown number of components (with discussion),” J. R. Stat. Soc. Ser. B. 59,731–792 (1997). [CrossRef]

] and the appendix A). All prior distributions are chosen to be conjugate, as is classically done, to allow closed form computation in the optimization procedure which will be detailed in the sequel. Conjugacy may loosely be defined as follows. A prior p(θ) is conjugate to the likelihood p(y∣θ) if the posterior distribution p(y∣θ) is in the same class of distributions as the prior p(θ). Conjugate priors are often good approximations and they simplify computations. In the 10

10. J. Bernardo and A. Smith, Bayesian Theory, (Wiley, 2000).

, 11

11. A. Gelman, J. Carlin, H. Stern, and D. Rubin, Bayesian data analysis, Second ed., (CRC Press, 2003).

] for extensive details on this topic. Let us emphasize that all probability laws derive directly from the Gaussian model used for the noise and from the MRF model used for I 0. Considering these assumptions and the conjugacy property, the other laws are fixed. This explains for example why the μ(jn) k’s have a Gaussian distribution and why the σ−2(jn) k’s follow a Gamma distribution.

The corresponding joint probability distribution π can be written as:

π=p(ImΘ,Φ,I0,ρ)p(Θα4,K)p(α4)p(ΦK)p(I0β0,K)p(β0)p(ρ)p(K)
(14)

where dependencies upon the hyperparameters were dropped for sake of clarity.

The stochastic dependencies are formulated as follows:

{{p(ImΘ,Φ,I0,ρ)=jy,jx,jnp(Im(jy,jx,jn)Θ,Φ,I0,ρ)ImjyjxjnΘ,Φ,I0,ρN(Idjyjxjn+μI0jyjx(jn),σI0jyjx(jn))(seeeqs.(8,12)){p(Θα4,K)=jn,kp(μk(jn))p(σk(jn)α4)μk(jn)Nα1α2(σk(jn))‒2Gaα3α4α4Gaα41α42{p(Φ)=kp(ϕ(:,k))p(ϕ(:,k))[je12πγ2exp(12γ2(ϕjekγ1)2)](Ck)p(I0β0,K)=[β0]n1[1β0K1]n2[1K]n3β0Beβ1β2ρBeδ1δ2KUKminKmax
(15)

Analogous choices for the distribution of Q were made in Ref. [15

15. S. Richardson and P. Green, “On Bayesian analysis of mixtures with an unknown number of components (with discussion),” J. R. Stat. Soc. Ser. B. 59,731–792 (1997). [CrossRef]

].

Notation ℘( ) designates the indicator function, taking values 0 or 1 whether the condition in parentheses is false or true. The condition Ck is the conjunction of conditions ϕ(0, k) ≥ 0 and ϕ0k2je=13φ(je,k)2 ensuring the physical admissibility of the estimated Stokes vectors. U, Ga, and Be define respectively the uniform, Gamma, and Beta laws, as:

U(Kmax,Kmin):p(z)=1KmaxKmin+1(zKminKmax)
Gaλ1λ2:p(z)=λ2Γ(λ1)zλ11exp(λ2z)(z0)
Beλ1λ2)p(z)=Γ(λ1+λ2)Γ(λ1)Γ(λ2)zλ11(1z)λ21(z0,1)
(16)

3. Estimation procedure

All information of this estimation problem is contained in a stochastic point of view in the posterior distribution p(Θ, Φ, I 0) ρ α 4 K β0I m which verifies:

π=p(Θ,Φ,I0,ρ,α4,K,β0Im)p(Im)
(17)

We retain here the maximum a posteriori (MAP) estimate, which is given by the maximization of the posterior or equivalently by the joint distribution. There is no analytical expression for the solution to this optimization problem. We have to resort to an iterative algorithm, where each variable is optimized in turn. This yields a local maximum of the posterior distribution. Details are omitted since each elementary optimization step is straightforward. Let us merely mention that I 0 is updated using a single raster scan (Iterated Conditional Modes, see Ref. [16

16. J. Besag, “On the statistical analysis of dirty pictures (with discussion),” J. R. Stat. Soc. B 48,259–302 (1986).

]) and that a local optimization method is used to update ϕ (:, k) if the solution to the unconstrained problem, e.g., not accounting for Ck, violates the conditions Ck. The number K of classes is not updated during the procedure but optimizations considering all values of K ∈[Kmin, Kmax are achieved.

It is well known that the initialization stage is very important for such procedures. The DAG is here initialized as the min-norm estimate Ŝin obtained by Eq. (3). The initialization of I 0 relies on the Ŝin clustering by a variant of the C-means algorithms family where the used distances were redefined in relation with the Stokes images specificities [8

8. S. Ainouz, J. Zallat, A. de Martino, and C. Collet., “Physical interpretation of polarization-encoded images by color preview,” Opt. Express 14,5916–5927 (2006). [CrossRef] [PubMed]

]. Finally, Φ is initialized by the mean values of Ŝin corresponding to each class. Initializing the other variables follows straightforwardly as given in the Appendix.

The proposed model can handle different practical situations that may arise in the framework of Stokes imaging measurements. Accounting for a number of probing states greater than four or considering different noise characteristics is straightforward.

4. Discussion and analysis

4.1 Simulation results

To illustrate the relevance and the efficiency of the Bayesian approach in polarimetric data reduction, we synthesized a 64-pixel × 64-pixel Stokes image by using the four class label Fig. 2(a). The Stokes vector s1 =(1.0,0.5,0,0.866)t was assigned to the black pixels, s2 =(0.8,0,0,0.8)t to the heavy gray pixels, s3 =(0.9,0.39, −0.675,0.45)t to the light gray pixels, and s4 =(1.2,0.85,0.85,0) to the white pixels in the label map. Fig. 2(b) shows these polarization state locations on the Poincaré sphere.

The PMM of an optimal rotating-retarder polarimeter as defined in Ref. [7

7. J. S. Tyo, “Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error,” Appl. Opt. 41619–630 (2002). [CrossRef] [PubMed]

] was used to generate the corresponding intensity images by mean of Eq. (1). The parameter vector η that controls the PMM is given by η = (δ = 132°,{θi = ±15.12°,±51.7°}) where δ stands for the retardance of the birifringent wave-plate while θi represent its angular positions.

Practically, acquired intensity images are not instantaneous snapshots but result from photons accumulation due to exposure time settings. Moreover, each intensity image combines many sources of errors and noises that make reasonable to assume that ε follows a centred Gaussian distribution. Hence, a zero mean white Gaussian noise with variance of σ 2 n was added to each intensity image for this simulation providing the necessary inputs to our model, e.g. intensity images as well as the PMM P. The aim of this section is to compare the estimated Stokes image ŜLS resulting from Eq. (3) with the one resulting from our Bayesian model (ŜB).

Figure 3 shows the four channels images corresponding to ŜLS for a noise variance σ 2 n = 0.01. Figures 4(a) and 4(b) show the polarization states locations corresponding to ŜLS over the Poincaré sphere for σ 2 n = 0.01(SNR ∼ 20dB) and σ 2 n = 0.1(SNR ∼ 10dB). Points that lie outside the Poincaré sphere correspond to unphysical estimated states. The percentage of unphysical estimated pixel by the min-norm method are respectively 54 % for σ 2 n = 0.01 and 57 % for σ 2 n = 0.1.

Fig. 2. (a). The label map that was used to generate the simulated Stokes image. The s1 Stokes vector was assigned to black pixels (1215 pixels), s2 was assigned to heavy gray pixels (436 pixels), s3 was assigned to light gray pixels (997 pixels), and S4 was assigned to white pixels (1448 pixels). (b) Red circles indicate the positions of the four polarization states corresponding to the si vectors on the Poincaré sphere.

A look at Figs 4(a) and 4(b) shows that even at low level noises, the min-norm inversion yields unphysical estimated Stokes vectors (Degree of Polarization (DOP) greater than one). Moreover, if the points remain well grouped into four different classes for a 20 dB SNR [Fig.4(a)], this is no longer the case for a 10 dB SNR. This suggests that classical clustering algorithms may fail to provide a workable label map [see Fig. 4(b)] that allows an accurate physical interpretation of the Stokes image content.

We apply now the Bayesian approach introduced in the preceding sections to the same estimation problem. We note that our model provides one Stokes estimate per class. The noise estimated variances are 0.0103 for 0.01 and 0.1015 for σ 2 n = 0.1. Table 1 summarizes the estimated Stokes vectors for the different classes. The last two rows lists the RMSE defined as 100×∥si −ŝi/si∥ as well as the ratio of misclassified pixels of each class for the two considered variances. As expected the estimated ρ factor is practically zero. Figure 5 shows the true polarization locations over the Poincaré sphere as well as the estimated ones for the two considered noise variances.

Furthermore, we point out that contrary to the min-norm solution, our model ensures always the physical admissibility of the estimated Stokes parameters. This can be seen clearly by comparing Figs. (4) and (5).

Fig. 3. From upper left to bottom right, the four Stokes channels (s0 (total intensity), s1, s2, and s3) obtained by the min-norm solution ŜLS for a noise variance of 0.01.
Fig. 4. Polarization states locations corresponding to ŜLS over the Poincaré sphere for σ 2 n =0.01(a) and σ 2 n = 0.1 (b). Points that lie outside the Poincaré sphere correspond to unphysical estimated states. The ratios of unphysical estimated states are 54% (a) and 57% (b).

Table 1. Estimated Stokes vectors of each class. The last two rows lists the RMSE and the ratio of misclassified pixels for each class.

table-icon
View This Table
Fig. 5. Locations of the true polarization states (circles) and the estimated ones for σ 2 n = 0.01 (+ sign) and for σ 2 n = 0.1 (crosses).

Figure (6) presents the estimated noise variance versus the true one for different noise variance levels. Circles represent the estimated variance by using the pixels that belong to the largest class. Horizontal bars correspond to the variations of the estimates over the four classes. We conclude that the proposed Bayesian approach has an excellent behavior with regard to different SNR.

Table 1 and Fig. 5 emphasize the accuracy and the performance of the proposed model in estimating the Stokes image as well as the noise parameters.

At this stage, we relax the hypothesis of a uniform noise field over the whole image and consider the case of different noises that contaminate independently each class. One can imagine easily situations where the polarization state of the light emergent from a particular class of pixels to be nearly orthogonal to the analyzer state which reduces the photon flux impinging on the detector which in turn decreases the SNR for the considered pixels. So we carried out simulations with the following noise variances (σ 2 n)i=1,4= 0.1,0.01,0.05, and 0.2 that attain the four classes defined in Fig. 2(a).

Fig. 6. Estimated noise variance vs. true noise variance. Circles represent the estimates that correspond to the largest class. Horizontal bars correspond to the variations of the estimates over the four classes.

Estimated variances were found to be (σ̂2 i=1,4)= 0.1,0.01,0.049, and 0.203. Figure 7 shows the location of the estimated polarization states and the true ones over the Poincaré sphere. Again, we observe the excellent behaviour of our model for the case of different noises that reach each class.

Fig. 7. Locations of the true polarization states (circles) and the estimated ones (crosses) where different noises reach each class.

4.2 True measurement case

Here, the developed model is applied and validated with real measured images acquired by a Stokes imaging polarimeter. The used object to carry out these measurements consists of two dichroic patches (Polaroid) contacted on a diffusing glass slide with a drop of water. The overall mount was backlight illuminated by a vertical polarized beam and the intensity images were sensed by a four probing states rotating quarter-wave-plate analyzer in front of a 12-bits CCD scientific camera. We note that the different orientations were given to transmission axes of the Polaroid shapes to obtain different signatures at the output.

Figure 8 shows the intensity images issued from this configuration; Fig. 9 shows the min-norm estimate of Stokes images; and Fig. 10 in conjunction with Eq. (18) show the solution obtained with our Bayesian model.

Fig. 8. Intensity images corresponding to four polarization probing states used to retrieve the Stokes image. Gray values are scaled for display purposes.
Fig. 9. Stokes channels images issued from intensity images of Fig. 8 by min-norm solution. Gray values are scaled for display purposes.

As expected, the noise that is present in raw intensity data propagates into the Stokes channels. Additional treatments (clustering, validations, etc.) are needed to make the most of these images. Figure 11 illustrates clearly this fact. Indeed, the clutters of points that lie inside the Poincaré sphere do not permit a clear interpretation of polarization properties of the scene.

Fig. 10. The label map obtained with the proposed Bayesian approach. Different gray values correspond to different polarization signatures.
Fig. 11. Locations of the estimated polarization states obtained by min-norm solution (a) and by our Bayesian approach (b).

On the other side, our Bayesian approach yields a robust and accurate solution as illustrated by Figs. (10) and 11(b). The obtained polarization-encoded images are noise free and a neat solution is provided to the polarization states estimation problem. The three locations inside the Poincaré sphere [Fig. 11(b)] correspond to the three physical classes of the object, i.e., the two dichroic patches and the diffusing glass slide. The three estimated states are normalized to the s0 element and are given by:

ŝ1=[1.00.0390.030.003]
ŝ2=[1.00.4720.3630.002]
ŝ3=[1.00.7820.0070.025]
(18)

ŝ1 corresponds to the smallest patch, ŝ2 to the trapezoidal shape, and ŝ3 to the background.

By looking at Eq. (18) and Figs. 10 and 11, one can observe the following:

  • The polarization state corresponding to ŝ1 is located near the center (Fig. 10) indicates that the transmission axis of this patch is oriented horizontally which extinguish a large amount of the vertically polarized incident beam.
  • The polarization state corresponding to ŝ2 lies in the (S1-S2) plan indicating a linear state as expected.
  • The incident beam undergoes isotropic depolarization induced by the diffusing glass slide. This is confirmed by observing that the degree of polarization (DOP) of ŝ2 and ŝ3 are nearly equal DOP(ŝ1) ≈ DOP(ŝ2).

5. Conclusion and future extensions

In this work we addressed the problem of estimating Stokes vectors from indirect noisy intensity measurements. Min-norm least squares estimate represents the state of the art method in the field. Because of the influence of measurement noise in the procedure, the estimate may violate the physical admissibility conditions. Moreover, the Stokes vectors vary in a given class, which is in contradiction with the prior knowledge. These shortcomings may lead to severe degradation of the estimated Stokes images, thereby hindering their interpretation and exploitation.

We propose here an alternative markovian Bayesian model accounting for both prior informations (the physical admissibility constraints). The MAP estimate was retained. The proposed optimization algorithm yields robust and accurate estimates on both synthetic and real data considered here.

Work is in progress to incorporate of such priors in the Mueller imaging framework.

Appendix: hyperparameters choice

The hyperparameters are chosen so as to yield weakly informative priors (see for example Ref. [15

15. S. Richardson and P. Green, “On Bayesian analysis of mixtures with an unknown number of components (with discussion),” J. R. Stat. Soc. Ser. B. 59,731–792 (1997). [CrossRef]

], p. 735 where equivalent choices are made). We set:

α1=(max(Im)+min(Im))2,α2=(max(Im)min(Im))3,
α3=1.2,
α41=0.95,
α42=2(max(Im)min(Im))2,
β1=β2=1,
γ1=(max(ŜLS)+min(ŜLS))2,γ2=(max(ŜLS)min(ŜLS))3,
δ1=δ2=1.01,
Kmin=2,Kmax=10

We remind the reader that ŜLS is the min-norm estimate of the Stokes image.

References and Links

1.

D. Miyazaki, M. Saito, Y. Sato, and K. Ikeuchi, “Determining surface orientations of transparent objects based on polarization degrees in visible and infrared wavelengths,” J. Opt. Soc. Am. A 19,687–694 (2002). [CrossRef]

2.

D. Miyazaki, M. Kagesawa, and K. Ikeuchi, “Transparent surface modeling from a pair of polarization images,” IEEE Trans. PAMI 26,920–932 (2004). [CrossRef]

3.

J. M. Bueno and P. Artal, “Double-pass imaging polarimetry in the human eye,” Opt. Letters. 2464–66 (1999) [CrossRef]

4.

S. D. Giattina, et al., “Assessment of coronary plaque collagen with polarization sensitive optical coherence tomography (PS-OCT),” Int. J. Cardiol. 107,400–409 (2006). [CrossRef] [PubMed]

5.

D. H. Goldstein and D. B. Chenault, and Society of Photo-optical Instrumentation Engineers, Polarization: measurement, analysis, and remote sensing II, 19–21 July, 1999, Denver, Colorado. 1999, Bellingham, Washington: SPIE. ix, 426 p.

6.

M. H. Smith, “Optimizing a dual-rotating-retarder Mueller matrix polarimeter,” in Polarization Analysis and Measurements IV, SPIE (2001).

7.

J. S. Tyo, “Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error,” Appl. Opt. 41619–630 (2002). [CrossRef] [PubMed]

8.

S. Ainouz, J. Zallat, A. de Martino, and C. Collet., “Physical interpretation of polarization-encoded images by color preview,” Opt. Express 14,5916–5927 (2006). [CrossRef] [PubMed]

9.

S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng. 41,965–972 (2002). [CrossRef]

10.

J. Bernardo and A. Smith, Bayesian Theory, (Wiley, 2000).

11.

A. Gelman, J. Carlin, H. Stern, and D. Rubin, Bayesian data analysis, Second ed., (CRC Press, 2003).

12.

S. Z. Li, Markov random field modeling in image analysis, Second ed., (Springer, 2001).

13.

A. Gray, J. Kay, and D. Titterington, “An empirical study of the simulation of various models used for images,” IEEE Trans. PAMI ,16,507–513 (1994). [CrossRef]

14.

A. Dunmur and D. Titterington, “Computational Bayesian analysis of hidden Markov mesh models,” IEEE PAMI. 19,1296–1300 (1997). [CrossRef]

15.

S. Richardson and P. Green, “On Bayesian analysis of mixtures with an unknown number of components (with discussion),” J. R. Stat. Soc. Ser. B. 59,731–792 (1997). [CrossRef]

16.

J. Besag, “On the statistical analysis of dirty pictures (with discussion),” J. R. Stat. Soc. B 48,259–302 (1986).

OCIS Codes
(000.3860) General : Mathematical methods in physics
(100.3190) Image processing : Inverse problems
(110.2960) Imaging systems : Image analysis
(120.5410) Instrumentation, measurement, and metrology : Polarimetry

ToC Category:
Imaging Systems

History
Original Manuscript: September 7, 2006
Manuscript Accepted: October 16, 2006
Published: January 8, 2007

Citation
Jihad Zallat and Christian Heinrich, "Polarimetric data reduction: a Bayesian approach," Opt. Express 15, 83-96 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-1-83


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References

  1. D. Miyazaki, M. Saito, Y. Sato, and K. Ikeuchi, "Determining surface orientations of transparent objects based on polarization degrees in visible and infrared wavelengths," J. Opt. Soc. Am. A 19, 687-694 (2002). [CrossRef]
  2. D. Miyazaki, M. Kagesawa, and K. Ikeuchi, "Transparent surface modeling from a pair of polarization images," IEEE Trans. PAMI 26,920-932 (2004). [CrossRef]
  3. J. M. Bueno and P. Artal, "Double-pass imaging polarimetry in the human eye," Opt. Letters. 2464-66 (1999). [CrossRef]
  4. S. D. Giattina,  et al., "Assessment of coronary plaque collagen with polarization sensitive optical coherence tomography (PS-OCT)," Int. J. Cardiol. 107, 400-409 (2006). [CrossRef] [PubMed]
  5. D. H. Goldstein, D. B. Chenault, and Society of Photo-optical Instrumentation Engineers, Polarization: measurement, analysis, and remote sensing II, 19-21 July, 1999, Denver, Colorado. 1999, Bellingham, Washington: SPIE. ix, 426 p.
  6. M. H. Smith, "Optimizing a dual-rotating-retarder Mueller matrix polarimeter," in Polarization Analysis and Measurements IV, SPIE (2001).
  7. J. S. Tyo, "Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error," Appl. Opt. 41619-630 (2002). [CrossRef] [PubMed]
  8. S. Ainouz, J. Zallat, A. de Martino, and C. Collet., "Physical interpretation of polarization-encoded images by color preview," Opt. Express 14, 5916-5927 (2006). [CrossRef] [PubMed]
  9. S. N. Savenkov, "Optimization and structuring of the instrument matrix for polarimetric measurements," Opt. Eng. 41, 965-972 (2002). [CrossRef]
  10. J. Bernardo and A. Smith, Bayesian Theory, (Wiley, 2000).
  11. A. Gelman, J. Carlin, H. Stern, and D. Rubin, Bayesian data analysis, Second ed., (CRC Press, 2003).
  12. S. Z. Li, Markov random field modeling in image analysis, Second ed., (Springer, 2001).
  13. A. Gray, J. Kay, and D. Titterington, "An empirical study of the simulation of various models used for images," IEEE Trans. PAMI,  16, 507-513 (1994). [CrossRef]
  14. A. Dunmur and D. Titterington, "Computational Bayesian analysis of hidden Markov mesh models," IEEE PAMI. 19, 1296-1300 (1997). [CrossRef]
  15. S. Richardson and P. Green, "On Bayesian analysis of mixtures with an unknown number of components (with discussion)," J. R. Stat. Soc. Ser. B. 59, 731-792 (1997). [CrossRef]
  16. J. Besag, "On the statistical analysis of dirty pictures (with discussion)," J. R. Stat. Soc. B 48, 259-302 (1986).

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