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| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 9, Iss. 5 — Apr. 29, 2014
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Optimal estimation in polarimetric imaging in the presence of correlated noise fluctuations

Julien Fade, Swapnesh Panigrahi, and Mehdi Alouini  »View Author Affiliations


Optics Express, Vol. 22, Issue 5, pp. 4920-4931 (2014)
http://dx.doi.org/10.1364/OE.22.004920


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Abstract

We quantitatively analyze how a polarization-sensitive imager can overcome the precision of a standard intensity camera when estimating a parameter on a polarized source over an intense background. We show that the gain is maximized when the two polarimetric channels are perturbed with significantly correlated noise fluctuations. An optimal estimator is derived and compared to standard intensity and polarimetric estimators.

© 2014 Optical Society of America

1. Introduction and posing of the problem

1.1. Introduction

Polarimetric-sensitive detectors (PSD) have long been implemented and have proved efficient in many application fields, such as biomedical imaging [1

1. M. P. Rowe, J. S. Tyo, N. Engheta, and E. N. Pugh, “Polarization-difference imaging: a biologically inspired techniquefor observation through scattering media,” Opt. Lett. 20, 608–610 (1995). [CrossRef] [PubMed]

,2

2. S. Demos, H. Savage, A. S. Heerdt, S. Schantz, and R. Alfano, “Time resolved degree of polarization for human breast tissue,” Opt. Commun. 124, 439–442 (1996). [CrossRef]

], and vision/contrast enhancement through turbid media [3

3. O. Emile, F. Bretenaker, and A. L. Floch, “Rotating polarization imaging in turbid media,” Opt. Lett. 21, 1706–1708 (1996). [CrossRef] [PubMed]

5

5. J. Guan and J. Zhu, “Target detection in turbid medium using polarization-based range-gated technology,” Opt. Express 21, 14152–14158 (2013). [CrossRef] [PubMed]

]. In this context, the benefits of polarimetric imaging have been thoroughly investigated by considering various imaging architectures and noise models [6

6. G. D. Lewis, D. L. Jordan, and P. J. Roberts, “Backscattering target detection in a turbid medium by polarization discrimination,” Appl. Opt. 38, 3937–3944 (1999). [CrossRef]

9

9. M. Boffety, F. Galland, and A.-G. Allais, “Influence of polarization filtering on image registration precision in underwater conditions,” Opt. Lett. 37, 3273–3275 (2012). [CrossRef] [PubMed]

]. However, the gain in measurement precision that can be reached when a PSD is used instead of a standard intensity detector (ID), in the presence of significantly correlated noise fluctuations in each polarimetric channel, is still unexplored to the best of our knowledge. Indeed, for practical reasons it is usually assumed that these noise fluctuations are uncorrelated. As a result, considering the most favorable situation of a perfectly polarized source (or polarizing object) embedded in unpolarized background, the polarimetric channel which offers the best contrast is the one corresponding to the polarization state of the source. In this channel, the mean intensity level of the source is thus preserved, whereas that of the unpolarized background is reduced by a factor of two, leading to a doubling of contrast as compared to standard intensity detection [1

1. M. P. Rowe, J. S. Tyo, N. Engheta, and E. N. Pugh, “Polarization-difference imaging: a biologically inspired techniquefor observation through scattering media,” Opt. Lett. 20, 608–610 (1995). [CrossRef] [PubMed]

, 10

10. M. Dubreuil, P. Delrot, I. Leonard, A. Alfalou, C. Brosseau, and A. Dogariu, “Exploring underwater target detection by imaging polarimetry and correlation techniques,” Appl. Opt. 52, 997–1005 (2013). [CrossRef] [PubMed]

]. Nevertheless, the assumption of uncorrelated noise fluctuations is not representative of most real field scenarios especially when the polarimetric channels are acquired simultaneously [11

11. A. Bénière, M. Alouini, F. Goudail, and D. Dolfi, “Design and experimental validation of a snapshot polarization contrast imager,” Appl. Opt. 48, 5764–5773 (2009). [CrossRef] [PubMed]

]. For instance, a polarized source appearing through fog or haze is a situation where the background mean level is time-varying [12

12. N. Hautiere and D. Aubert, “Contrast restoration of foggy images through use of an onboard camera,” in Proceedings of 2005 IEEE Intelligent Transportation Systems(2005), pp 601–606.

] especially when the imaging system is moving or vibrating. More generally, similar situations might be encountered when imaging objects through turbid media, as in the fields of underwater imaging [13

13. N. Gracias, S. Negahdaripour, L. Neumann, R. Prados, and R. Garcia, “A motion compensated filtering approach to remove sunlight flicker in shallow water images,” in OCEANS 2008, (2008), pp. 1–7. [CrossRef]

, 14

14. M. Darecki, D. Stramski, and M. Sokólski, “Measurements of high-frequency light fluctuations induced by sea surface waves with an underwater porcupine radiometer system,” J. Geophys. Res. 116, C00H09 (2011). [CrossRef]

], or infrared target detection [15

15. F. A. Sadjadi and C. L. Chun, “Automatic detection of small objects from their infrared state-of-polarization vectors,” Opt. Lett. 28, 531–533 (2003). [CrossRef] [PubMed]

]. Imaging a static scene might also be subject to intensity fluctuations of the illuminating source, as often encountered in polarimetric microscopy [16

16. B. Laude-Boulesteix, A. D. Martino, B. Drévillon, and L. Schwartz, “Mueller polarimetric imaging system with liquid crystals,” Appl. Opt. 43, 2824–2832 (2004). [CrossRef] [PubMed]

]. Thus, one can wonder whether the noise correlation properties of the different polarimetric channels could be properly exploited in order to optimize, in terms of contrast, the representation of the polarimetric image.

In this article, we intend to rigorously quantify the gain in measurement precision that can be reached when a PSD is used in the presence of significantly correlated noise fluctuations in each polarimetric channel. This article is organized as follows: in the remainder of the first Section, we describe the general polarimetric image formation model addressed, as well as the correlated-noise statistical model considered throughout this article. Within the theoretical framework of information theory, the benefit of using PSD instead of a standard ID is then derived in Section 2, for a general estimation problem consisting in measuring a parameter (intensity, absorbance, location, etc.) on a polarized source over an intense background. The expression of this gain in optimal estimation precision is then thoroughly analyzed in Section 3 in relation with realistic experimental imaging conditions. Lastly, optimal estimation procedures are derived and discussed in Section 4, before providing conclusions of the article in Section 5.

1.2. Image formation model

We will consider a general framework consisting in the estimation of a given parameter (intensity, location, etc.) from a polarized signal contribution, denoted si at location i, with a degree of polarization (DOP) denoted by P ∈ [0, 1], which is either emitted by an active source or backscattered by an object of interest. Using a simple classical but realistic illumination model [9

9. M. Boffety, F. Galland, and A.-G. Allais, “Influence of polarization filtering on image registration precision in underwater conditions,” Opt. Lett. 37, 3273–3275 (2012). [CrossRef] [PubMed]

, 10

10. M. Dubreuil, P. Delrot, I. Leonard, A. Alfalou, C. Brosseau, and A. Dogariu, “Exploring underwater target detection by imaging polarimetry and correlation techniques,” Appl. Opt. 52, 997–1005 (2013). [CrossRef] [PubMed]

, 17

17. J. Jaffe, “Computer modeling and the design of optimal underwater imaging systems,” IEEE J. Oceanic Eng. 15, 101–111 (1990). [CrossRef]

], the intensity XiI detected at location i is assumed to also comprise a back-ground contribution bi, with a DOP denoted by β ∈ [0, 1]. This background contribution is due to ambient light scattering through a turbid medium (atmosphere, water, or biological tissue). For the sake of generality, we shall analyze any couple of polarization parameters P and β which can correspond to many different experimental conditions. Although in most experiments the signal contribution is highly polarized in comparison to an unpolarized background (Pβ), some situations can involve opposite physical conditions (βP), such as underwater imaging as mentioned in [10

10. M. Dubreuil, P. Delrot, I. Leonard, A. Alfalou, C. Brosseau, and A. Dogariu, “Exploring underwater target detection by imaging polarimetry and correlation techniques,” Appl. Opt. 52, 997–1005 (2013). [CrossRef] [PubMed]

].

A non-polarimetric ID with N pixels gives access to a sample XiI={XiI}i=1,,N, with XiI=si+bi, whereas a PSD provides a bidimensional vector XiP=[Xi//,Xi]T at each location i of the detector, obtained from the intensities recorded along two orthogonal polarization directions [11

11. A. Bénière, M. Alouini, F. Goudail, and D. Dolfi, “Design and experimental validation of a snapshot polarization contrast imager,” Appl. Opt. 48, 5764–5773 (2009). [CrossRef] [PubMed]

], as sketched in Fig. 1. With the above illumination model, the average value of XiP is simply given by
XiP=[1+P2si+1+β2bi1P2si+1β2bi].
(1)

Fig. 1 Sketch of the image formation model: a polarization-splitting analyzing device (PSAD) can be any suitable birefringent crystal in case of simultaneous acquisitions of images X// and X [11], or a rotating polarizer or liquid crystal device for sequential acquisitions. Image formation optics are not represented for the sake of clarity.

1.3. Noise model

Throughout this article, we shall consider a Gaussian noise model, which makes it possible to take into account various sources of noise in realistic situations. In addition, such model provides closed-form expressions which is in favour of physical interpretation. At a given location i, the second order statistical properties of the bidimensional measurement vector XiP are modeled by a covariance matrix Γi=δXiP(δXiP)T, with δXiP=XiPXiP, of the following form:
Γi=[σ//,i2ciciσ,i2]=[1+β2εi2+σ02ρ1β22εi2ρ1β22εi21β2εi2+σ02].
The Gaussian probability density function of a N–pixels measurement sample is then given by PX(XP)=i=1Nexp{12(δXiP)TΓi1δXiP}/2πdet[Γi].

Table 1. List and description of symbols and acronyms. Dependency in scene location i has been omitted for the sake of concision.

table-icon
View This Table

Let us focus on the parallel channel: through this statistical description, we assume that the noise variance can be written σ//,i2=(1+β)εi2/2+σ02, with the detector electronic noise contribution σ02 being rationally independent from the location i in the image, and from the illumination level or polarization properties. The first term in the expression of σ//,i2 accounts for a multiplicative “optical” noise, introduced by background optical intensity fluctuations, and hence depends on the background DOP β. This noise contribution, proportional to the background average level bi, can model the effect of turbulence or variations of scatterers density, as well as photon noise in the high background intensity limit.

Due to these scene-dependent optical fluctuations, the intensity measurements in the two polarimetric channels are likely to be correlated, especially in the case of simultaneous acquisition of the polarimetric images with a polarization-splitting analyzing device (PSAD), as sketched in Fig. 1 or as extensively described in [11

11. A. Bénière, M. Alouini, F. Goudail, and D. Dolfi, “Design and experimental validation of a snapshot polarization contrast imager,” Appl. Opt. 48, 5764–5773 (2009). [CrossRef] [PubMed]

]. Such partial correlation will be modeled by a non-null covariance term ci in Γi. We assume that the scene-dependent noise contributions only are partially correlated through a correlation parameter ρ, whereas the detector noise is assumed to be uncorrelated between the two channels.

2. Gain in optimal estimation performance

2.1. Principle

To characterize the gain in terms of estimation precision when PSDs are used instead of classical IDs, we propose to resort to information theory, by determining and comparing the Fisher Information (FI) associated to each imaging modality. The FI characterizes the amount of information available in a sample X for the estimation of a parameter y, and is defined as [18

18. P. Garthwaite, I. Jolliffe, and B. Jones, Statistical Inference (Prentice Hall, 1995).

]
IF(y)=2lnPX(X)y2.
(2)
According to the well-known Cramer-Rao theorem, its inverse value IF−1(y) defines a lower bound (Cramer-Rao bound (CRB)) on the minimum variance expectable for estimating parameter y with an unbiased estimation procedure [18

18. P. Garthwaite, I. Jolliffe, and B. Jones, Statistical Inference (Prentice Hall, 1995).

]. In the following, we shall limit ourselves to the estimation of the mean signal intensity si at location i for the sake of simplicity but without loss of generality. Indeed, it is possible to extrapolate the results of this article to other physical situations since one has IF(z) = IF(y) [dy/dz]2 from simple variable transformation relations. For instance, for the estimation of an atmospheric transmittance τ such that s = e, the FI is directly obtained with IF(τ) = L2s2IF(s), which simply involves the FI for the estimation of the mean signal intensity IF(s). Another illutration is the interesting case of image registration addressed in [9

9. M. Boffety, F. Galland, and A.-G. Allais, “Influence of polarization filtering on image registration precision in underwater conditions,” Opt. Lett. 37, 3273–3275 (2012). [CrossRef] [PubMed]

], in which a translation parameter η is to be estimated over the whole image such that s = {s(xiη)}i=1,...,N. In this latter case, the above relation yields IF(η)=i=1N[s(xiη)]2IF(si), which again only involves the FI for the estimation of the mean intensity at each location i.

2.2. Expression of the gain

The FI in the case of polarimetric and intensity measurements are derived in Appendix A, and are not recalled here for the sake of concision. We propose to define a gain in optimal precision by comparing the FI available with a polarimetric setup over the FI available with a standard intensity detector, for given experimental conditions. This definition, which has been used in other references [9

9. M. Boffety, F. Galland, and A.-G. Allais, “Influence of polarization filtering on image registration precision in underwater conditions,” Opt. Lett. 37, 3273–3275 (2012). [CrossRef] [PubMed]

, 19

19. J. Fade, N. Treps, C. Fabre, and P. Réfrégier, “Optimal precision of parameter estimation in images with local sub-Poissonian quantum fluctuations,” Eur. Phys. J. D 50, 215–227 (2008). [CrossRef]

], yields:
μ(ω,P,β,ρ)=IFP(s)IFI(s)=(1+ω2)[1+P22+Q4ω2]1+ω2+(1ρ2)(1β2)4ω4,
(3)
where
Q=(12βP+P2)ρ(1P2)1β2,
(4)
and with ω2=ε2/σ02. This last parameter ω2 gives the relative value of the noise contributions variances, allowing one to identify the dominant noise term. Thus, “optical” noise ε2 dominates when ω2 ≫ 1, whereas electronic fluctuations are the main source of noise when ω2 ≪ 1. As an illustration, the evolution of the gain μ(ω, P, β, ρ) given in Eq. (3) is plotted in Fig. 2 as a function of ρ for various values of ω, and for a partially polarized source (P = 0.4) and background (β = 0.1). It can be immediately checked that the gain does not depend on ρ when electronic noise dominates (ω ≪ 1), and that it increases as ω increases.

Fig. 2 Evolution of μ(ω, P, β, ρ) for P = 0.4 and β = 0.1 as a function of ρ for ω = {10−3, 1, 5, 104}.

As will be shown in the following, such definition of a gain in optimal estimation precision can provide insightful results on the physical estimation problem at hands, regardless of the actual estimation procedure used, since derived from information theory. In addition, it can have practical implications if optimal estimators can be identified, as will be shown in Section 4.

3. Physical analysis of the gain μ(ω, P, β, ρ)

In this section, we derive and analyze a number of properties of the gain in optimal precision μ(ω, P, β, ρ) defined above. These results will allow us to study the benefits of using PSDs for estimation tasks in the presence of intense background and potentially correlated measurements.

3.1. Influence of ambient illumination level

Let us first study how the gain evolves as a function of the ambient background illumination level b. For that purpose, we analyze the behaviour of the gain μ(ω, P, β, ρ) as a function of ω = ε/σ0, since ε has been assumed proportional to b. A tractable but tedious calculus sketched in Appendix B leads to this first property:

Property 1 The gain μ(ω, P, β, ρ) is a monotonically increasing function of ω.

This is an interesting result, showing that increasing the relative amount of “optical” noise with respect to electronic noise tends to favour a polarimetric setup in terms of estimation performance, even if the polarimetric measurements are totally uncorrelated (ρ → 0).

When electronic noise dominates, the gain falls down below unity, since μ(ω ≪ 1, P, β, ρ) → (1 + P2)/2 ≤ 1. Indeed, for a given amount of light energy entering the imaging system, the PSAD reduces the signal-to-noise ratio (SNR) on the detectors in comparison to a standard ID since energy is splitted into two polarization channels. This property can be checked in Fig. 2 where μ(ω, P, β, ρ) is plotted as a function of ρ, when P = 0.4 and β = 0.1.

3.2. Asymptotic behaviour in the high intensity regime

Focusing on the high intensity regime by setting ω → ∞, we obtain a simpler expression
μ(P,β,ρ)=μ(ω1,P,β,ρ)=Q(1ρ2)(1β2),
(5)
which will be referred to as asymptotic gain subsequently.

Let us analyze the evolution of the asymptotic gain as a function of the correlation between polarimetric channels. Surprisingly, it can be shown that μ(P, β, ρ) is not a monotonically increasing function of the correlation parameter ρ, as can be observed in Fig. 2. The following property can indeed be demonstrated (see Appendix C):

Property 2 The asymptotic gain μ(P, β, ρ) reaches a minimum value μ,min for a correlation parameter ρmin , such that
{μ,min=(1+P)22(1+β)andρmin=1P1+P1+β1β,ifβ2P1+P2μ,min=(1P)22(1β)andρmin=1+P1P1β1+β,otherwise
(6)
This property is rather counter-intuitive but can be interpreted as follows. First, when the two acquired polarimetric images are uncorrelated (ρ ≃ 0), gain in estimation precision only occurs if SNR reduction caused by intensity splitting between the two polarization channels is compensated by the increase in size of the statistical sample considered (Two sets of N measures with a PSD, instead of one in an standard ID). Though, as soon as ρ ≠ 0, the polarimetric measures are no longer independent, and thus the available FI is necessarily lower than the one available with two independent sets of N measurements. This remains true for smaller values of ρ. However, for values of ρ > ρmin, the strongly correlated noise perturbing each polarization channel can be partly cancelled out by taking profit of the two acquired images, leading to a potentially strong increase in the gain. This is indeed possible if signal and background contributions exhibit different relative intensity levels on the two acquired images. In this case, an optimal estimation procedure, such as the one described in Section 4, can take profit of this relative contrast mismatch to estimate the desired parameter on the signal contribution with a high precision.

Using the expression of the asymptotic gain given in Eq. (5), let us now analyze in which physical conditions one should favour using a PSD rather than a standard ID. For that purpose, the two following properties can be established. A sketch of the demonstration of these properties is given in Appendix D.

Property 3 For a given value of P, the asymptotic gain μ(P, β, ρ) is greater or equal to a minimum gain value K (with K ≥ 1) for any value of the correlation parameter ρ provided
β(1+P)22K1,ifβP,
(7)
β1(1P)22KifβP.
(8)
Property 4 When the conditions of Property 3 are not verified, the asymptotic gain μ(P, β, ρ) is greater or equal to a minimum gain value K (with K ≥ 1) provided the correlation parameter ρ verifies
ρρlimK=1P22K1β2+Φ,
(9)
where
Φ=[112K(1P)21β]×[112K(1+P)21+β].
(10)

3.3. Discussion

We obviously start focusing on the case of unitary gain (i.e., K = 1) which delimitates situations in which polarimetric imaging systems can bring an improvement in estimation precision. In this case, the conditions of Eqs. (7) and (8) respectively read β ≤ (1 + P)2/2 −1 when βP, and β ≥ 1− (1− P)2/2 when βP. For a fully depolarized background (β = 0), for instance, this means that a polarimetric imaging system can improve the quality of estimation, whatever be the value of ρ, as long as a moderately polarized source is used with a minimum value of P=210.414. On the other hand, when the source is totally unpolarized, a gain can be expected for any value of ρ provided β ≥ 1/2. In the two-dimensional plot of Fig. 3(a) as a function of polarization parameters P and β, the conditions of Eqs. (7) and (8) for K = 1 are represented with continuous green curves and delimitate two regions. When the conditions hold (greyed region in Fig. 3(a)), the values of μ,min and ρmin are represented in contour plots, respectively in blue dashed lines and green dot-dashed lines. In the second region, i.e., when the inequalities of Eqs. (7) and (8) are not verified, the correlation parameter ρ has to be greater than a minimum value denoted ρlimK=1 so as to ensure μ(P, β, ρ) ≥ 1. The values of ρlimK=1 are plotted in Fig. 3(a) in red continuous lines, as a function of P and β.

Fig. 3 Contour plots of ρlimK for various values of K as a function of P and β. Additional contour plots of ρmin and μ,min are provided when relations (7) and (8) hold. The yellow circles correspond to the situation addressed in Fig. 2 (P = 0.4 and β = 0.1).

The same graphical representation has been used in Fig. 3(b)–3(d) in the case of K = {2, 5, 10} respectively, to plot the values of ρmin and μ,min when the relation of Eq. (8) holds, and the values of ρlimK otherwise. It is interesting to notice that when Pβ, a limit value ρlimK>0 always has to be ensured for any couple of parameters P and β as long as K ≥ 2 since condition of Eq. (7) cannot be fulfilled in this case. On the other hand, when a highly polarized background is considered and βP, a high asymptotic gain value K can be reached with uncorrelated measurements (i.e., ρ = 0) provided P is small enough. This property can be understood by noticing that a high value of β implies a low background contribution on one of the two acquired images, thus facilitating estimation of a parameter on the low polarized signal contribution. This result must be however mitigated since the detector noise has been neglected to derive Properties 3 and 4, but should be taken into account in this latter case involving low background illumination levels.

In terms of practical application, the charts given in Fig. 3 provide insightful information about the expectable gain in precision using a PSD for a given set of physical parameters P, β and ρ. As could be expected, the best performance gain is obtained when a high polarimetric contrast can be observed between the background and signal contributions (high P and low β, or high β and low P). However, these charts clearly evidence that the gain in performance increases also when the measurements are significantly correlated. Yet, these charts may be of great use to assess the optimal performance of a real field polarimetric imaging system, in which all intermediate situations are likely to occur. For instance, the degradation of the DOP of a highly polarized source could be taken into account in the dimensioning of an experiment. The influence of unwanted or unexpected polarization/depolarization of the background could be also analyzed with the above results.

4. Optimal estimation procedure

For a fair comparison, the estimation samples should involve the same number of pixels. Thus, a PSD with N pixels in each polarimetric channels must be compared to a 2N-pixels standard ID. In this case, the relative performance of the two estimators can be directly assessed from the chart plotted in Fig. 3(b), which gives conditions for a minimum gain value of μ(P, β, ρ) ≥ K = 2. The analyzis of this chart interestingly shows that PSDs are not systematically preferable to standard ID if the correlation between the fluctuations lies below a lower limit ρlimK=2 determined above. As a result, the chart plotted in Fig. 3(b) turns out to be a useful tool for determining the optimal estimation procedure, depending on the experimental conditions.

Lastly, it can be interesting to compare the ML estimator with other estimation procedures which are classically used in polarimetric imaging. For instance, when polarimetric measurements along orthogonal polarization directions are available, a simple difference image is classically obtained by substraction of the two polarimetric channels [1

1. M. P. Rowe, J. S. Tyo, N. Engheta, and E. N. Pugh, “Polarization-difference imaging: a biologically inspired techniquefor observation through scattering media,” Opt. Lett. 20, 608–610 (1995). [CrossRef] [PubMed]

]. For the estimation of the parameter s, such difference estimator would simply read s^ΔP=[X^//X^βb]/P. However, it can be shown that this standard estimator is not optimal, in general, in the situation addressed in this article. Its variance, derived in Appendix F, is indeed greater that var(s^MLP) (and thus greater than the CRB) except when ρ = (1 − β P)/(1 − β2), in which case the difference estimator s^ΔP identifies with s^MLP.

5. Conclusion

As a conclusion, the theoretical results derived in this article quantitatively demonstrate that polarimetric imagers can significantly improve the estimation precision, provided noise fluctuations in each polarimetric channels are significantly correlated. Hence, this confirms the interest of snapshot polarimetric imagers as described in [11

11. A. Bénière, M. Alouini, F. Goudail, and D. Dolfi, “Design and experimental validation of a snapshot polarization contrast imager,” Appl. Opt. 48, 5764–5773 (2009). [CrossRef] [PubMed]

] since they may favour correlated background/noise fluctuations in the two polarimetric channels, which are acquired simultaneously. In these conditions, we have also shown that the optimal estimation procedure differs from a natural difference image, but can be simply implemented. These results can be useful for the design of polarimetric imaging systems involving estimation through turbid media, or in other fields of application, for post-processing of polarimetric images exhibiting temporally or spatially correlated fluctuations.

Appendix

To derive the expressions presented in this article, it is interesting to introduce the following simplified notations: a = (1 + P)/2 and α = (1−P)/(1 + P) on the one hand, and, on the other hand, c2 = (1 + β)/2 and γ2 = (1 − β)/(1 + β). In a single pixel configuration (N = 1) for the sake of simplicity, the polarimetric measurement considered is XP = [X//, X]T, such that 〈XP〉 = [as + c2b, α as + γ2c2b]T and
Γ=δXδXT=c2[ε2+ς2ργε2ργε2γ2ε2+ς2],
with ς2=2σ02/(1+β). It is easily checked that the conditions P ∈ [0, 1], β ∈ [0, 1] and βP are equivalent to α ∈ [0, 1], γ ∈ [0, 1] and γ2α.

A. Fisher informations calculations

With the Gaussian noise model used in this article, the loglikelihood of the polarimetric measure XP can be written (XP) = lnPX(XP) = −(δXP)T Γ−1δXP/2 up to an additive term independent of s. An application of Eq. (2) leads to the FI for the estimation of s, which reads IFP(s) = [〈XP]T Γ−1XP, with 〈XP = XP /∂s = 〈XP /∂s〉 = [a, a α]T. A direct calculation gives:
IFP(s)=a2c2ε2(α22ραγ+γ2)+(1+α2)u2γ2(1ρ2)+(1+γ2)u2+u4,
(12)
with u2 = ε2/ς2 = (1 + β)ω2/2.

The FI for the estimation of s from the total intensity of the beam (non-polarimetric measurement) is a standard result under Gaussian fluctuations hypothesis. One has
IFI(s)=a2c2ε2(1+α2)(1+γ2)+u2=σ21+ω2.
(13)
The gain μ(u, α, γ, ρ) = IFP(s)/IFI(s) can then be easily derived, leading to Eq. (3) with appropriate changes of variables.

B. Monotonicity of μ(u, α, γ, ρ) as a function of u

To demonstrate Property 1, let us first rewrite μ(u, α, γ, ρ) as
μ(u,α,γ,ρ)=Au2+BDu4+Cu2+1×Cu2+1E,
(14)
with A = (α2 − 2ραγ + γ2), B = 1 + α2, C = 1 + γ2, D = γ2(1 − ρ2) and E = (1 + α)2, all these expressions being greater or equal to zero. Let us notice that μ(0, α, γ, ρ) = B/E = (1 + P2)/2.

The derivative of μ(u, α, γ, ρ) as a function of u is thus
[μ(u,α,γ,ρ)]u=2u(u)[1+Cu2+Du4]2E,
(15)
with (u) = A + 2(ACBD)u2 + (AC2ADBCD)u4.

The function μ(u, α, γ, ρ) is monotonically increasing on u ∈ [0; ∞[ if [μ(u, α, γ, ρ)]/∂u ≥ 0 ⇔ (u) ≥ 0, ∀u ∈ [0; ∞[. Noticing that (0) = A ≥ 0, and that (u) is a second-order polynomial in u2, we can compute the discriminant
Δ=4D(A2+DB2ABC)=4γ2(1ρ2)[α(1γ2)ργ(1α2)]2,
(16)
which is negative. As a consequence, (u) does not admit real root on u ∈ [0; ∞[, and hence, (u) is positive on u ∈ [0; ∞[. As a result, μ(u, α, γ, ρ) is a positive, monotonically increasing function of u for u ∈ [0; ∞[.

C. Minimum value of the asymptotic gain μ(α, γ, ρ):

The asymptotic gain is obtained by setting u → ∞:
μ(α,γ,ρ)=(α22ραγ+γ2)(1+γ2)γ2(1ρ2)(1+α2)
(17)

It is easily shown that μ(α, γ, ρ) reaches a minimum if (αργ)(αργ) = 0. Since ρ ∈ [0, 1], the only admissible root is ρmin = α/γ when γα and thus μ,min(α, γ) = (1 + γ2)/(1 + α)2. When γα, the only admissible root is ρmin = γ/α, and in this case μ,min(α, γ) = α2(1 + γ2)/γ2(1 + α)2. The expressions of ρmin and μ,min(P, β) given in the article can be recovered with an appropriate change of variables.

From the above results, the conditions for μ,min(α, γ) ≥ K are directly derived as
γ2K(1+α)21whenγα
(18)
γ2α2/[K(1+α)2α2]whenγα.
(19)

D. Condition for minimum gain μ(α, γ, ρ) = K:

Solving μ(α, γ, ρ) = K leads to two roots ρ1/2K=[α(1+γ2)Φ]/Kγ(1+α)2 verifying ρ1/2K[0,1], with Φ = [1 + γ2K(1 + α)2] [α2(1 + γ2) − K(1 + α)2γ2], or with the notations of Appendix B, Φ = [CKE] [α2CKEγ2].

Let us focus on the greatest root, denoted ρlimK=ρ2K in the following, and which defines the minimum value of ρ such that μ(α, γ, ρ) ≥ K, ρρlimK. It can first be checked that the expressions of ρlimK and Φ respectively given in the Eqs. (9) and (10) of Property 4 can be retrieved with appropriate changes of variables. In particular, one has Φ = Ψ/[K2γ2(1 + α)4].

Let us now study in which conditions this upper root ρlimK actually defines a real-valued limit on the correlation parameter ρ. It is clear that ρlimK is imaginary if Φ ≤ 0, which occurs when one of the two following inequalities is verified: (a): α2C/γ2EKC/E; or (b): α2C/γ2EKC/E. When γα, inequality (b) is impossible, and (a) is verified if γ2K(1 + α)2 − 1. When γα, inequality (a) is impossible, and (b) is verified if γ2α2/[K(1 + α)2α2]. These conditions obviously correspond to those derived above in Appendix C in Eqs. (18) and (19) for ensuring μ,min(α, γ) ≥ K.

For the sake of physical interpretation, we can rewrite these conditions as
γ2K(1+α)21,ifαγ2,and
(20)
γ2α2/[K(1+α)2α2],ifαγ2,
(21)
which allows relations (7) and (8) of Property 3 to be retrieved by appropriate change of variables. Indeed, it can be checked that none of the conditions of Eqs. (18) and (19) apply when γ2αγ.

E. ML estimator s^MLP

We derive the ML estimator in a situation of negligible detector noise, i.e., u → ∞ (or ω → ∞). From the expression of (XP) given in Appendix A, we derive s^MLP by solving ∂ℓ(XP)/∂s = 0, leading to equation (δXP)T Γ−1X + (〈X)T Γ−1δXP = 0. A straightforward but tedious calculation finally gives the expression of Eq. (11), with U = γ [γαρ], V = [αγρ], Z = bc2γ [ρ(α + γ2) − γ(1 + α)], and W = a[α2 − 2ραγ + γ2]. It is easily checked that this estimator is unbiased s^MLP=s, since 〈//〉 = as + c2b and 〈〉 = aαs + c2γ2b. Moreover, the variance of this estimator necessarily reaches the CRB since ML estimator is efficient under Gaussian fluctuations [18

18. P. Garthwaite, I. Jolliffe, and B. Jones, Statistical Inference (Prentice Hall, 1995).

]. This can be checked by noticing that var(s^MLP)=[U2var(X^//)+V2var(X^)+UVcov(X^//,X^)]/W2 which, after a simplification step, is equal to limu→∞{1/IFP(s)}.

F. Difference image estimator s^ΔP

With the notations used in this appendix, the estimator s^ΔP given in Section 4 reads s^ΔP=[(X^//X^)bc2(1γ2)]/[a(1+α2)]. One easily checks that it is unbiased and that var(s^ΔP)=[var(X^//)+var(X^)2cov(X^//,X^)]/[a2(1+α2)], which is equal to var(ŝΔ) = c2ε2[1 + γ2 − 2ργ]/[a2(1 + α)2].

Lastly, it can be shown that var(s^ΔP)=var(s^MLP) if α + γ2ρ (1 + α) γ = 0, i.e., if ρ = (α + γ2)/γ (1 + α) = (1 − βP)/(1 − β2), in which case s^ΔP=s^MLP.

Acknowledgments

This work has been supported by the TDF company and by Rennes Metropole through Dr. J. Fade’s AIS grant.

References and links

1.

M. P. Rowe, J. S. Tyo, N. Engheta, and E. N. Pugh, “Polarization-difference imaging: a biologically inspired techniquefor observation through scattering media,” Opt. Lett. 20, 608–610 (1995). [CrossRef] [PubMed]

2.

S. Demos, H. Savage, A. S. Heerdt, S. Schantz, and R. Alfano, “Time resolved degree of polarization for human breast tissue,” Opt. Commun. 124, 439–442 (1996). [CrossRef]

3.

O. Emile, F. Bretenaker, and A. L. Floch, “Rotating polarization imaging in turbid media,” Opt. Lett. 21, 1706–1708 (1996). [CrossRef] [PubMed]

4.

H. Ramachandran and A. Narayanan, “Two-dimensional imaging through turbid media using a continuous wave light source,” Opt. Commun. 154, 255–260 (1998). [CrossRef]

5.

J. Guan and J. Zhu, “Target detection in turbid medium using polarization-based range-gated technology,” Opt. Express 21, 14152–14158 (2013). [CrossRef] [PubMed]

6.

G. D. Lewis, D. L. Jordan, and P. J. Roberts, “Backscattering target detection in a turbid medium by polarization discrimination,” Appl. Opt. 38, 3937–3944 (1999). [CrossRef]

7.

P. Réfrégier, M. Roche, and F. Goudail, “Cramer-Rao lower bound for the estimation of the degree of polarization in active coherent imagery at low photon levels,” Opt. Lett. 31, 3565–3567 (2006). [CrossRef] [PubMed]

8.

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Degree of polarization estimation in the presence of nonuniform illumination and additive gaussian noise,” J. Opt. Soc. Am. A 25, 919–929 (2008). [CrossRef]

9.

M. Boffety, F. Galland, and A.-G. Allais, “Influence of polarization filtering on image registration precision in underwater conditions,” Opt. Lett. 37, 3273–3275 (2012). [CrossRef] [PubMed]

10.

M. Dubreuil, P. Delrot, I. Leonard, A. Alfalou, C. Brosseau, and A. Dogariu, “Exploring underwater target detection by imaging polarimetry and correlation techniques,” Appl. Opt. 52, 997–1005 (2013). [CrossRef] [PubMed]

11.

A. Bénière, M. Alouini, F. Goudail, and D. Dolfi, “Design and experimental validation of a snapshot polarization contrast imager,” Appl. Opt. 48, 5764–5773 (2009). [CrossRef] [PubMed]

12.

N. Hautiere and D. Aubert, “Contrast restoration of foggy images through use of an onboard camera,” in Proceedings of 2005 IEEE Intelligent Transportation Systems(2005), pp 601–606.

13.

N. Gracias, S. Negahdaripour, L. Neumann, R. Prados, and R. Garcia, “A motion compensated filtering approach to remove sunlight flicker in shallow water images,” in OCEANS 2008, (2008), pp. 1–7. [CrossRef]

14.

M. Darecki, D. Stramski, and M. Sokólski, “Measurements of high-frequency light fluctuations induced by sea surface waves with an underwater porcupine radiometer system,” J. Geophys. Res. 116, C00H09 (2011). [CrossRef]

15.

F. A. Sadjadi and C. L. Chun, “Automatic detection of small objects from their infrared state-of-polarization vectors,” Opt. Lett. 28, 531–533 (2003). [CrossRef] [PubMed]

16.

B. Laude-Boulesteix, A. D. Martino, B. Drévillon, and L. Schwartz, “Mueller polarimetric imaging system with liquid crystals,” Appl. Opt. 43, 2824–2832 (2004). [CrossRef] [PubMed]

17.

J. Jaffe, “Computer modeling and the design of optimal underwater imaging systems,” IEEE J. Oceanic Eng. 15, 101–111 (1990). [CrossRef]

18.

P. Garthwaite, I. Jolliffe, and B. Jones, Statistical Inference (Prentice Hall, 1995).

19.

J. Fade, N. Treps, C. Fabre, and P. Réfrégier, “Optimal precision of parameter estimation in images with local sub-Poissonian quantum fluctuations,” Eur. Phys. J. D 50, 215–227 (2008). [CrossRef]

OCIS Codes
(030.6600) Coherence and statistical optics : Statistical optics
(110.4280) Imaging systems : Noise in imaging systems
(110.0113) Imaging systems : Imaging through turbid media
(110.3055) Imaging systems : Information theoretical analysis
(110.5405) Imaging systems : Polarimetric imaging

ToC Category:
Imaging Systems

History
Original Manuscript: November 8, 2013
Revised Manuscript: December 22, 2013
Manuscript Accepted: December 23, 2013
Published: February 24, 2014

Virtual Issues
Vol. 9, Iss. 5 Virtual Journal for Biomedical Optics

Citation
Julien Fade, Swapnesh Panigrahi, and Mehdi Alouini, "Optimal estimation in polarimetric imaging in the presence of correlated noise fluctuations," Opt. Express 22, 4920-4931 (2014)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-22-5-4920


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References

  1. M. P. Rowe, J. S. Tyo, N. Engheta, E. N. Pugh, “Polarization-difference imaging: a biologically inspired techniquefor observation through scattering media,” Opt. Lett. 20, 608–610 (1995). [CrossRef] [PubMed]
  2. S. Demos, H. Savage, A. S. Heerdt, S. Schantz, R. Alfano, “Time resolved degree of polarization for human breast tissue,” Opt. Commun. 124, 439–442 (1996). [CrossRef]
  3. O. Emile, F. Bretenaker, A. L. Floch, “Rotating polarization imaging in turbid media,” Opt. Lett. 21, 1706–1708 (1996). [CrossRef] [PubMed]
  4. H. Ramachandran, A. Narayanan, “Two-dimensional imaging through turbid media using a continuous wave light source,” Opt. Commun. 154, 255–260 (1998). [CrossRef]
  5. J. Guan, J. Zhu, “Target detection in turbid medium using polarization-based range-gated technology,” Opt. Express 21, 14152–14158 (2013). [CrossRef] [PubMed]
  6. G. D. Lewis, D. L. Jordan, P. J. Roberts, “Backscattering target detection in a turbid medium by polarization discrimination,” Appl. Opt. 38, 3937–3944 (1999). [CrossRef]
  7. P. Réfrégier, M. Roche, F. Goudail, “Cramer-Rao lower bound for the estimation of the degree of polarization in active coherent imagery at low photon levels,” Opt. Lett. 31, 3565–3567 (2006). [CrossRef] [PubMed]
  8. A. Bénière, F. Goudail, M. Alouini, D. Dolfi, “Degree of polarization estimation in the presence of nonuniform illumination and additive gaussian noise,” J. Opt. Soc. Am. A 25, 919–929 (2008). [CrossRef]
  9. M. Boffety, F. Galland, A.-G. Allais, “Influence of polarization filtering on image registration precision in underwater conditions,” Opt. Lett. 37, 3273–3275 (2012). [CrossRef] [PubMed]
  10. M. Dubreuil, P. Delrot, I. Leonard, A. Alfalou, C. Brosseau, A. Dogariu, “Exploring underwater target detection by imaging polarimetry and correlation techniques,” Appl. Opt. 52, 997–1005 (2013). [CrossRef] [PubMed]
  11. A. Bénière, M. Alouini, F. Goudail, D. Dolfi, “Design and experimental validation of a snapshot polarization contrast imager,” Appl. Opt. 48, 5764–5773 (2009). [CrossRef] [PubMed]
  12. N. Hautiere, D. Aubert, “Contrast restoration of foggy images through use of an onboard camera,” in Proceedings of 2005 IEEE Intelligent Transportation Systems(2005), pp 601–606.
  13. N. Gracias, S. Negahdaripour, L. Neumann, R. Prados, R. Garcia, “A motion compensated filtering approach to remove sunlight flicker in shallow water images,” in OCEANS 2008, (2008), pp. 1–7. [CrossRef]
  14. M. Darecki, D. Stramski, M. Sokólski, “Measurements of high-frequency light fluctuations induced by sea surface waves with an underwater porcupine radiometer system,” J. Geophys. Res. 116, C00H09 (2011). [CrossRef]
  15. F. A. Sadjadi, C. L. Chun, “Automatic detection of small objects from their infrared state-of-polarization vectors,” Opt. Lett. 28, 531–533 (2003). [CrossRef] [PubMed]
  16. B. Laude-Boulesteix, A. D. Martino, B. Drévillon, L. Schwartz, “Mueller polarimetric imaging system with liquid crystals,” Appl. Opt. 43, 2824–2832 (2004). [CrossRef] [PubMed]
  17. J. Jaffe, “Computer modeling and the design of optimal underwater imaging systems,” IEEE J. Oceanic Eng. 15, 101–111 (1990). [CrossRef]
  18. P. Garthwaite, I. Jolliffe, B. Jones, Statistical Inference (Prentice Hall, 1995).
  19. J. Fade, N. Treps, C. Fabre, P. Réfrégier, “Optimal precision of parameter estimation in images with local sub-Poissonian quantum fluctuations,” Eur. Phys. J. D 50, 215–227 (2008). [CrossRef]

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