## Optimal estimation in polarimetric imaging in the presence of correlated noise fluctuations |

Optics Express, Vol. 22, Issue 5, pp. 4920-4931 (2014)

http://dx.doi.org/10.1364/OE.22.004920

Acrobat PDF (1450 KB)

### 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

### 1.2. Image formation model

*s*at location

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

*i*is assumed to also comprise a back-ground contribution

*b*, with a DOP denoted by

_{i}*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 (

*P*≫

*β*), some situations can involve opposite physical conditions (

*β*≫

*P*), 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]

*N*pixels gives access to a sample

*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]

### 1.3. Noise model

*i*, the second order statistical properties of the bidimensional measurement vector

*N*–pixels measurement sample is then given by

*i*in the image, and from the illumination level or polarization properties. The first term in the expression of

*β*. This noise contribution, proportional to the background average level

*b*, can model the effect of turbulence or variations of scatterers density, as well as photon noise in the high background intensity limit.

_{i}## 2. Gain in optimal estimation performance

### 2.1. Principle

**X**for the estimation of a parameter

*y*, and is defined as [18] According to the well-known Cramer-Rao theorem, its inverse value I

_{F}

^{−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]. In the following, we shall limit ourselves to the estimation of the mean signal intensity

*s*at location

_{i}*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 I

_{F}(

*z*) = I

_{F}(

*y*) [

*dy/dz*]

^{2}from simple variable transformation relations. For instance, for the estimation of an atmospheric transmittance

*τ*such that

*s*=

*e*

^{−Lτ}, the FI is directly obtained with I

_{F}(

*τ*) =

*L*

^{2}

*s*

^{2}I

_{F}(

*s*), which simply involves the FI for the estimation of the mean signal intensity I

_{F}(

*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]

*η*is to be estimated over the whole image such that

**s**= {

*s*(

*x*−

_{i}*η*)}

_{i}_{=1,...,}

*. In this latter case, the above relation yields*

_{N}*i*.

### 2.2. Expression of the gain

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. 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]

*ω*

^{2}gives the relative value of the noise contributions variances, allowing one to identify the dominant noise term. Thus, “optical” noise

*ε*

^{2}dominates when

*ω*

^{2}≫ 1, whereas electronic fluctuations are the main source of noise when

*ω*

^{2}≪ 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.

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

### 3.1. Influence of ambient illumination level

*b*. For that purpose, we analyze the behaviour of the gain

*μ*(

*ω*,

*P*,

*β*,

*ρ*) as a function of

*ω*=

*ε/σ*

_{0}, 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*.

*μ*(

*ω*≪ 1,

*P*,

*β*,

*ρ*) → (1 +

*P*

^{2})/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

*ω*→ ∞, we obtain a simpler expression which will be referred to as

*asymptotic gain*subsequently.

*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 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*

**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*

*where*

### 3.3. Discussion

*K*when using PSDs instead of standard imagers. In this subsection, we propose to quantitatively analyze these theoretical results.

*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

*β*≤

*P*, and

*β*≥ 1− (1−

*P*)

^{2}/2 when

*β*≥

*P*. 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

*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*

_{∞}

*and*

_{,min}*ρ*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

_{min}*ρ*has to be greater than a minimum value denoted

*μ*

_{∞}(

*P*,

*β*,

*ρ*) ≥ 1. The values of

*P*and

*β*.

## 4. Optimal estimation procedure

*efficient*estimation procedures, i.e., estimators ensuring unbiased estimation and a minimum variance which reaches the CRB studied above. Let us thus consider estimators of

*s*in the maximum likelihood (ML) sense, since ML estimators are known to be

*efficient*under Gaussian fluctuations [18], which is the noise model considered throughout this article. Limiting ourselves to the high intensity regime (

*ω*→ ∞), and assuming that the background mean value

*b*is

*a priori*known, the ML estimator of

*s*using a standard intensity detector is simply given by

*s*is detailed in Appendix E and leads to where

*U*,

*V*,

*W*and

*Z*are functions of

*P*,

*β*,

*ρ*and

*b*, which parameters are assumed

*a priori*known. These functions can be easily derived from Appendix E with appropriate changes of variable, but are not detailed here for brevity reasons. Both ML estimators are unbiased, i.e.,

*N*pixels in each polarimetric channels must be compared to a 2

*N*-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

*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]

*s*, such

*difference*estimator would simply read

*ρ*= (1 −

*β*

*P*)/(1 −

*β*

^{2}), in which case the difference estimator

## 5. Conclusion

*a*= (1 +

*P*)/2 and

*α*= (1−

*P*)/(1 +

*P*) on the one hand, and, on the other hand,

*c*

^{2}= (1 +

*β*)/2 and

*γ*

^{2}= (1 −

*β*)/(1 +

*β*). In a single pixel configuration (

*N*= 1) for the sake of simplicity, the polarimetric measurement considered is

**X**

*= [*

^{P}*X*

^{//},

*X*

^{⊥}]

*, such that 〈*

^{T}**X**

*〉 = [*

^{P}*as*+

*c*

^{2}

*b*,

*α*

*as*+

*γ*

^{2}

*c*

^{2}

*b*]

*and with*

^{T}*P*∈ [0, 1],

*β*∈ [0, 1] and

*β*≤

*P*are equivalent to

*α*∈ [0, 1],

*γ*∈ [0, 1] and

*γ*

^{2}≥

*α*.

## A. Fisher informations calculations

**X**

*can be written*

^{P}*ℓ*(

**X**

*) = ln*

^{P}*P*

**(**

_{X}**X**

*) = −(*

^{P}*δ*

**X**

*)*

^{P}*Γ*

^{T}^{−1}

*δ*

**X**

*/2 up to an additive term independent of*

^{P}*s*. An application of Eq. (2) leads to the FI for the estimation of

*s*, which reads I

_{F}

*(*

^{P}*s*) = [〈

**X**〉

*′*]

^{P}*Γ*

^{T}^{−1}〈

**X**〉

*′*, with 〈

^{P}**X**〉

*′*=

^{P}*∂*〈

**X**〉

^{P}*/∂s*= 〈

*∂*

**X**

^{P}*/∂s*〉 = [

*a*,

*a*

*α*]

*. A direct calculation gives: with*

^{T}*u*

^{2}=

*ε*

^{2}/

*ς*

^{2}= (1 +

*β*)

*ω*

^{2}/2.

*s*from the total intensity of the beam (non-polarimetric measurement) is a standard result under Gaussian fluctuations hypothesis. One has The gain

*μ*(

*u*,

*α*,

*γ*,

*ρ*) = I

_{F}

*(*

^{P}*s*)/I

_{F}

*(*

^{I}*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*

*μ*(

*u*,

*α*,

*γ*,

*ρ*) as with

*A*= (

*α*

^{2}− 2

*ραγ*+

*γ*

^{2}),

*B*= 1 +

*α*

^{2},

*C*= 1 +

*γ*

^{2},

*D*=

*γ*

^{2}(1 −

*ρ*

^{2}) 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 +

*P*

^{2})/2.

*μ*(

*u*,

*α*,

*γ*,

*ρ*) as a function of

*u*is thus with

*ℋ*(

*u*) =

*A*+ 2(

*AC*−

*BD*)

*u*

^{2}+ (

*AC*

^{2}−

*AD*−

*BCD*)

*u*

^{4}.

*μ*(

*u*,

*α*,

*γ*,

*ρ*) is monotonically increasing on

*u*∈ [0; ∞[ if

*∂*[

*μ*(

*u*,

*α*,

*γ*,

*ρ*)]/

*∂u*≥ 0 ⇔

*ℋ*(

*u*) ≥ 0, ∀

*u*∈ [0; ∞[. Noticing that

*ℋ*(0) =

*A*≥ 0, and that

*ℋ*(

*u*) is a second-order polynomial in

*u*

^{2}, we can compute the discriminant which is negative. As a consequence,

*ℋ*(

*u*) does not admit real root on

*u*∈ [0; ∞[, and hence,

*ℋ*(

*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*):

*μ*

_{∞}(

*α*,

*γ*,

*ρ*) reaches a minimum if (

*αρ*−

*γ*)(

*α*−

*ργ*) = 0. Since

*ρ*∈ [0, 1], the only admissible root is

*ρ*=

_{min}*α/γ*when

*γ*≥

*α*and thus

*μ*

_{∞}

*(*

_{,min}*α*,

*γ*) = (1 +

*γ*

^{2})/(1 +

*α*)

^{2}. When

*γ*≤

*α*, the only admissible root is

*ρ*=

_{min}*γ/α*, and in this case

*μ*

_{∞}

*(*

_{,min}*α*,

*γ*) =

*α*

^{2}(1 +

*γ*

^{2})/

*γ*

^{2}(1 +

*α*)

^{2}. The expressions of

*ρ*and

_{min}*μ*

_{∞}

*(*

_{,min}*P*,

*β*) given in the article can be recovered with an appropriate change of variables.

## D. Condition for minimum gain *μ*_{∞}(*α*, *γ*, *ρ*) = *K*:

*μ*

_{∞}(

*α*,

*γ*,

*ρ*) =

*K*leads to two roots

*γ*

^{2}−

*K*(1 +

*α*)

^{2}] [

*α*

^{2}(1 +

*γ*

^{2}) −

*K*(1 +

*α*)

^{2}

*γ*

^{2}], or with the notations of Appendix B, Φ = [

*C*−

*KE*] [

*α*

^{2}

*C*−

*KEγ*

^{2}].

*ρ*such that

*μ*

_{∞}(

*α*,

*γ*,

*ρ*) ≥

*K*,

*K*

^{2}

*γ*

^{2}(1 +

*α*)

^{4}].

*ρ*. It is clear that

*α*

^{2}

*C/γ*

^{2}

*E*≤

*K*≤

*C/E*; or (b):

*α*

^{2}

*C/γ*

^{2}

*E*≥

*K*≥

*C/E*. When

*γ*≥

*α*, inequality (b) is impossible, and (a) is verified if

*γ*

^{2}≥

*K*(1 +

*α*)

^{2}− 1. When

*γ*≤

*α*, inequality (a) is impossible, and (b) is verified if

*γ*

^{2}≤

*α*

^{2}/[

*K*(1 +

*α*)

^{2}−

*α*

^{2}]. These conditions obviously correspond to those derived above in Appendix C in Eqs. (18) and (19) for ensuring

*μ*

_{∞}

*(*

_{,min}*α*,

*γ*) ≥

*K*.

## E. ML estimator
s ^ M L P

*u*→ ∞ (or

*ω*→ ∞). From the expression of

*ℓ*(

**X**

*) given in Appendix A, we derive*

^{P}*∂ℓ*(

**X**

*)/*

^{P}*∂s*= 0, leading to equation (

*δ*

**X**

*)*

^{P}*Γ*

^{T}^{−1}〈

**X**〉

*′*+ (〈

**X**〉

*′*)

*Γ*

^{T}^{−1}

*δ*

**X**

*= 0. A straightforward but tedious calculation finally gives the expression of Eq. (11), with*

^{P}*U*=

*γ*[

*γ*−

*αρ*],

*V*= [

*α*−

*γρ*],

*Z*=

*bc*

^{2}

*γ*[

*ρ*(

*α*+

*γ*

^{2}) −

*γ*(1 +

*α*)], and

*W*=

*a*[

*α*

^{2}− 2

*ραγ*+

*γ*

^{2}]. It is easily checked that this estimator is unbiased

*X̂*

^{//}〉 =

*as*+

*c*

^{2}

*b*and 〈

*X̂*

^{⊥}〉 =

*aαs*+

*c*

^{2}

*γ*

^{2}

*b*. Moreover, the variance of this estimator necessarily reaches the CRB since ML estimator is efficient under Gaussian fluctuations [18]. This can be checked by noticing that

_{u}_{→∞}{1/I

_{F}

*(*

^{P}*s*)}.

## F. Difference image estimator
s ^ Δ P

*ŝ*

_{Δ}) =

*c*

^{2}

*ε*

^{2}[1 +

*γ*

^{2}− 2

*ργ*]/[

*a*

^{2}(1 +

*α*)

^{2}].

*α*+

*γ*

^{2}−

*ρ*(1 +

*α*)

*γ*= 0, i.e., if

*ρ*= (

*α*+

*γ*

^{2})/

*γ*(1 +

*α*) = (1 −

*βP*)/(1 −

*β*

^{2}), in which case

## Acknowledgments

*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. |

2. | S. Demos, H. Savage, A. S. Heerdt, S. Schantz, and R. Alfano, “Time resolved degree of polarization for human breast tissue,” Opt. Commun. |

3. | O. Emile, F. Bretenaker, and A. L. Floch, “Rotating polarization imaging in turbid media,” Opt. Lett. |

4. | H. Ramachandran and A. Narayanan, “Two-dimensional imaging through turbid media using a continuous wave light source,” Opt. Commun. |

5. | J. Guan and J. Zhu, “Target detection in turbid medium using polarization-based range-gated technology,” Opt. Express |

6. | G. D. Lewis, D. L. Jordan, and P. J. Roberts, “Backscattering target detection in a turbid medium by polarization discrimination,” Appl. Opt. |

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. |

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 |

9. | M. Boffety, F. Galland, and A.-G. Allais, “Influence of polarization filtering on image registration precision in underwater conditions,” Opt. Lett. |

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. |

11. | A. Bénière, M. Alouini, F. Goudail, and D. Dolfi, “Design and experimental validation of a snapshot polarization contrast imager,” Appl. Opt. |

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. |

15. | F. A. Sadjadi and C. L. Chun, “Automatic detection of small objects from their infrared state-of-polarization vectors,” Opt. Lett. |

16. | B. Laude-Boulesteix, A. D. Martino, B. Drévillon, and L. Schwartz, “Mueller polarimetric imaging system with liquid crystals,” Appl. Opt. |

17. | J. Jaffe, “Computer modeling and the design of optimal underwater imaging systems,” IEEE J. Oceanic Eng. |

18. | P. Garthwaite, I. Jolliffe, and B. Jones, |

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 |

**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

- 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]
- 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]
- O. Emile, F. Bretenaker, A. L. Floch, “Rotating polarization imaging in turbid media,” Opt. Lett. 21, 1706–1708 (1996). [CrossRef] [PubMed]
- H. Ramachandran, A. Narayanan, “Two-dimensional imaging through turbid media using a continuous wave light source,” Opt. Commun. 154, 255–260 (1998). [CrossRef]
- J. Guan, J. Zhu, “Target detection in turbid medium using polarization-based range-gated technology,” Opt. Express 21, 14152–14158 (2013). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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]
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