## Full analytical solution of adapted polarisation state contrast imaging |

Optics Express, Vol. 19, Issue 25, pp. 25188-25198 (2011)

http://dx.doi.org/10.1364/OE.19.025188

Acrobat PDF (1375 KB)

### Abstract

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

© 2011 OSA

## 1. Introduction

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

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

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

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

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

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

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

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

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

- What are the ensembles of solutions for the specific excitations states depending on the polarimetric scenes considered.
- What is the effect of the shot noise on these solutions and
- for a given polarimetric scene, what is the theoretical maximal distance in the Poincaré sphere between the two Stokes vectors describing the field scattered by the object and background region.

2. S. Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A **13**, 1106–1113 (1996). [CrossRef]

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

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

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

### 2.1. The Mathematical Modeling of the Problem

**M**and

_{O}**M**

*fully describes the polarimetric properties of the two disjoint regions*

_{B}*𝒪*and

*ℬ*respectively for the object and the background. Here

**M**

_{O}*and*

_{ij}**M**

_{B}*belongs to*

_{ij}**R**∀ (

*i*,

*j*) ∈ [0, 3]. Considering experimental situation, the scene is considered to be a priori unknown and hence we need an initial estimation of the Mueller matrices of the object

**M̃**and of the background

_{O}**M̃**by Classical Mueller Imaging (CMI) before using APSCI method. Here similarly, from the physicality criterion

_{B}**M̃**and

_{O}**M̃**belongs to

_{B}**R**

^{4}X

**R**

^{4}. During CMI, we consider that each pixel of the detector indexed by (

*u*,

*v*) receive an intensity

**I**(

*u*,

*v*) perturbed by a shot noise of Poisson distribution. Then from noisy intensity matrices

**Ĩ**(

*u*,

*v*), the Mueller matrices

**M̃**(

**u**,

**v**) are retrieved for each pixel. The noise at any two pixels of the detector are also considered to be mutually independent.

*S*used to illuminate the scene after the initial evaluation of Mueller matrices for

*𝒪*and

*ℬ*. Here

*S*∈

*B*

^{4}.

*B*

^{4}is the closed ball in 4 dimensions. Which implies,

*S*∈

**R**

^{4}: ||

*S*||

^{2}≤ 2. The estimations of the Stokes vectors of the field scattered by the object

*S̃*and background

_{O}*S̃*can be expressed as :

_{B}*S*||

^{2}= 2 i.e.

*S*∈

*𝒫*, where

*𝒫*is the set of fully polarised Stokes vectors corresponding to all the possible states on the surface of Poincaré sphere.

*S̃*and

_{O}*S̃*in the Poincaré sphere by the Euclidean distance

_{B}*D*considering their last three parameters in following way:

### 2.2. The Solution

*S*= (

*s*

_{0},

*s*

_{1},

*s*

_{2},

*s*

_{3})

*, the difference vector is*

^{T}*s*’s, ∀

_{i}*i*∈ 0, 1, 2, 3. Let us denote this quadratic function by

*𝒬*(

*s*

_{0},

*s*

_{1},

*s*

_{2},

*s*

_{3}). Then we get,

*a*and which is unique. It can be calculated from Eq. (6) that

_{ij}*X*4 matrix, which implies that all its eigen values and eigen vectors are real and can be diagonalised to

*A*by an orthogonal matrix transformation of form

^{d}*O*

^{T}*AO*, where O is orthogonal matrix (columns of which are eigen vectors of A) and

*A*is real and diagonal (having the corresponding eigenvalues of A on the diagonal).

^{d}### 2.3. 3-Distance Eigen Space

*S*

_{0},

*S*

_{1},

*S*

_{2},

*S*

_{3}space to the eigen space of matrix corresponding to 3-Distance Quadratic Function is equivalent to a rotation in 4D space. Moreover, from a set of orthogonal basis (here it is

*S*

_{0},

*S*

_{1},

*S*

_{2},

*S*

_{3}) if we move to the eigen bases of a symmetric matrix (in this case it is A), the new bases will also remain orthogonal. The nomenclature for this new rotated space is from now on will be 3-Distance Eigen Space (3DES). 3DES has two intrinsic properties. Firstly, it depends on the quadratic form we work with, which can be very general and irrespective of the imaging problem we are addressing here (in our case the Cartesian distance function on Poincaré space of Stokes vectors) Eq. (6) and can be broadened further and secondly it depends on the exact Mueller matrices of the scene we work with and this is what makes the 3DES adaptive to our imaging. Now if we concentrate on the trace of

*A*, we can see that:

*λ*’s (∀

_{i}*i*∈ (0, 1, 2, 3)) are the eigen values of

*A*. As we know under similarity transformation the trace is conserved and hence Eq. (11). In the result section we analyse some properties of 3DES and it becomes immensely helpful to understand underlying physics of APSCI.

## 3. The Illumination Technique of APSCI

*λ*and

_{max}*S*are real, hence we can reach a

_{max}**physically realisable solution**. Though due to the solution space (i.e. ||

*S*||

^{2}≤ 2) we work with, we may come up with solutions where input Stokes vectors are not fully polarised. For the sake of APSCI, only fully polarised solutions are selected. As we can readily see from Eq. (10), the maximum 3-Distance is proportional to

*λ*, but it might not be achievable in most of the imaging situations. In fact here, we are obtaining the direction of the illumination state using mathematical algorithm and then we are stretching out Stokes vector norm to unity in order to achieve specific fully polarised incident light. So, the algorithm we use to get

_{max}*S*is as follow :

_{max}*S*and –

_{max}*S*both will yield the maximum of 3-Distance Quadratic Function, and hence we choose the physically possible solution so that

_{max}*S*

_{max0}is positive.

*S̃*and

_{O}*S̃*are not normalized and hence they can exhibit different rates of depolarization. As a consequence, the maximization of the 3-Distance mentioned above takes into account both the physical entities: the polarization state and the intensity of the polarized part of the scattered field.

_{B}## 4. The Detection Technique of APSCI

*𝒪*and

*ℬ*into some state

*S*on the surface of Poincaré sphere. Then to maximise the imaging contrast we have to maximise the difference of projection. Hence we have to maximize the following function

_{d}*F*(

*S*) where ||

_{d}*S*||

_{d}^{2}= 2

*dS*is orthogonal to

_{d}*S*||

_{d}^{2}= 2 we get,

*F*| will be maximum for

*S*parallel and antiparallel to

_{d}*i*∈ (1, 2, 3). Thus,

*S*

_{out}_{1}and

*S*

_{out}_{2}will increase the detected intensities corresponding respectively to the object and background pixels while imaging.

*u*,

*v*) as :

*I*

_{1}(

*u*,

*v*) and

*I*

_{2}(

*u*,

*v*) are the detected intensities after projection respectively on the 2 states of polarization

*S*

_{out}_{1}and

*S*

_{out}_{2}.

## 5. Results and Analysis

*S*for different realizations of shot noise, the ensemble of solutions is on and around a great circle over Poincaré sphere. From theoretical solutions, we can see in this case if we consider pure Mueller matrices (not affected by shot noise), then the eigen space corresponding to

_{max}*λ*is degenerate. In this kind of cases, two eigen vectors, let’s say

_{max}*λ*= 0.0979) considering pure Mueller matrices and modified eigen directions ([1, 0.0184, 0.2636, −0.9644] using the dark green arrow corresponds to eigen value = 0.0887 and [1, 0.0107, 0.9645, 0.2639] using the light green arrow corresponds to eigen value = 0.0991) considering Mueller matrices perturbed by a single instance of shot noise. We can easily verify from the Fig. 1(a) that the two eigen planes have a small angle between them in (

_{max}*S*

_{1},

*S*

_{2},

*S*

_{3}) co-ordinate system and this statistics of the angular shift between the eigen planes in 3DES can provide the effect of noise during measurements. To provide an intuitive understanding we investigate the case graphically in Fig. 1(b). In case of birefringence, the direction of the birefringence vector in Poincaré sphere is defined by corresponding azimuth and ellipticity, while the scalar birefringence is defined by the rotation angle around it. Now as in our scene the object and the background only differ in scalar birefringence, hence the angles of rotation (for the object denoted by dark blue, for the background denoted by bluish green) of a polarimetric state around the same birefringence axis (denoted by the red arrow) are different. The distance between the final states (heads of arrows) will in turn denote the 3-Distance (denoted by the pink arrow) in the Poincaré sphere. For any point

*S*not lying in the equatorial plane perpendicular to the birefringence vector, the same angular rotation will always yield lesser length of the pink arrow compared to any point

_{in}*D⃗*using the blue arrow, while the background dichroic vector is by

_{O}*D⃗*using the greenish blue arrow in Fig. 1(d). Any incident Stokes vector

_{B}*S*on Poincaré sphere will provide more or less similar 3-Distance. Only the Stokes vector, that is bisecting the angle between

_{in}*D⃗*and

_{B}*D⃗*, will provoke a scattering in almost opposite direction for object and background and hence will yield the maximum 3-Distance(denoted by the pink arrow) and that’s the reason why we get the ensemble of solutions as a cluster of points.

_{O}*D⃗*using the blue arrow, while the background dichroic vector is by

_{O}*D⃗*using the greenish blue arrow in Fig. 1(d). Any incident Stokes vector

_{B}*S*on Poincaré sphere will provide more or less similar 3-Distance. Only the Stokes vector, that is bisecting the angle between

_{in}*D⃗*and

_{B}*D⃗*, will provoke a scattering in almost opposite direction for object and background and hence will yield the maximum 3-Distance(denoted by the pink arrow) and that’s the reason why we get the ensemble of solutions as a cluster of points.

_{O}*B*(

*M*),

*B*(

*M*

_{Δ}),

*B*(Δ),

*B*(

*APSCI*) and

*B*(

*MAPSCI*) are the Bhattacharyya distance available from best element of Mueller Matrix (the one element out of the 16 elements of total Mueller matrices that provides maximum contrast), from the best element of Dichroic matrix after Lu-Chipman forward decomposition [2

2. S. Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A **13**, 1106–1113 (1996). [CrossRef]

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

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

**17**, 14199–14210 (2009). [CrossRef] [PubMed]

## 6. Conclusion

## Acknowledgments

## References and links

1. | J. L. Pezzaniti and R. A. Chipman, “Mueller matrix imaging polarimetry (journal paper),” Opt. Eng. |

2. | S. Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A |

3. | J. E. Wolfe and R. A. Chipman, “Polarimetric characterization of liquid-crystal-on-silicon panels,” Appl. Opt. |

4. | R. Ossikovski, M. Anastasiadoua, and A. De Martino, “Product decompositions of depolarizing Mueller matrices with negative determinants,” Opt. Commun. |

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

6. | M. Richert, X. Orlik, and A. De Martino, “Adapted polarization state contrast image,” Opt. Express |

7. | F. Goudail, “Optimization of the contrast in active Stokes images,” Opt. Lett. |

8. | F. Goudail and A. Béniére, “Optimization of the contrast in polarimetric scalar images,” Opt. Lett. |

9. | L. Meng and J. P. Kerekes, “Adaptive target detection with a polarization-sensitive optical system,” Appl. Opt. |

10. | H. Poincaré, |

11. | A. Bhattacharyya, “On a measure of divergence between two statistical populations defined by probability distributions,” Bull. Calcutta Math. Soc. |

12. | F. Goudail, P. Réfrégier, and G. Delyon, “Bhattacharyya distance as a contrast parameter for statistical processing of noisy optical images,” J. Opt. Soc. Am. A |

13. | D. Upadhyay, M. Richert, E. Lacot, A. De Martino, and X. Orlik, “Effect of speckle on APSCI method and Mueller imaging,” Opt. Express |

14. | R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett. |

**OCIS Codes**

(110.2970) Imaging systems : Image detection systems

(120.5410) Instrumentation, measurement, and metrology : Polarimetry

(260.5430) Physical optics : Polarization

(110.5405) Imaging systems : Polarimetric imaging

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: May 17, 2011

Revised Manuscript: June 24, 2011

Manuscript Accepted: July 5, 2011

Published: November 23, 2011

**Citation**

Debajyoti Upadhyay, Sugata Mondal, Eric Lacot, and Xavier Orlik, "Full analytical solution of adapted polarisation state contrast imaging," Opt. Express **19**, 25188-25198 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-25-25188

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### References

- J. L. Pezzaniti and R. A. Chipman, “Mueller matrix imaging polarimetry (journal paper),” Opt. Eng. 34(06), 1558–1568 (1995). [CrossRef]
- S. Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13, 1106–1113 (1996). [CrossRef]
- J. E. Wolfe and R. A. Chipman, “Polarimetric characterization of liquid-crystal-on-silicon panels,” Appl. Opt. 45, 1688–1703 (2006). [CrossRef] [PubMed]
- R. Ossikovski, M. Anastasiadoua, and A. De Martino, “Product decompositions of depolarizing Mueller matrices with negative determinants,” Opt. Commun. 281(9), 2406–2410 (2008). [CrossRef]
- M. Richert, X. Orlik, and A. De Martino, “Optimized orthogonal state contrast image,” Proceedings of the 21th Congress of the Int. Comm. for Opt. (2008), Vol. 139.
- M. Richert, X. Orlik, and A. De Martino, “Adapted polarization state contrast image,” Opt. Express 17, 14199–14210 (2009). [CrossRef] [PubMed]
- F. Goudail, “Optimization of the contrast in active Stokes images,” Opt. Lett. 34, 121–123 (2009). [CrossRef] [PubMed]
- F. Goudail and A. Béniére, “Optimization of the contrast in polarimetric scalar images,” Opt. Lett. 34, 1471–1473 (2009). [CrossRef] [PubMed]
- L. Meng and J. P. Kerekes, “Adaptive target detection with a polarization-sensitive optical system,” Appl. Opt. 50, 1925–1932 (2011). [CrossRef] [PubMed]
- H. Poincaré, Théorie mathématique de la lumière (GABAY, 1892). [PubMed]
- A. Bhattacharyya, “On a measure of divergence between two statistical populations defined by probability distributions,” Bull. Calcutta Math. Soc. 35, 99–109 (1943).
- F. Goudail, P. Réfrégier, and G. Delyon, “Bhattacharyya distance as a contrast parameter for statistical processing of noisy optical images,” J. Opt. Soc. Am. A 21, 1231–1240 (2004). [CrossRef]
- D. Upadhyay, M. Richert, E. Lacot, A. De Martino, and X. Orlik, “Effect of speckle on APSCI method and Mueller imaging,” Opt. Express 19(5), 4553–4559 (2010). [CrossRef]
- R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett. 32, 689–691 (2007). [CrossRef] [PubMed]

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