## Effect of speckle on APSCI method and Mueller Imaging |

Optics Express, Vol. 19, Issue 5, pp. 4553-4559 (2011)

http://dx.doi.org/10.1364/OE.19.004553

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### Abstract

The principle of the polarimetric imaging method called APSCI (** A**dapted

**olarization**

*P***tate**

*S***ontrast**

*C***maging) is to maximize the polarimetric contrast between an object and its background using specific polarization states of illumination and detection. We perform here a comparative study of the APSCI method with existing Classical Mueller Imaging(CMI) associated with polar decomposition in the presence of fully and partially polarized circular Gaussian speckle. The results show a noticeable increase of the Bhattacharyya distance used as our contrast parameter for the APSCI method, especially when the object and background exhibit several polarimetric properties simultaneously.**

*I*© 2011 Optical Society of America

## 1. Introduction

**[1**

*APSCI*1. M. Richert, X. Orlik, and A. De Martino, “Adapted Polarization state contrast imaging,” Opt. Express **17**, 14199–14210 (2009). [CrossRef] [PubMed]

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

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

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

## 2. Brief review of the APSCI method

**M**and

_{O}**M**

*, respectively for the object and the background. As the scene is considered to be a priori unknown, we need an initial estimation of Mueller matrices of the object*

_{B}**M̃**and of the background

_{O}**M̃**by CMI before implementing APSCI method. During the Mueller imaging process, we consider that each pixel of the detector indexed by (

_{B}*u*,

*v*) receives an intensity

**I**(

*u*,

*v*) perturbed by a Poisson distribution in order to take into account the shot noise. The Mueller matrix

**M̃**(

*u*,

*v*) at each pixel is then calculated from the noisy detected intensities Ĩ(

*u*,

*v*).

*S⃗*is used to illuminate the scene after CMI. The estimations of the Stokes vectors of the field scattered by the object

*D*between their last three parameters. Then, we determine numerically using a simplex search algorithm the specific incident Stokes vector

*S̃*that maximizes this Euclidean distance. It is worthy to emphasize that the Stokes vectors

^{in}*S⃗*, into 2 states of polarization

^{in}*S⃗*

^{out}^{1}and

*S⃗*

^{out}^{2}, that are defined to maximize respectively (

*Ĩ*–

_{O}*Ĩ*) and (

_{B}*Ĩ*–

_{B}*Ĩ*), where

_{O}*Ĩ*and

_{O}*Ĩ*are the evaluations of the mean intensity detected respectively from the object and background scattering. From simple calculation it can be shown that: where with

_{B}*u*,

*v*) as : where

*I*

_{1}(

*u*,

*v*) and

*I*

_{2}(

*u*,

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

*S⃗*

^{out}^{1}and

*S⃗*

^{out}^{2}.

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

## 3. Characteristics of the Speckle noise

*p*(

_{I}*I*) of a completely developed circular Gaussian speckle that depends on the degree of polarization

*P*[5]: where

*Ī*is the average intensity.

## 4. Results and analysis

*B*(

*M*) is defined to be the selected element between

**M̃**and

_{O}**M̃**that provides the best Bhattacharyya distance over the 16 possible elements. In a similar way,

_{B}*B*(

*M*),

_{R}*B*(

*M*) and

_{D}*B*(

*M*

_{Δ}) represent respectively the best Bhattacharyya distance obtained from the selected element of the birefringence, the dichroism and the depolarization matrices extracted from

**M̃**and

_{O}**M̃**using the forward polar decomposition.

_{B}*B*(

*R*),

*B*(

*D*) and

*B*(

*DOP*). Finally,

_{L}*B*represents the Bhattacharyya distance of the APSCI parameter as defined in section 2.

_{APSCI}*B*(

*M*) (green curve) selected from the raw data of

**M̃**and

_{O}**M̃**. Below this threshold, the noise introduced by this decomposition worsen the situation (as can be seen in case (a)) because

_{B}**M̃**and

_{O}**M̃**are insufficiently determined. Secondly, we observe that the parameter

_{B}*B*(blue curve) exhibits the highest Bhattacharyya distances for all the SNR studied in (a) (c) and (e). However, for the case (c), we notice that it exhibits also higher uncertainty bars associated to lower mean values of Bhattacharyya distances compared to cases (a) and (e). This lower performance of the AP-SCI method in the case of dichroism is coming from 2 phenomena : the absorption of energy due to the dichroism effect and the cartesian distance between the matrices of the object and background defined here as the square root of the sum of the square of the element-wise differences. Indeed, as previously discussed in [1

_{APSCI}**17**, 14199–14210 (2009). [CrossRef] [PubMed]

*APSCI*still remains the more pertinent parameter to distinguish the object from the background even if its standard deviation noticeably increases due to the presence of speckle. The effect of the same speckle noise on situation (c) is plotted on (d). We observe that

*B*(

*M*) and

_{D}*B*(

*M*) fall to very low values even for high SNR and as a consequence become unusable for imaging. As a result, from the raw data of the Mueller matrices

**M̃**and

_{O}**M̃**associated to the polar decomposition, only

_{B}*B*(

*D*) reaches an order of magnitude similar to

*B*. Moreover, we notice that the standard deviation of

_{APSCI}*B*has considerably increased due to the speckle noise. After a deeper analysis, we have observed that

_{APSCI}**M̃**and

_{O}**M̃**are particularly poorly estimated for the case of dichroism (for the 2 reasons mentioned above) and that the addition of speckle noise worsen noticeably this situation. As a consequence, selective states of excitation

_{B}*S*spread near all over the Poincaré sphere, showing only a weak increase of density of probability in the theoretical optimum region.

^{in}*P*=0.78 and

_{obj}*P*=0.71 for the object and background that exhibit a difference of 10% in their ability to depolarize linear polarized light. We observe only a weak decrease of all the Bhattacharyya distances due to the fact that the speckle, only partially polarized, exhibits a lower contrast (

_{back}*C*= 0.90 and

_{obj}*C*= 0.87) than in previous situations. Moreover, the APSCI parameter gives rise to Bhattacharyya distances much higher than using the CMI alone or associated with the polar decomposition.

_{back}7. I. Jung, M. Vaupel, M. Pelton, R. Piner, D. A. Dikin, S. Stankovich, J. An, and R. S. Ruoff, “Characterization of Thermally Reduced Graphene Oxide by Imaging Ellipsometry,” J. Phys. Chem. C , **112**(23), 8499–8506 (2008). [CrossRef]

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

9. J. S. Tyo, M. P. Rowe, E. N. Pugh Jr., and N. Engheta, “Target detection in optically scattering media by polarization-difference imaging,” Appl. Opt. **35**, 1855–1870 (1996). [CrossRef] [PubMed]

## 5. Conclusion

## Acknowledgments

## References and links

1. | M. Richert, X. Orlik, and A. De Martino, “Adapted Polarization state contrast imaging,” Opt. Express |

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

3. | H. Poincaré, |

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

5. | J. W. Goodman, Speckle Phenomena in optics (Roberts & Company Pub., Colorado, 2007) pp. 48–50. |

6. | P. Drude, “Über Oberflächenschichten,” Ann. der Physik |

7. | I. Jung, M. Vaupel, M. Pelton, R. Piner, D. A. Dikin, S. Stankovich, J. An, and R. S. Ruoff, “Characterization of Thermally Reduced Graphene Oxide by Imaging Ellipsometry,” J. Phys. Chem. C , |

8. | M. P. Rowe, E. N. Pugh Jr., J. S. Tyo, and N. Engheta, “Polarization-difference imaging: a biologically inspired technique for observation through scattering media,” Opt. Lett. |

9. | J. S. Tyo, M. P. Rowe, E. N. Pugh Jr., and N. Engheta, “Target detection in optically scattering media by polarization-difference imaging,” Appl. Opt. |

**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: October 25, 2010

Revised Manuscript: December 15, 2010

Manuscript Accepted: December 17, 2010

Published: February 24, 2011

**Citation**

Debajyoti Upadhyay, Micheal Richert, Eric Lacot, Antonello De Martino, and Xavier Orlik, "Effect of speckle on APSCI method and Mueller Imaging," Opt. Express **19**, 4553-4559 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-5-4553

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### References

- M. Richert, X. Orlik, and A. De Martino, “Adapted Polarization state contrast imaging,” Opt. Express 17, 14199–14210 (2009). [CrossRef] [PubMed]
- 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]
- 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).
- J. W. Goodman, Speckle Phenomena in optics (Roberts & Company Pub., Colorado, 2007) pp. 48–50.
- P. Drude, “Über Oberflächenschichten,” Ann. Phys. 36, 86597 (1889).
- I. Jung, M. Vaupel, M. Pelton, R. Piner, D. A. Dikin, S. Stankovich, J. An, and R. S. Ruoff, “Characterization of Thermally Reduced Graphene Oxide by Imaging Ellipsometry,” J. Phys. Chem. C 112(23), 8499–8506 (2008). [CrossRef]
- M. P. Rowe, and E. N. Jr, “Pugh, J. S. Tyo, and N. Engheta, “Polarization-difference imaging: a biologically inspired technique for observation through scattering media,” Opt. Lett. 20, 608–610 (1995). [CrossRef] [PubMed]
- J. S. Tyo, M. P. Rowe, E. N. Pugh, Jr., and N. Engheta, “Target detection in optically scattering media by polarization-difference imaging,” Appl. Opt. 35, 1855–1870 (1996). [CrossRef] [PubMed]

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