## Optimal discrimination of multiple regions with an active polarimetric imager |

Optics Express, Vol. 19, Issue 25, pp. 25367-25378 (2011)

http://dx.doi.org/10.1364/OE.19.025367

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### Abstract

Until now, most studies about polarimetric contrast optimization have focused on the discrimination of two regions (a target and a background). In this paper, we propose a methodology to determine the set of polarimetric measurements that optimize discrimination of an arbitrary number of regions with different polarimetric properties. We show on real world examples that in some situations, a few number of optimized polarimetric measurements can overcome the performance of full Mueller matrix imaging.

© 2011 OSA

## 1. Introduction

1. J. E. Solomon, “Polarization imaging,” Appl. Opt. **20**, 1537–1544 (1981). [CrossRef] [PubMed]

8. A. Pierangelo, B. Abdelali, M.-R. Antonelli, T. Novikova, P. Validire, B. Gayet, and A. De Martino, “Ex-vivo characterization of human colon cancer by mueller polarimetric imaging,” Opt. Express **19**, 1582–1593 (2011). [CrossRef] [PubMed]

9. J. S. Tyo, Z. Wang, S. J. Johnson, and B. G. Hoover, “Design and optimization of partial mueller matrix polarimeters,” Appl. Opt. **49**, 2326–2333 (2010). [CrossRef] [PubMed]

10. F. Goudail, “Comparison of the maximal achievable contrast in scalar, stokes and mueller images,” Opt. Lett. **35**, 2600–2602 (2010). [CrossRef] [PubMed]

11. M. Dubreuil, S. Rivet, B. Le Jeune, and J. Cariou, “Snapshot mueller matrix polarimeter by wavelength polarization coding,” Opt. Express **15**, 13660–13668 (2007). [CrossRef] [PubMed]

12. A. A. Swartz, H. A. Yueh, J. A. Kong, L. M. Novak, and R. T. Shin, “Optimal polarizations for achieving maximal constrast in radar images,” J. Geophys. Res. **93**, 15252–15260 (1988). [CrossRef]

13. J. Yang, “Numerical methods for solving the optimal problem of contrast enhancement,” IEEE transactions on geoscience and remote sensing **38**, 965–971 (2000). [CrossRef]

14. M. Floc’h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, “Polarimetric considerations to optimize lidar detection of immersed targets,” Pure Appl. Opt. **7**, 1327–1340 (1998). [CrossRef]

16. F. Goudail and A. Bénière, “Optimization of the contrast in polarimetric scalar images,” Opt. Lett. **34**, 1471–1473 (2009). [CrossRef] [PubMed]

10. F. Goudail, “Comparison of the maximal achievable contrast in scalar, stokes and mueller images,” Opt. Lett. **35**, 2600–2602 (2010). [CrossRef] [PubMed]

## 2. Polarimetric imaging and region discrimination

### 2.1. Polarimetric measurements

**S**generated thanks to a Polarization State Generator (PSG) (see Fig. 1). In the practical implementation we use, the PSG is composed of two Liquid Crystal Variable Retarders and one polarizer. The output beam allows illuminating the scene uniformly in polarization and intensity. The polarimetric properties of a region of the scene corresponding to a pixel in the image is characterized by its Mueller matrix

*M*. The Stokes vector of the light scattered by this region is

**S**′ =

*M*

**S**. It is analyzed by a Polarization State Analyser (PSA), which is a generalized polarizer whose eigenstate is the Stokes vector

**T**. As for the PSG, in the experimental setup we use, the PSA is composed of two Liquid Crystal Variable Retarders and one polarizer. The number of photoelectrons measured at a pixel of the sensor is: where the superscript

*denotes matrix transposition. In this equation,*

^{T}**S**and

**T**are unit intensity, purely polarized Stokes vectors,

*I*

_{0}is a number of photons and

*η*is the conversion efficiency between photons and electrons. The total field of view of the imager, using a 480 × 640 CCD camera, is about 10°. The measurements are done in a spectral band of 10 nm centered on 640 nm and selected thanks to an interference filter.

### 2.2. Maximum Likelihood (ML) region classification

*K*of regions with different polarimetric properties, indexed as

*k*∈ [1,

*K*]. We assume that we have a database containing sets of Mueller matrices associated to these different regions. One acquires

*N*scalar polarimetric images with N different couples of illumination Stokes vectors (𝒮 = [

**S**

_{1}, ...,

**S**

*]) and analysis Stokes vectors (𝒯 = [*

_{N}**T**

_{1}, ...,

**T**

*]). For each pixel*

_{N}*p*of the scene, the measures can be gathered in a vector

*x*of dimension

_{p}*N*defined as: with

*M*, the Mueller matrix of the region of the scene corresponding to pixel

^{p}*p*and

*n*∈ [1,

*N*], the intensity associated to the pixel

*p*projected on the polarization states

**S**

*and*

_{n}**T**

*. Each pixel of the scene is thus represented by a point in a*

_{n}*N*-dimensional space, and our goal is to discriminate the different regions in this space.

*k*is thus: where det returns the determinant of a square matrix,

*x̄*and Γ

_{k}*respectively the mean and covariance matrix of*

_{k}*x*associated to the class

*k*. From this expression of the PDF, it is possible to define the log-likelihood: The Maximum Likelihood (ML) classifier consists in deciding that a pixel of the scene belongs to class

*k̂*as:

*k*are estimated from a database composed of sets Ω

*containing*

_{k}*P*training Mueller matrices. For each of these datasets,

_{k}*P*values of

_{k}*p*∈ [1,

*P*], can be computed as in the Eq. (2). The estimate of

_{k}*x̄*and Γ

_{k}*are obtained by the following formula: We will now illustrate this classification method on full Mueller matrix data.*

_{k}### 2.3. Classification on full Mueller matrix data and discussion

*p*of the scene, which is thus represented by a vector

*x*of dimension

_{p}*N*= 16 (Eq. (2)). The set of illumination and analysis states is usually chosen to minimize the error propagation during this inversion [17

17. J. S. Tyo, “Design of optimal polarimeters : maximization of the signal-to-noise ratio and minimization of systematic error,” Appl. Opt. **41**, 619–630 (2002). [CrossRef] [PubMed]

18. F. Goudail, “Noise minimization and equalization for stokes polarimeters in the presence of signal-dependent poisson shot noise,” Opt. Lett. **34**, 647–649 (2009). [CrossRef] [PubMed]

19. S. Ainouz, O. Morel, and F. Meriaudeau, “Geometric-based segmentation of polarization-encoded images,” in “IEEE International Conference on Signal Image Technology and Internet Based System,” (2008). [CrossRef]

20. J. Ahmad and Y. Takakura, “Improving segmentation maps using polarization imaging,” in “IEEE International Conference on Image Processing,” (2007). [CrossRef]

*t*,

_{i}*i*= {1, 2, 3} and

*b*in the following and are represented in Fig. 2(a). The intensity image of the scene is presented in Fig. 2(b). We can see that the four regions cannot be discriminated on it, since they have similar intensity reflectances. The Mueller matrices and covariances matrices are estimated from the database and these estimates are used to design a ML classifier (see Eq. (5)). The classification results are presented in Fig. 3(b). We can see that the different objects are globally well discriminated, although some errors are present.

*t*

_{0}available to perform the acquisition is constant. Consequently, the larger the number of projections, the smaller the integration time for each measurement. For example, for the Mueller matrix acquisition, we have to acquire 16 projections thus the acquisition time is

*t*

_{0}/16 for each image. In Fig. 3(a), it is seen that some Mueller images contain no information useful for discrimination. The time used for acquiring them would thus have been better spent on other more informative projections. To reduce the number of images that need to be acquired, one solution is to estimate only the coefficients of the Mueller matrix containing relevant information to discriminate the regions. This selection of specific coefficients has been the subject of recent works [9

9. J. S. Tyo, Z. Wang, S. J. Johnson, and B. G. Hoover, “Design and optimization of partial mueller matrix polarimeters,” Appl. Opt. **49**, 2326–2333 (2010). [CrossRef] [PubMed]

## 3. Discrimination using optimal projections

21. J. Zallat, S. Ainouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters : impact of image noise and systematic errors.” J. Opt. A **8**, 807–814 (2006). [CrossRef]

**m**is the 16-component vector formed by reading the Mueller matrix

*M*in the lexicographic order: , and

**q**=

**T**⊗

**S**(⊗ denotes the Kronecker product). Using this formulation, each pixel of the scene can be seen as a point in a 16-dimension space defined by each component of the Mueller matrix and, as all the pixels associated with the same object have not exactly the same polarimetric properties, each region is represented by a point cloud in this 16-dimension space. Using a vecteur

**q**, it is then possible to project these points on a direction of the space in which the separability of the regions is maximal. The problem is thus of finding the optimal set of linear projections that maximizes the separability of the different point clouds. This issue has long been solved in pattern recognition: it is the well-known Linear Discriminant Analysis (LDA) [22]. The optimal projection vectors are the

*K*– 1 generalized eigenvectors associated to the interclass and intraclass covariances matrices. In our case, it would lead to

*K*– 1 vectors

**q**, that, after having projected the data, allow maximizing the separability of the different regions. However, this solution is valid when the domain of definition of the projection vectors is the set of real valued 16-component vectors with unit norm. In our problem, the vector

**q**=

**T**⊗

**S**does not span this space (it has 4 degrees of freedom instead of 15). Consequently, the classical LDA technique cannot be used, and we describe in this section an approach to solve this problem. The key point is to define a tractable separability criterion for multi-region discrimination, which is not obvious. We then briefly describe the optimization algorithm that we use to determine the optimal configuration and illustrate this approach on a real-world imaging scenario.

### 3.1. Separability criterion

*k*, and thus the performance of the classifier, depends on the sets of illumination and analysis states 𝒮 = [

**S**

_{1}, ...,

**S**] and 𝒯 = [

_{N}**T**

_{1}, ...,

**T**

*]. Our goal in the following will be to optimize these states in order to maximize the separability of the different classes. For that purpose, one has to define a separability criterion, that is, a function*

_{N}*C*(𝒮, 𝒯) of the illumination and analysis states that quantifies the discrimination performance. The optimal projection parameters will be obtained as: The adequate separability criterion for a multi-class discrimination problem is the Bayesian probability of error, which involves a sum of integrals of the PDF over the decision regions, weighted by the relative importance of the different types of errors (misclassification between pairs of classes) [23]. However, this criterion is difficult to calculate and optimize. This is why we will use a separability criterion which is suboptimal, but easier to compute. Recently, it has been shown that for such real-world discrimination tasks as target detection and object segmentation, the Bhattacharyya distance is a good candidate for separability criterion [24

24. 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]

*B*) is an asymptotic exponent on the probability of error in discrimination problems [25

25. T. M. Cover and J. A. Thomas, *Elements of Information Theory* (John Wiley and Sons, New York, 1991). [CrossRef]

26. A. Jain, P. Moulin, M. I. Miller, and K. Ramchandran, “Information-theoretic bounds on target recognition performance based on degraded image data,” IEEE Trans. Pattern Anal. Mach. Intell. **24**, 1153–1166 (2002). [CrossRef]

*P*(

_{k}*x*) and

*P*(

_{l}*x*). In our case, these pdf correspond to the noise statistics of the pixels in regions

*k*and

*l*. If

*n*denotes the size of the sample, the probability of error in deciding wether the observed data has been generated with

*P*(

_{k}*x*) and

*P*(

_{l}*x*) behaves as exp[−

*nB*] as

*n*tends to infinity [26

26. A. Jain, P. Moulin, M. I. Miller, and K. Ramchandran, “Information-theoretic bounds on target recognition performance based on degraded image data,” IEEE Trans. Pattern Anal. Mach. Intell. **24**, 1153–1166 (2002). [CrossRef]

*P*(

_{k}*x*) and

*P*(

_{l}*x*), the Bhattacharyya distance is defined as: with 𝒟 the definition domain of

*P*and

_{k}*P*. The Bhattacharyya distance is thus a scalar value that quantifies the similarity between the pdf

_{l}*P*(

_{k}*x*) and

*P*(

_{l}*x*). It belongs to the interval [0; +∞[, is equal to zero when the pdf are identical and infinite when the pdf supports do not overlap.

*k*and

*l*has the following expression: We have to note that, if it can be considered that the average values of the projected pixels in each class

*x̄*are sufficiently different, we can neglect the second term associated only to the difference of covariance matrices Γ

_{k}*and Γ*

_{k}*. This simplification is particularly valuable since a large number of iterations have to be done to compute the optimal sets of images. We have checked that in the example of Fig. 4, taking into account the second term does not change the sets of optimal polarization states.*

_{l}*N*-dimensional space. Our goal is now to determine the optimal set of parameters (𝒮, 𝒯) maximizing this criterion.

### 3.2. Computational issue for the optimization

*N*of projections chosen, one has to search for the optimal combination of projections maximizing the separability. The main characteristic of this problem is the number of parameters that have to be optimized simultaneously. Indeed, each projection depends on 4 parameters, e.g., the azimuth and ellipticity of the illumination and analysis states. Searching for

*N*projections thus involves optimization on 4

*N*parameters, which is quite large even for low values of

*N*and it is thus likely that the separability criterion will have local maxima. It is thus necessary to use an algorithm robust to the presence of local maxima. After having compared different solutions, we have chosen to use the Shuffled Complex Evolution (SCE-UA) Method [27

27. Q. Y. Duan, V. K. Gupta, and S. Sorooshian, “A shuffled complex evolution approach for effective and efficient global minimization,” J. Optim. Theory Appl. **76**, 501–521 (1993). [CrossRef]

### 3.3. Application to a real-world imaging example

*t*

_{0}. If only one projection is acquired, the integration time is thus

*t*

_{0}. It can been seen in Fig. 4(a) that the obtained image is indeed much less noisy than the Mueller images in Fig. 3, since the latter corresponds to an integration time that is 16 times smaller for each projection. However, for this 4-region scenario, one projection - although optimal - is clearly insufficient since many classification errors are observed in Fig. 4(a). To visualize the detection efficiency, we have represented the estimation of the PDF (histogram) associated to the four classes of object in Fig. 5. We can observe that the average values of the four classes are well separated (which explains why we can globally discriminate the objects in the scene), but there is also a large overlap between the different PDF that leads to a high number of classification errors as we can see in Fig. 4(a). Consequently, the contrast criterion (𝒞 = 2.0) is low compared to that obtained using the full Mueller matrix (𝒞 = 6.6). It is thus necessary to increase the number of projections to enhance the classification performance.

*t*

_{0}/2. We observe that the optimal projections are different from that obtained by trying to optimize the separability on only one projection (Fig. 4a). Indeed, in the image 4.b, the three regions are separated from the background but are not discriminated between themselves. This discrimination is done thanks to the second image (Fig. 4.c). To visualize the detection efficiency, we have plotted, in Fig. 6, the pixel value distributions of the different objects in the 2-dimensional space defined by the two optimal projections. We can see that the different point clouds are well separated that leads to good discrimination performance, as we can see in Fig. 4(b). Indeed, the obtained value of the separability criterion (𝒞 = 9.8) is higher than that obtained with full Mueller matrix data. We can now ask the question: is it possible to increase further the contrast using a third projection ?

*t*

_{0}/3. These images are different from all the previously obtained images and correspond to a different way to discriminate the objects. Indeed the first image discriminates the regions

*t*

_{1}and

*t*

_{3}from the background

*b*and the region

*t*

_{2}. The second image separates the region

*t*

_{1}from the region

*t*

_{3}and finally the third image isolates all the regions from the background. This set of images leads to a high value of the separability criterion (𝒞 = 13.1) that corresponds to a good separability of the classes in the 3-dimensional space (see Fig. 7) and thus to excellent discrimination results, as we can see in Fig. 4(c). Theses results are better than those obtained with the full Mueller matrix because we acquire only images that contain information relevant to classification: by decreasing the number of images, we increase the integration time associated with each image and thus increase the signal to noise ratio.

*t*

_{0}∼ 80

*ms*. We can see in Fig. 8 that the contrast begins by increasing and reaches its maximum for 3 images. This evolution can be explained by the fact that each extra image brings enough new information to compensate the decrease of the signal to noise ratio per image due to the reduction of the integration time. With 3 images, all the information useful for discrimination is gathered.

*K*classes have to be discriminated,

*K*– 1 linear projection are sufficient to obtain optimal discrimination [22]. If the number of acquired images is increased while keeping the total acquisition time constant, the information brought by these new images is no longer sufficient to compensate for the reduction of the signal to noise ratio and the contrast decreases. We can also notice that with 16 optimal projections, we obtain the same contrast as with the raw Mueller matrix data (𝒞 = 6.6).

## 4. Conclusion

## References and links

1. | J. E. Solomon, “Polarization imaging,” Appl. Opt. |

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

3. | S. Breugnot and P. Clémenceau, “Modeling and performances of a polarization active imager at |

4. | S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. |

5. | Y. Y. Schechner, S. G. Narasimhan, and S. K. Nayar, “Polarization-based vision through haze,” Appl. Opt. |

6. | F. Boulvert, B. Boulbry, G. Le Brun, B. Le Jeune, S. Rivet, and J. Cariou, “Analysis of the depolarizing properties of irradiated pig skin,” J. Opt. A Pure Appl. Opt. |

7. | J. M. Bueno, J. Hunter, C. Cookson, M. Kisilak, and M. Campbell, “Improved scanning laser fundus imaging using polarimetry,” J. Opt. Soc. Am. A |

8. | A. Pierangelo, B. Abdelali, M.-R. Antonelli, T. Novikova, P. Validire, B. Gayet, and A. De Martino, “Ex-vivo characterization of human colon cancer by mueller polarimetric imaging,” Opt. Express |

9. | J. S. Tyo, Z. Wang, S. J. Johnson, and B. G. Hoover, “Design and optimization of partial mueller matrix polarimeters,” Appl. Opt. |

10. | F. Goudail, “Comparison of the maximal achievable contrast in scalar, stokes and mueller images,” Opt. Lett. |

11. | M. Dubreuil, S. Rivet, B. Le Jeune, and J. Cariou, “Snapshot mueller matrix polarimeter by wavelength polarization coding,” Opt. Express |

12. | A. A. Swartz, H. A. Yueh, J. A. Kong, L. M. Novak, and R. T. Shin, “Optimal polarizations for achieving maximal constrast in radar images,” J. Geophys. Res. |

13. | J. Yang, “Numerical methods for solving the optimal problem of contrast enhancement,” IEEE transactions on geoscience and remote sensing |

14. | M. Floc’h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, “Polarimetric considerations to optimize lidar detection of immersed targets,” Pure Appl. Opt. |

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

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

17. | J. S. Tyo, “Design of optimal polarimeters : maximization of the signal-to-noise ratio and minimization of systematic error,” Appl. Opt. |

18. | F. Goudail, “Noise minimization and equalization for stokes polarimeters in the presence of signal-dependent poisson shot noise,” Opt. Lett. |

19. | S. Ainouz, O. Morel, and F. Meriaudeau, “Geometric-based segmentation of polarization-encoded images,” in “IEEE International Conference on Signal Image Technology and Internet Based System,” (2008). [CrossRef] |

20. | J. Ahmad and Y. Takakura, “Improving segmentation maps using polarization imaging,” in “IEEE International Conference on Image Processing,” (2007). [CrossRef] |

21. | J. Zallat, S. Ainouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters : impact of image noise and systematic errors.” J. Opt. A |

22. | K. Fukunaga, |

23. | H. L. Van Trees, |

24. | 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 |

25. | T. M. Cover and J. A. Thomas, |

26. | A. Jain, P. Moulin, M. I. Miller, and K. Ramchandran, “Information-theoretic bounds on target recognition performance based on degraded image data,” IEEE Trans. Pattern Anal. Mach. Intell. |

27. | Q. Y. Duan, V. K. Gupta, and S. Sorooshian, “A shuffled complex evolution approach for effective and efficient global minimization,” J. Optim. Theory Appl. |

**OCIS Codes**

(100.0100) Image processing : Image processing

(110.5405) Imaging systems : Polarimetric imaging

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: October 24, 2011

Revised Manuscript: November 17, 2011

Manuscript Accepted: November 17, 2011

Published: November 28, 2011

**Citation**

Guillaume Anna, François Goudail, and Daniel Dolfi, "Optimal discrimination of multiple regions with an active polarimetric imager," Opt. Express **19**, 25367-25378 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-25-25367

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### References

- J. E. Solomon, “Polarization imaging,” Appl. Opt.20, 1537–1544 (1981). [CrossRef] [PubMed]
- J. S. Tyo, M. P. Rowe, E. N. Pugh, and N. Engheta, “Target detection in optical scattering media by polarization-difference imaging,” Appl. Opt.35, 1855–1870 (1996). [CrossRef] [PubMed]
- S. Breugnot and P. Clémenceau, “Modeling and performances of a polarization active imager at λ = 806 nm,” Opt. Eng.39, 2681–2688 (2000). [CrossRef]
- S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt.7, 329–340 (2002). [CrossRef] [PubMed]
- Y. Y. Schechner, S. G. Narasimhan, and S. K. Nayar, “Polarization-based vision through haze,” Appl. Opt.42, 511–525 (2003). [CrossRef] [PubMed]
- F. Boulvert, B. Boulbry, G. Le Brun, B. Le Jeune, S. Rivet, and J. Cariou, “Analysis of the depolarizing properties of irradiated pig skin,” J. Opt. A Pure Appl. Opt.7, 21–28 (2005). [CrossRef]
- J. M. Bueno, J. Hunter, C. Cookson, M. Kisilak, and M. Campbell, “Improved scanning laser fundus imaging using polarimetry,” J. Opt. Soc. Am. A24, 1337–1348 (2007). [CrossRef]
- A. Pierangelo, B. Abdelali, M.-R. Antonelli, T. Novikova, P. Validire, B. Gayet, and A. De Martino, “Ex-vivo characterization of human colon cancer by mueller polarimetric imaging,” Opt. Express19, 1582–1593 (2011). [CrossRef] [PubMed]
- J. S. Tyo, Z. Wang, S. J. Johnson, and B. G. Hoover, “Design and optimization of partial mueller matrix polarimeters,” Appl. Opt.49, 2326–2333 (2010). [CrossRef] [PubMed]
- F. Goudail, “Comparison of the maximal achievable contrast in scalar, stokes and mueller images,” Opt. Lett.35, 2600–2602 (2010). [CrossRef] [PubMed]
- M. Dubreuil, S. Rivet, B. Le Jeune, and J. Cariou, “Snapshot mueller matrix polarimeter by wavelength polarization coding,” Opt. Express15, 13660–13668 (2007). [CrossRef] [PubMed]
- A. A. Swartz, H. A. Yueh, J. A. Kong, L. M. Novak, and R. T. Shin, “Optimal polarizations for achieving maximal constrast in radar images,” J. Geophys. Res.93, 15252–15260 (1988). [CrossRef]
- J. Yang and , “Numerical methods for solving the optimal problem of contrast enhancement,” IEEE transactions on geoscience and remote sensing38, 965–971 (2000). [CrossRef]
- M. Floc’h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, “Polarimetric considerations to optimize lidar detection of immersed targets,” Pure Appl. Opt.7, 1327–1340 (1998). [CrossRef]
- F. Goudail, “Optimization of the contrast in active stokes images,” Opt. Lett.34, 121–123 (2009). [CrossRef] [PubMed]
- F. Goudail and A. Bénière, “Optimization of the contrast in polarimetric scalar images,” Opt. Lett.34, 1471–1473 (2009). [CrossRef] [PubMed]
- J. S. Tyo, “Design of optimal polarimeters : maximization of the signal-to-noise ratio and minimization of systematic error,” Appl. Opt.41, 619–630 (2002). [CrossRef] [PubMed]
- F. Goudail, “Noise minimization and equalization for stokes polarimeters in the presence of signal-dependent poisson shot noise,” Opt. Lett.34, 647–649 (2009). [CrossRef] [PubMed]
- S. Ainouz, O. Morel, and F. Meriaudeau, “Geometric-based segmentation of polarization-encoded images,” in “IEEE International Conference on Signal Image Technology and Internet Based System,” (2008). [CrossRef]
- J. Ahmad and Y. Takakura, “Improving segmentation maps using polarization imaging,” in “IEEE International Conference on Image Processing,” (2007). [CrossRef]
- J. Zallat, S. Ainouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters : impact of image noise and systematic errors.” J. Opt. A8, 807–814 (2006). [CrossRef]
- K. Fukunaga, Introduction to statistical pattern recognition (Academic Press, San Diego, 1990).
- H. L. Van Trees, Detection, Estimation and Modulation Theory (John Wiley and Sons, Inc., New York, 1968).
- F. Goudail, P. Réfrégier, and G. Delyon, “Bhattacharyya distance as a contrast parameter for statistical processing of noisy optical images,” J. Opt. Soc. Am. A21, 1231–1240 (2004). [CrossRef]
- T. M. Cover and J. A. Thomas, Elements of Information Theory (John Wiley and Sons, New York, 1991). [CrossRef]
- A. Jain, P. Moulin, M. I. Miller, and K. Ramchandran, “Information-theoretic bounds on target recognition performance based on degraded image data,” IEEE Trans. Pattern Anal. Mach. Intell.24, 1153–1166 (2002). [CrossRef]
- Q. Y. Duan, V. K. Gupta, and S. Sorooshian, “A shuffled complex evolution approach for effective and efficient global minimization,” J. Optim. Theory Appl.76, 501–521 (1993). [CrossRef]

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