## Fundamental estimation bounds for polarimetric imagery

Optics Express, Vol. 16, Issue 16, pp. 12018-12036 (2008)

http://dx.doi.org/10.1364/OE.16.012018

Acrobat PDF (313 KB)

### Abstract

Precise channel-to-channel registration is a prerequisite for effective exploitation of passive polarimetric imagery. In this paper, the Cramer-Rao bound is employed to determine the limits of registration precision in the presence of scene polarization diversity, channel noise, and random translational registration errors between channels. The effects of misregistration on Stokes image estimation are also explored in depth. Algorithm bias is discussed in the context of the bound, without being estimator specific. Finally, case studies are presented for polarization insensitive imagery (a special case) and linear polarization imaging systems with three and four channels. An optimum polarization channel arrangement is proposed in the context of the bound.

© 2008 Optical Society of America

## 1. Introduction

3. S. Lin, K. Yemelyanov, E. Pugh Jr, and N. Engheta, “Separation and contrast enhancement of overlapping cast
shadow components using polarization,” Opt. Express **14**, 7099–7108 (2006). [CrossRef] [PubMed]

11. B. Zitova and J. Flusser, “Image registration methods: a survey,” Image and Vision Computing **21**, 977–1000 (2003). [CrossRef]

### 1.1. Stokes vector formulation

*ϕ*, between the

*x*and

*y*electric field components of the incident signal. The second element, a polarizer, transmits the portion of this field along angle

*θ*(measured with respect to the x axis) and completely attenuates the field everywhere else. The intensity measured at the detector is given by:

*S*

_{0}is the total incident intensity,

*S*

_{1}and

*S*

_{2}together describe the inclination angle of the polarization ellipse, and

*S*

_{3}and

*S*

_{0}together describe its ellipticity. The Stokes formalism is also capable of describing partially polarized, broadband radiation whereas a field based representation (e.g. the Jones formalism) is not. As stated previously, system responsiveness to the ellipticity of the incident field is not considered in this research. Consequently, the phase delay parameter

*ϕ*will be assumed to be zero for each imaging channel. In other words,

*I*is not a function of

*S*

_{3}and all measurable polarization effects will be due to linear polarization.

*ϕ*= 0, one must still contend with polarizing optical elements that are less than ideal. That is to say, the preferred signal may be partially attenuated while, at the same time, the remaining signal is not completely suppressed. Consequently, the intensity equation is better represented with weights on the Stokes parameters that are determined via lab calibration.

*x*refers to the specific channel in question. For an ensemble of channels, it is useful to define vectors for intensity measurements,

**I**, and Stokes parameters,

**S**, such that

**I**=

*A*

**S**and matrix

*A*consists of the channel Stokes weighting parameters:

### 1.2. A quick example

*°*, 60

*°*, or −60

*°*. The target polarization is rotated slightly (≈ 10

*°*) out of the sensor reference frame so that each bar appears (to a greater or lesser extent) in each channel. False color and an arrow are used to accentuate the weak signal of the top bar in the 0

*°*channel.

*°*has been misregistered to the top bars in the 60

*°*and −60° channels. The Stokes parameter images of this scene (figure 2) provide an explanation of this behavior.

*S*

_{1}and

*S*

_{2}can take on negative values; the reader should interpret the dark regions in the

*S*

_{1}and

*S*

_{2}images in this way. This behavior is unlike

*S*

_{0}, which is strictly positive. The comparatively higher contrast in the

*S*

_{1}image occurs because the signal in

*S*

_{2}is much weaker than in

*S*

_{1}. If the sensor were not sensitive to polarization then the captured image in each channel would be the

*S*

_{0}image.

*°*channel image is bright because its preferred polarization state is nearly parallel to the preferred polarization state of the channel. In this region

*I*

*S*

_{1}is a positive quantity. The small contribution from

*S*

_{2}in this channel is ignored. The top bar is orthogonally polarized to channel 0

*°*and in this region the form of the intensity equation is the same but

*S*

_{1}is

*negative*. A similar analysis could be conducted for the remaining channels but the point of this section has already been made: the phenomenology of polarization imagery is different than that of traditional intensity imagery and as such, the rules developed for image registration must be reevaluated in this new context.

## 2. Bound definition and data model

### 2.1. Definition of the Cramer-Rao bound

**Z**be a vector of random variables that is parameterized by vector

*θ*. Define

*θ*̂ to be any unbiased estimate of these parameters and

**z**to be one realization of

**Z**. The Cramer-Rao in-equality provides the lower bound on this estimator’s error covariance matrix in terms of the Fisher Information Matrix (FIM),

*J*:

*L*(

*θ*,

**z**) is the data log-likelihood function [12]. Note that

*E*[…] represents the expected value operation over

**Z**. The minimum variance for the estimate of each parameter in

*θ*is given by the diagonal elements of

*J*

^{−1}. When

*J*is not positive definite the Cramer-Rao bound is not defined.

*θ*changes. Hence, calculation of the likelihood function requires knowledge of the probability density function for

**Z**,

*p*

_{θ}(

**z**), at each measured

**z**. The log-likelihood function is simply the natural log of the likelihood function:

### 2.2. Data model

**Z**, parameter vector,

*θ*, and their corresponding log-likelihood function,

*L*(

*θ*,

**z**). The purpose of this section is to define these items for the specific problem of generating polarimetric imagery from noisy, misregistered data.

**Z**. Consistent with [9

9. D. Robinson and P. Milanfar, “Fundamental performance limits in image registration.” IEEE Trans. Image Process **13**, 1185–1199 (2004). [CrossRef] [PubMed]

10. A. Yetik and I.S. Nehorai, “Performance bounds on image registration,” IEEE Trans. Signal Process **54**, 1737–1749 (May 2006). [CrossRef]

*f*

_{1}, are taken to be the reference by which the remaining channels,

*f*

_{1}to

*f*, are specified. Mathematically, the collected image

_{N}*z*for channel

_{i}*i*is given by:

**v**

*is the 2-dimensional translational misregistration between*

_{i}**f**

*and*

_{i}**f**

_{1}.

*σ*

_{2}is the noise variance and ξ is a constant term that is not dependent on

*θ*. It is clear from this equation that the log-likelihood is dependent on the region of overlap defined by the intersection of all images

**f**

*. This region of intersection is, in turn, dependent on the relative misregistration between the images. Following the lead of Yetik, this overlap region is assumed to be constant. The efficacy of this assumption is greatest when the relative misregistration between images is small when compared to the dimensions of the overlap region; it is reasonable to assume that a multi-channel polarimeter will be operating in this regime.*

_{i}*θ*. Since the goal is to place a bound on a joint estimator of the translational shifts between images and the values of the Stokes parameters at each pixel in the image, this parameter vector will be very large indeed:

**f**

*, each Stokes parameter vector is given by:*

_{i}*R*, and mean,

*f*:

*m*= 1) for each of

*N*random images. Each image

*i*has a covariance matrix

*R*=

*σ*

^{2}

*I*and mean

**f**

*. It is straightforward to show that the FIM in this case is simply a sum of*

_{i}*N*instances of (13):

## 3. Fisher information for the joint estimator

*J*into submatrices. Matrix partitioning is used to exploit the inherent symmetry in the Fisher information matrix and to lend insight into the physical interpretation of

*J*. In [13

13. L. L. Scharf and L. T. McWhorter, “Geometry of the Cramer-Rao bound” Signal Processing **31**, 301–311 (1993). [CrossRef]

*V*describes correlations amongst the translational registration parameters and the sub-FIM

*S*describes correlations between Stokes parameters. Define:

*H*, to use Scharf and McWhorter’s parlance, relates the intercorrelations between the registration and Stokes partitions in

*θ*. In what follows, each of

*V*,

*H*, and

*S*are described in detail.

*V*, is actually the FIM for an unbiased estimator of the misregistration between channels when the underlying Stokes images are known a priori. The inverse of

*V*is the Cramer-Rao bound for the registration estimator under this “known prior” condition. In the two channel polarization insensitive case,

*V*is the bound from Robinson and Milanfar.

*V*is composed of (

*N*−1)

^{2}submatrices of the form:

**0**

_{2×2}is a 2×2 zero matrix. Though it does not appear explicitly in (15), it is also useful to define

*Ṽ*with the same form as

*V*but with entries:

*p*

^{2}×3

*p*

^{2}matrix

*S*. Note that

*S*is the FIM that would be used to estimate the bound on a unbiased Stokes estimator if the relative shifts between the collected images were known a priori.

*S*divides into 3×3 submatrices corresponding to combinations of Stokes parameters (images).

*I*

_{p2×p2}is an identity matrix of rank

*p*

^{2}. The block symmetry in

*S*allows for substantial further simplification by defining the matrix:

*S*matrix.

*p*

^{2}×3

*p*

^{2}matrix

*S*is solved by simply inverting the 3×3 matrix

*C*.

*V*and

*S*is the 3

*p*

^{2}×2(

*N*−1) matrix

*H*. Physically, if

*H*were the zero matrix then the bounds on the registration and Stokes parameters could be determined independently of each other. Proof of this statement is provided in section (4). In the process of defining

*H*, it becomes obvious that this independence condition is never met.

*H*is composed of 3×(

*N*−1) readily identifiable submatrices:

*H*is non-zero and the covariance bounds must be determined jointly. More on the relationship between

*H*,

*V*and

*Ṽ*can be found in appendix A.

## 4. Bound derivation

*N*−1)+3

*p*

^{2}]×[2(

*N*−1)+3

*p*

^{2}] entries. To put the enormity of this matrix into perspective, consider the bound calculation for a four channel polarimeter used to estimate the first three Stokes parameters. Assuming a 512×512 overlap region, the corresponding FIM has approximately 6.18×10

^{11}entries. Inversion of such a large matrix is prohibitive. In this section, the partitioning of the Fisher information matrix from the previous section is exploited to make this inversion problem tractable.

### 4.1. Block matrix inversion

*θ*is given by the diagonal entries of

*J*

^{−1}. Consequently, computational expense can be significantly reduced by avoiding the unnecessary calculation of many of the off diagonal terms in the inverse. A trivial rearrangement of the partitioned inverse of a block matrix in [18] provides the following:

*B*and

_{ν}*B*, which are submatrices of J

_{S}^{−1}, identify the bounding covariance matrices on the shift parameters and Stokes images. As an aside, if

*H*=

**0**then

*B*=

_{ν}*V*

^{−1}and

*B*=

_{S}*S*

^{−1}, thus demonstrating the physical interpretation of

*H*put forth in the previous section.

*H*=

**0**case, if the underlying Stokes images are known perfectly then the Cramer-Rao bound on the misregistration estimates would simply be

*B*=

_{ν}*V*

^{−1}. Likewise, if perfect knowledge of the registration parameters existed then

*B*=

_{S}*S*

^{−1}. Consequently, it is clear that

*B*and

_{ν}*B*are always larger than

_{S}*V*

^{−1}and

*S*

^{−1}in the absence of perfect knowledge.

### 4.2. Simplified registration parameter bound

*B*is addressed first because it is required to calculate

_{ν}*B*. As preliminary work, note that:

_{S}*H*

^{T}*S*

^{−1}

*H*can itself be partitioned into a matrix,

*D*, such that:

*V*into the definition of

*D*such that:

*M*is formed of the 2 to

*N*rows of

*A*, the matrix of per channel Stokes weighting parameters from (3). The bound on the shift estimates can now be expressed concisely:

*Ṽ*has indeed provided the constituents for the cross-covariance terms in

*B*.

_{ν}### 4.3. Simplified Stokes parameter bound

*B*. We propose that the variance bound on the Stokes parameter estimate for any one pixel in the image is of less interest than the average bound across the image. In turn, this calculation is significantly simplified by applying properties of the trace and of the Kronecker product as shown in appendix B, where the bound is derived in detail. What follows are the highlights of this derivation.

_{S}*B*, the covariance matrix for an estimator of Stokes image

_{Si}**S**

*, to be a submatrix of*

_{i}*B*. Equivalently, let Γ

_{S}*be the submatrix of*

_{i}*S*corresponding to the Stokes image

*S*

*:*

_{i}**S**

*is defined to be:*

_{i}*V*and

*Ṽ*:

*B*⟩ and

_{Si}*B*are clear. The terms in

_{ν}*B*that are not contained in

_{S}*B*can be ignored because they do not influence the trace. Also, it is interesting to note how the sensor itself (realized by

_{Si}*W*and

_{Si}*W*in

_{ν}*B*) plays opposing roles in equation (38) analogous to multiplication and division if this were a purely scalar case.

_{ν}## 5. Biased estimators

*J*, is the Fisher information matrix (identical to the unbiased case),

*is the estimate of*θ ^

*θ*and

*is estimator specific (i.e. registration algorithm specific) and, furthermore, the CRLB of a biased estimator may be higher or lower than that of an unbiased estimator. Note that in the unbiased case, Δ =*θ ^

**I**and equation (39) reduces to equation (4). In what follows, a specific partition of Δ is defined via subscript. For instance,Δ

*refers to a partition of Δ dealing strictly with estimates of the registration parameters whereas Δ*

_{ν}_{S0}refers to estimates of the Stokes image

*S*

_{0}.

*B̃*, is easily achieved by combining (30) with (39):

_{ν}*J*

^{−1}in (39) with its submatrix

*B*(from the unbiased case). In what appears to be a very small step, this equation shows that the CRLB can be decomposed into an scene specific part,

_{ν}*V*+

*Ṽ*, a sensor specific part

*W*, and an estimator specific part, Δ

_{ν}_{ν}. This separation may not be complete in that the channel spacing effects

*V*+

*Ṽ*and, more than likely, bias in the estimator will be to some extent affected by scene polarimetric content.

^{T}

_{Si}**Φ**

^{−}

*in for*

_{i}**Φ**

^{−1}

*in appendix B and noting that:*

_{i}9. D. Robinson and P. Milanfar, “Fundamental performance limits in image registration.” IEEE Trans. Image Process **13**, 1185–1199 (2004). [CrossRef] [PubMed]

## 6. Example bound calculations

### 6.1. Bounds on polarization insensitive imagery

*N*noisy samples of this image. Recall that

*S*

_{0}represents total scene intensity, therefore,

*S*

_{0}is the only Stokes image of interest. In this scenario, the matrix in equation (19) reduces to a scalar,

*C*=

*N*, because there is only one parameter per pixel to be estimated. The channel transmission coefficient,

*a*

_{i0}, is assumed to be unity for each channel. In that case,

*M*= 1

_{(N−1)×1}. Therefore, each of

*Wν*and

*W*

_{S0}are the 2(

*N*−1)×2(

*N*−1) matrices:

*V*and

*Ṽ*are image dependent. That being said, the overall bound’s behavior with increasing

*N*can be predicted by the behavior of

*W*and

_{ν}*W*

_{S0}for any image. Equation (45) shows that

*B*= 2

_{ν}*V*

^{−1}in the two channel case and

*B*=

_{ν}*V*

^{−1}in the limit of

*N*. Essentially, the

*Ṽ*matrix is suppressed by the 1/

*N*terms in

*W*as

_{ν}*N*increases. These suppressed terms represent the decreasing influence of each individual image as the true underlying image emerges. In this limit, the bound parameters for each image asymptotically achieve the bound predicted by Robinson and Milanfar. Assuming a few images do not differ substantially from the rest of the ensemble (e.g. due to a large translational error or parallax) then the results follow a

_{S0}⟩. Again, we consider the endpoints. In the two channel case:

*N*case:

_{S0}⟩, and the average misregistration bound i.e.

*tr*(

*B*)/(

_{ν}*N*−1)) are normalized and plotted against the number of frames,

*N*. Normalization is carried out to illustrate the 1/

*N*behavior of the intensity variance bound and the

*N*is confirmed for these disparate examples. The deviation of average registration parameter bound in the G.G. Stokes image from the expected trend demonstrates the slight influence of changes in image overlap area on the results.

### 6.2. The four channel, three Stokes case

16. J. Tyo, “Optimum linear combination strategy for an N-channel polarization-sensitive imaging or vision system,” J. Opt. Soc. Am. A **15**, 359–366 (1998). [CrossRef]

*°*. Since his work was brought about from different assumptions and desired outcomes, it is of interest to compare this result to the optimum configuration as defined by the Cramer-Rao bound.

*°*corresponds to an orientation of 5

*°*, 10

*°*, and 15

*°*for the second, third, and fourth channels with respect to the first. Figure (6) shows the average bound on the Stokes parameters for each of the three test cases. Each plot is normalized by

*σ*

^{−2}since it may be divided out of equation (38).

*C*

^{−1}

*with increasing channel spacing. This result is significant for two reasons. First, there appears to be global agreement as to which channel spacing is most preferable, at least for the test cases sampled here. Second, it would appear that each*

_{ii}*B*is very well approximated by direct interrogation of

_{Si}*S*

_{−1}, at least in an average sense. In other words, the Stokes parameter bounds are effectively scene independent with respect to channel spacing. Recall that

*S*

^{−1}is, by itself, the Cramer-Rao bound on the Stokes estimates when perfect knowledge of the misregistration parameters is available, consequently, these observations apply equally to that scenario.

*S*

_{1}and

*S*

_{2}estimates meet at Tyo’s predicted optimal channel spacing of 45

*°*, however,

*S*

_{1}has a minimum bound at a somewhat closer channel spacing. Consequently, there is no global minimum bound for all Stokes parameters. Rather, the 45

*°*spacing is the point where there is no preferred parameter. This point is significant because, as stated previously, the Stokes parameters are defined with respect to some coordinate system and, as this system changes in relation to the target (e.g. through camera motion) then scene content can shift between

*S*

_{1}and

*S*

_{2}. With this qualifier, a 45

*°*channel spacing can be said to be optimal for a four channel system.

*σ*

^{2}has been normalized out. Unlike the bounds on the Stokes parameters, these bounds depend both on scene content and channel orientation. Consistent with the Robinson and Milanfar analysis of

*V*in the polarization insensitive two channel case, the difference in bound magnitude correlates with the amount of high spatial frequency content in the test images. The PC board image, with its multi-faceted geometric features, generates the lowest bound while the galaxy Markarian 3, which has largely diffuse features, generates the highest bound.

### 6.3. The three channel, three Stokes case

*S*

_{0},

*S*

_{1},

*S*

_{2}). The interested reader may easily verify this three channel prerequisite for themselves by attempting to calculate

*B*with only two channels. Three channel polarimeters are also common in practice; all examples in the previous section were originally collected by different three channel systems. In this section, the three channel polarimeter is examined from a joint estimation perspective.

_{S}*B*for some combination of three polarimeter channels represented by weight matrix

_{ν}*A*. Immediately, a mathematical difficulty arises:

*S*

^{−1}will dominate equation (24), analogous to the four channel joint estimation examples. Following the prescribed method in the previous section, the diagonal terms in

*J*

^{−1}

*will be jointly minimized for*

_{S}*S*

_{1}and

*S*

_{1}when the angular separation between channels is 60

*°*. In this sense, the Cramer-Rao bound for the Stokes estimates can be compared across the three and four channel cases. Note from the table below that

*S*

_{0}follows the

*S*

_{1}and

*S*

_{2}go by

## 7. Conclusion

*N*polarization insensitive images. The bound itself is a useful evaluation metric because it incorporates both sensor and estimation algorithm effects and can be used to describe these effects theoretically in a way that pure simulation can not. In addition, the following general conclusions can be drawn from the results:

*°*and 45

*°*respectively in the three and four channel cases. Optimum, in this context, refers to a joint minimum bound for

*S*

_{1}and

*S*

_{2}. These arrangements do not guarantee that the bound on the registration estimator is minimized.

## A. Appendix: More on the *V* and *Ṽ* matrices

*V*and

*Ṽ*play a central role in the bound calculation. In addition, the

*V*matrix provides the connection between the previous work by Robinson and Milanfar [9

9. D. Robinson and P. Milanfar, “Fundamental performance limits in image registration.” IEEE Trans. Image Process **13**, 1185–1199 (2004). [CrossRef] [PubMed]

*V*,

*Ṽ*, and

*H*:

_{ij}*i*≠

*j*.

**v**

*is established in the plane of the image. Therefore, a common direction vector,*

_{i}*x*, for all relevant derivatives can be defined via the chain rule:

*V*and

*Ṽ*is defined in a common coordinate system.

## B. Appendix: Detailed derivation of the Stokes parameter bound

*A*and

*G*are square matrices and

*C*,

*D*,

*E*, and

*F*are any matrices such that

*CGD*and

*EF*are square matrices. Operator

*tr*represents the trace and, for any matrix

*A*,

*vec*(

*A*) is defined to be an ordered stack of the columns of

*A*. For the Kronecker product (also from [19]):

*AGC*,

*AG*, and

*CD*are defined. Finally, note that, in contrast to regular matrix multiplication:

*H*(Φ

^{T}^{−T}

*Φ*

_{i}^{−1}

*)*

_{i}*H*is identical in form to equation (25) and can therefore be simplified in the same manner:

*W*,

_{Si}*V*, and

*Ṽ*:

## Acknowledgments

## References and links

1. | M. Kishimoto, L. E. Kay, R. Antonucci, T. W. Hurt, R. D. Cohen, and J. H. Krolik, “Ultraviolet Imaging Polarimetry of the Seyfert 2 Galaxy Markarian 3,” apj |

2. | W. G. Egan, “Polarization in remote sensing,” in |

3. | S. Lin, K. Yemelyanov, E. Pugh Jr, and N. Engheta, “Separation and contrast enhancement of overlapping cast
shadow components using polarization,” Opt. Express |

4. | C. M. Persons, D. B. Chenault, M. W. Jones, K. D. Spradley, M. G. Gulley, and C. A. Farlow, “Automated registration of polarimetric imagery using Fourier transform techniques,” in |

5. | X. Wang, S. Yang, J. Ma, and Y. Qiao, “Automated registration of polarimetric image using wavelet transform techniques,” vol. |

6. | D. A. LeMaster, “A Comparison of Template Matching Registration Methods for Polarimetric Imagery,” in |

7. | S. Guyot, M. Anastasiadou, E. Deléchelle, and A. De Martino, “Registration scheme suitable to Mueller matrix imaging for biomedical applications,” Opt. Express |

8. | S. M. Kay, |

9. | D. Robinson and P. Milanfar, “Fundamental performance limits in image registration.” IEEE Trans. Image Process |

10. | A. Yetik and I.S. Nehorai, “Performance bounds on image registration,” IEEE Trans. Signal Process |

11. | B. Zitova and J. Flusser, “Image registration methods: a survey,” Image and Vision Computing |

12. | L. L. Scharf, |

13. | L. L. Scharf and L. T. McWhorter, “Geometry of the Cramer-Rao bound” Signal Processing |

14. | H. Van Trees, |

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

16. | J. Tyo, “Optimum linear combination strategy for an N-channel polarization-sensitive imaging or vision system,” J. Opt. Soc. Am. A |

17. | E. Collett, |

18. | M. Healy, |

19. | A. Graham, |

**OCIS Codes**

(100.2000) Image processing : Digital image processing

(110.3925) Imaging systems : Metrics

(110.5405) Imaging systems : Polarimetric imaging

**ToC Category:**

Image Processing

**History**

Original Manuscript: February 28, 2008

Revised Manuscript: March 28, 2008

Manuscript Accepted: April 12, 2008

Published: July 25, 2008

**Citation**

Daniel A. LeMaster, "Fundamental estimation bounds for polarimetric imagery," Opt. Express **16**, 12018-12036 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-16-12018

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### References

- M. Kishimoto, L. E. Kay, R. Antonucci, T. W. Hurt, R. D. Cohen, and J. H. Krolik, "Ultraviolet Imaging Polarimetry of the Seyfert 2 Galaxy Markarian 3," apj 565, 155-162 (2002).
- W. G. Egan, "Polarization in remote sensing," in Polarization and Remote Sensing, W. G. Egan, ed., Proc. SPIE 1747, 2-48 (1992).
- S. Lin, K. Yemelyanov, E. Pugh, Jr, and N. Engheta, "Separation and contrast enhancement of overlapping cast shadow components using polarization," Opt. Express 14, 7099-7108 (2006). [CrossRef] [PubMed]
- C. M. Persons, D. B. Chenault, M. W. Jones, K. D. Spradley, M. G. Gulley, and C. A. Farlow, "Automated registration of polarimetric imagery using Fourier transform techniques," Proc. SPIE 4819, 107-117 (2002). [CrossRef]
- X. Wang, S. Yang, J. Ma, and Y. Qiao, "Automated registration of polarimetric image using wavelet transform techniques," (SPIE, 2005) Vol. 5832, pp. 695-702
- D. A. LeMaster, "A Comparison of Template Matching Registration Methods for Polarimetric Imagery," in Aerospace Conference, 2008 IEEE, Vol. 1, pp. 1-9 (2008).
- Guyot, S. and Anastasiadou, M. and Del�??echelle, E. and De Martino, A. , "Registration scheme suitable to Mueller matrix imaging for biomedical applications," Opt. Express 15, 7393-7400 (2007). [CrossRef] [PubMed]
- S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice Hall, Englewood Cliffs, New Jersey, 1993).
- D. Robinson and P. Milanfar, "Fundamental performance limits in image registration." IEEE Trans. Image Process 13, 1185-1199 (2004). [CrossRef] [PubMed]
- A. Yetik, and I. S. Nehorai, "Performance bounds on image registration," IEEE Trans. Signal Process 54, 1737-1749 (May 2006). [CrossRef]
- B. Zitova and J. Flusser, "Image registration methods: a survey," Image and Vision Computing 21, 977-1000 (2003). [CrossRef]
- L. L. Scharf, Statistical Signal Processing: detection, estimation, and time series analysis (Addison-Wesley, Reading, Massachusetts, 1991).
- L. L. Scharf and L. T. McWhorter, "Geometry of the Cramer-Rao bound," Signal Processing 31, 301-311 (1993). [CrossRef]
- H. Van Trees, Detection, estimation, and modulation theory. Part 1: detection, estimation, and linear modulation theory (Wiley, New York, 2001). [CrossRef]
- J. Tyo, "Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error," Appl. Opt 41, 619-630 (2002). [CrossRef] [PubMed]
- J. Tyo, "Optimum linear combination strategy for an N-channel polarization-sensitive imaging or vision system," J. Opt. Soc. Am. A 15, 359-366 (1998). [CrossRef]
- E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, Inc., New York, 1992).
- M. Healy, Matrices for Statistics (Oxford University Press, USA, 1986).
- A. Graham, Kronecker Products and Matrix Calculus With Applications. (Wiley, New York, 1982).

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