## Super-resolution for imagery from integrated microgrid polarimeters |

Optics Express, Vol. 19, Issue 14, pp. 12937-12960 (2011)

http://dx.doi.org/10.1364/OE.19.012937

Acrobat PDF (1874 KB)

### Abstract

Imagery from microgrid polarimeters is obtained by using a mosaic of pixel-wise micropolarizers on a focal plane array (FPA). Each distinct polarization image is obtained by subsampling the full FPA image. Thus, the effective pixel pitch for each polarization channel is increased and the sampling frequency is decreased. As a result, aliasing artifacts from such undersampling can corrupt the true polarization content of the scene. Here we present the first multi-channel multi-frame super-resolution (SR) algorithms designed specifically for the problem of image restoration in microgrid polarization imagers. These SR algorithms can be used to address aliasing and other degradations, without sacrificing field of view or compromising optical resolution with an anti-aliasing filter. The new SR methods are designed to exploit correlation between the polarimetric channels. One of the new SR algorithms uses a form of regularized least squares and has an iterative solution. The other is based on the faster adaptive Wiener filter SR method. We demonstrate that the new multi-channel SR algorithms are capable of providing significant enhancement of polarimetric imagery and that they outperform their independent channel counterparts.

© 2011 OSA

## 1. Introduction

1. J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. **45**(22), 5453–5469 (2006). [CrossRef] [PubMed]

2. B. M. Ratliff, C. F. LaCasse, and J. S. Tyo, “Interpolation strategies for reducing IFOV artifacts in microgrid polarimeter imagery,” Opt. Express **17**(11), 9112–9125 (2009). [CrossRef] [PubMed]

3. J. S. Tyo, C. F. LaCasse, and B. M. Ratliff, “Total elimination of sampling errors in polarization imagery obtained with integrated microgrid polarimeters,” Opt. Lett. **34**(20), 3187–3189 (2009). [CrossRef] [PubMed]

## 2. Observation model

*K*. Let the LR observed pixels from frame

*k*be expressed in lexicographical notation as

**g**(

*k*), where

*k*= 1, 2,...,

*K*. Breaking down each observed frame by polarimetric channel we get

**g**

*(*

_{θi}*k*) = [

*g*,

_{θi}_{1}(

*k*),

*g*,

_{θi}_{2}(

*k*),...,

*g*,

_{θi}*(*

_{M}*k*)]

*. Let the complete set of all observed pixels in lexicographical notation be specified by*

^{T}**g**= [

**g**

*(1),*

^{T}**g**

*(2),...,*

^{T}**g**

*(*

^{T}*K*)]

*= [*

^{T}*g*

_{1},

*g*

_{2},...,

*g*]

_{KMP}*.*

^{T}**p**to the observed frame

**g**(

*k*) is presented in Fig. 2. The ideal polarimatric data in

**p**is assumed to be sampled above the Nyquist rate. The motion between frames is parameterized with the vector

**s**(

*k*). Next, the model includes a linear shift invariant blurring using the discrete PSF for the optical system. This is followed by down-sampling by a factor of

*L*. Finally, the down-sampled images are mosaiced into the pattern in Fig. 1 and noise is introduced to obtain

**g**(

*k*). This observation model can be mathematically expressed such that each observed pixel is a weighted sum of the ideal pixels in

**p**plus noise. The particular weights depend on the system PSF, the motion between frames, and the microgrid layout. In particular, the LR pixels can be expressed in terms of ideal HR pixels as follows where

*h*(

_{θ}_{,i,j}*k*) is the contribution of

*p*to

_{θ}_{,j}*g*(

_{θ}_{,i}*k*) and

*n*(

_{θ}_{,i}*k*) is the noise term for

*g*(

_{θ}_{,i}*k*). This model can be express compactly for all frames jointly as where

**H**is a

*KMP*×

*NP*matrix and

**n**is a

*KMP*× 1 vector of noise samples.

8. R. Hardie, “A fast image super-resolution algorithm using an adaptive Wiener filter,” IEEE Trans. Image Process. **16**(12), 2953–2964 (2007). [CrossRef] [PubMed]

8. R. Hardie, “A fast image super-resolution algorithm using an adaptive Wiener filter,” IEEE Trans. Image Process. **16**(12), 2953–2964 (2007). [CrossRef] [PubMed]

8. R. Hardie, “A fast image super-resolution algorithm using an adaptive Wiener filter,” IEEE Trans. Image Process. **16**(12), 2953–2964 (2007). [CrossRef] [PubMed]

*f*(

*x,y,θ*) from

_{i}**g**. Then, restoration is used to estimate the corresponding samples of

*p*(

*x,y,*

*θ*).

_{i}*u*and

*v*are the horizontal and vertical spatial frequencies in cycles per millimeter,

*H*

_{dif}(

*u, v*) is from the diffraction-limited optics,

*H*

_{abr}(

*u, v*) models optical aberrations, and detector integration is included as

*H*

_{det}(

*u, v*). The OTF from diffraction-limited optics with a circular pupil function is given by [10] where

*F*/# of the optics. Since a purely diffraction-limited optical system is perhaps overly idealistic, we also include the following aberration OTF [11] where

*W*is the root mean square wavefront error [11]. The detector component of the OTF is found as the Fourier transform of a mask with the geometry of the active area of an individual detector in the FPA. Finally, the system PSF is the inverse Fourier transform of the overall OTF in Eq. (5).

_{RMS}*λ*= 7.8 – 9.8

*μ*m. We use a wavelength of

*λ*= 9

*μ*m for our PSF model. The system uses

*F*/2 optics and has a full FPA pixel pitch of 25

*μ*m with approximately 100% fill factor rectangular detectors. Note that for this microgrid imager, the pixel pitch for pixels of like polarization is actually 50

*μ*m. Thus, the effective active area for the like-polarization sub-array detectors is modeled as spanning half the 50

*μ*m like-polarization pixel pitch in each dimension. This equates to a 25% fill factor (taking into account the skipping of detectors in the FPA as shown in Fig. 1). For abberation modeling, we use

*W*= 1/14 [11]. Cross sections of the relevant 2-D modulation transfer functions (MTFs) are shown in Fig. 4(a) for this system. The continuous PSF is shown in Fig. 4(b). The discrete impulse invariant PSF, for use in the observation model operating on the HR image, can be found by sampling the continuous PSF with a sampling period of 50

_{RMS}*/L*

*μ*m.

*L*= 4 is a practical choice for SR processing for this sensor. The effective sampling frequency is then 80 cycles/mm, with a Nyquist frequency of 40 cycles/mm as shown in yellow in Fig. 4(a). In addition to aliasing reduction, we also hope to restore the imagery from the blurring effects of the overall MTF.

## 3. RLS SR for microgrid polarimeters

7. R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, “High resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system,” Opt. Eng. **37**(1), 247–260 (1998). [CrossRef]

7. R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, “High resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system,” Opt. Eng. **37**(1), 247–260 (1998). [CrossRef]

*s*

_{1,}

*} and {*

_{j}*s*

_{2,}

*} are known to generally have relatively low energy (i.e., low variance). Thus, applying a term favoring small high-frequency energy in these channels is sensible. The Stokes image {*

_{j}*s*

_{0,}

*} is the total intensity image. This should contain most of the energy. One may choose to use the smoothness constraint on this term minimally by employing a high*

_{j}*σ*

_{s}_{0}. The parameters

*σ*

_{s}_{1}and

*σ*

_{s}_{2}may be smaller, because of the expected lower energy in these images. Finally, because the Stokes terms interrelate the different polarization channels, it allows us to exploit their distinct spatial sampling positions better than if we applied a smoothness constraint on each channel independently. The approach of applying smoothness regularization to the Stokes images in polarimetric data may be viewed as akin to using luminance and chrominance coordinates in color processing [12–14]. It is interesting to note that the RLS estimate can be viewed as a maximum

*a posteriori*(MAP) estimator in the case of independent and identically distributed Gaussian noise [15

15. R. C. Hardie, K. J. Barnard, and E. E. Armstrong, “Joint MAP registration and high resolution image estimation using a sequence of undersampled images,” IEEE Trans. Image Process. **6**(12), 1621–1633 (1997). [CrossRef] [PubMed]

7. R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, “High resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system,” Opt. Eng. **37**(1), 247–260 (1998). [CrossRef]

*K*frames to a common reference. A convenient choice of reference is the most recent observed frame. Note that we need to register the frames to subpixel accuracy. We have found that we get better registration performance by deinterlacing the channels, applying the registration algorithm in [7

**37**(1), 247–260 (1998). [CrossRef]

**p**, denoted

**p̂**

^{0}. To improve this estimate using gradient descent, we need the gradient of Eq. (9) with respect to

**p**, denoted ∇

_{p}*C*(

**p**). This gradient is composed of the partial derivative of the cost function with respect to each of the HR samples in

**p**. This can be expressed as where The partial derivatives making up the gradient can be computed from Eq. (9), yielding where the observation model gradient term is given by and the model error component is The Stokes gradient term is given by Finally, the redundancy gradient term is Given the gradient, the gradient descent updates are computed as follows where

*ɛ*is the step size for iteration

^{n}*n*. The iterations may be performed a predetermined number of times, or may be stopped with a halting criterion, such as ||

**p̂**

*–*

^{n}**p̂**

^{n}^{−1}|| <

*T*, where

*T*is a threshold.

*C*(

**p̂**

*) as a function of*

^{n}*ɛ*, given

^{n}**p̂**

^{n}^{−1}. In particular, we set the derivative of

*C*(

**p̂**

*) with respect to*

^{n}*ɛ*

*equal to zero, and solve for*

^{n}*ɛ*. This calculation results in an optimum step size of the form The individual terms in the optimum step size are given by where and Computing the optimum step size in this fashion tends to be faster than performing a search. A final point to note about the RLS microgrid SR method is that we can easily treat “bad” pixels by simply setting the error,

^{n}*e*(

_{θ}_{,i}*k*) as defined in Eq. (18), to zero for any

*θ*,

*i*, and

*k*for which

*g*(

_{θ}_{,i}*k*) is known to be a bad pixel value. This is convenient as it obviates the need for a separate bad pixel correction algorithm, even when using a single frame. This is how we address bad pixels for the RLS SR results presented in Section 5. Other bad pixel replacement strategies that exploit microgrid redundancy have been proposed in [16

16. B. M. Ratliff, J. S. Tyo, J. K. Boger, W. T. Black, D. L. Bowers, and M. P. Fetrow, “Dead pixel replacement in LWIR microgrid polarimeters,” Opt. Express **15**(12), 7596–7609 (2007). [CrossRef] [PubMed]

## 4. AWF SR for microgrid polarimeters

**16**(12), 2953–2964 (2007). [CrossRef] [PubMed]

**16**(12), 2953–2964 (2007). [CrossRef] [PubMed]

**16**(12), 2953–2964 (2007). [CrossRef] [PubMed]

*L*= 4 and is populated with pixels from a single frame for simplicity. With multiple frames of data with motion between them, the observation window will contain additional sets of pixels from each polarization state, positioned according to the motion. In general, let the observation window span

*W*HR pixel spacings in the horizontal direction and

_{x}*W*HR pixel spacings in the vertical direction. All of the LR pixels of any polarization that lie within the span of this observation window are placed into an observation vector

_{y}**g**

*= [*

_{i}*g*

_{i}_{,1},

*g*

_{i}_{,2},...,

*g*]

_{i,Ji}*, where*

^{T}*i*is the window positional index and

*J*is the number of LR pixels within the window span. The samples within the observation window will be used to estimate the HR pixels within the generally smaller

_{i}*D*×

_{x}*D*sample estimation window at the center of the observation window, as shown in Fig. 5. Thus, the AWF SR method is effectively performing nonuniform interpolation and restoration simultaneously. One of the main differences here, versus the AWF in [8

_{y}**16**(12), 2953–2964 (2007). [CrossRef] [PubMed]

**d̂**

*= [*

_{i}*d̂*

_{i}_{,1},

*d̂*

_{i}_{,2},...,

*d̂*]

_{i,DxDyP}*and*

^{T}**W**

*is a*

_{i}*J*×

_{i}*D*matrix of weights. Each column of

_{x}D_{y}P**W**

*contains weights used to estimate the value of one particular HR pixel at one polarization state inside the estimation window. The observation window moves across the HR grid in a raster scan fashion stepping by increments of*

_{i}*D*and

_{x}*D*in the horizontal and vertical directions, respectively (non-overlapping estimation sub-windows).

_{y}**16**(12), 2953–2964 (2007). [CrossRef] [PubMed]

**d**

*and the observation vector. The goal now is to provide the statistics in*

_{i}**R**

*and*

_{i}**P**

*. As in [8*

_{i}**16**(12), 2953–2964 (2007). [CrossRef] [PubMed]

**f**

*be the noise-free version of*

_{i}**g**

*, such that*

_{i}**g**

*=*

_{i}**f**

*+*

_{i}**n**

*, where*

_{i}**n**

*is the random noise vector. We shall assume that the noise vector is zero-mean with independent and identically distributed elements of variance*

_{i}*r*(

_{dθdϕ}*x,y*). The variables

*θ*and

*ϕ*represent the polarization angles of the two samples involved, and

*x,y*represents the spatial separation. Here we propose the following model where

*ρ*controls the decay of the correlation with separation distance. We have obtained useful results using

_{θ,ϕ}*ρ*=

_{θ,ϕ}*ρ*for all

*θ, ϕ*, and Note that if

*α*= 0, we assume no correlation between samples of different polarizations regardless of their spatial proximity. This gives rise to an algorithm essentially equivalent to deinterlacing the microgrid data and applying independent AWF SR [8

**16**(12), 2953–2964 (2007). [CrossRef] [PubMed]

*α*= 1, this is equivalent to applying a single AWF to the mosaic microgrid data as if it were a standard FPA and each pixel represents the same physical quantity. In that case, spatial separation is the only factor altering the correlation model. This is also not likely the best approach as samples from the same polarization generally do exhibit higher correlation than those of different polarizations. Thus, one would want to select

*α*in the range 0 ≤

*α*≤ 1 based on expected scene content. A relatively high

*α*is appropriate when we have weakly polarized data, like we may see in LWIR. A lower

*α*may be used for more highly polarized data with a high average DoLP.

*r*(

_{dθdϕ}*x,y*), it can be shown that the cross-correlation function between

*p*(

*x,y,*

*θ*) and

*f*(

*x,y,*

*ϕ*), as shown in Fig. 3, can be expressed as The cross-correlation between

*f*(

*x,y,*

*θ*) and

*f*(

*x,y,ϕ*) is given by By registering the LR frames, the spatial locations and corresponding polarizations of all the LR pixels that make up

**g**

*, which are the same for*

_{i}**f**

*, are known. Thus, the horizontal and vertical displacements between these samples can be computed. Evaluating Eq. (38) using the displacements and the corresponding polarizations allows us to fill the correlation matrix*

_{i}**R**

*can be found from Eq. (33), given the noise variance. Similarly, we compute the displacements between the samples in the estimation window and the LR pixels in the observation window. With these displacements and the corresponding polarizations, we evaluate Eq. (37) and complete*

_{i}**P**

*as given in Eq. (34). Finally, the weights are found using Eq. (32).*

_{i}**16**(12), 2953–2964 (2007). [CrossRef] [PubMed]

*L*. If we have bad pixels, we can simply exclude those from the observation vector

**g**

*, so that they do not contribute to the output. However, this requires that a distinct set of weights be computed around any bad pixels, as it changes the spatial distribution of samples in the observation window. If processing speed is critical, a bad pixel replacement algorithm can be employed prior to AWF filtering [16*

_{i}16. B. M. Ratliff, J. S. Tyo, J. K. Boger, W. T. Black, D. L. Bowers, and M. P. Fetrow, “Dead pixel replacement in LWIR microgrid polarimeters,” Opt. Express **15**(12), 7596–7609 (2007). [CrossRef] [PubMed]

*α*and better exploit the spatial sampling diversity provided by the microgrid as shown in Fig. 1. To better describe this method, let

*a*(

*i, j*) represent a sample from the microgrid array, where

*i*= 1,2,...,

*M*is the 2 × 2 super-pixel index and

*j*is the polarization sample index within the super-pixel. Let

*j*= 1,2,3,4 correspond to

*θ*= 0

*°*, 45

*°*, 90

^{0}, 135

*°*, respectively. Local means, denoted

*ā*(

*i, j*), are estimated by deinterlacing the channels and applying a Gaussian low-pass filter with standard deviation of

*σ*pixels. Local standard deviations, denoted

*ã*(

*i, j*), are estimated by subtracting the local mean and then using a Gaussian low-pass filter on the squared difference, and taking the square root of the result. In terms of these variables, the preprocessing step is expressed as After SR processing, the local means and standard deviations are interpolated, shifted appropriately to match the HR grid, and reintroduced to the SR result to restore the estimated channels to their proper dynamic range. Even without SR processing, we have found that a local statistics fusion (LSF) method can be useful as a stand-alone demosaicing approach. In that case, the output is given by Here

*b*(

*i, j,k*) is the output for super-pixel

*i*= 1,2,...,

*M*, position

*j*= 1,2,3,4 within the super-pixel, and with polarization index

*k*= 1,2,3,4. The variables

*ā*(

_{j}*i, k*) and

*ã*(

_{j}*i, k*) are the mean and standard deviation, respectively, for super-pixel

*i*and polarization

*k*repositioned through interpolation to align with spatial position

*j*. Note that the low-pass nature of the statistic images make them more suited to interpolation than the raw data. This method works best when there is high correlation between the polarimetric channels, but perhaps a scale and bias difference.

## 5. Experimental results

### 5.1. LWIR microgrid polarimetric data

**p**

_{0}for

*L*= 4. In particular, the result of using single-frame channel-independent bicubic interpolation is shown in Fig. 7(b). The output using the single-frame frequency-domain method by Tyo

*et al*[3

3. J. S. Tyo, C. F. LaCasse, and B. M. Ratliff, “Total elimination of sampling errors in polarization imagery obtained with integrated microgrid polarimeters,” Opt. Lett. **34**(20), 3187–3189 (2009). [CrossRef] [PubMed]

2. B. M. Ratliff, C. F. LaCasse, and J. S. Tyo, “Interpolation strategies for reducing IFOV artifacts in microgrid polarimeter imagery,” Opt. Express **17**(11), 9112–9125 (2009). [CrossRef] [PubMed]

*σ*= 1 is shown in Fig. 7(d). The microgrid AWF SR method using only a single frame with

*ρ*=

*α*= 0.7 and

*σ*= 1 produces the result shown in Fig. 7(e). The microgrid RLS SR method using only a single frame with

*α*on the microgrid AWF SR method, we show the single frame result using

*α*= 0.0 and

*α*= 1.0 in Figs. 8(a) and 8(b), respectively. Note that for

*α*= 0.0, the polarimetric channels are treated independently [8

**16**(12), 2953–2964 (2007). [CrossRef] [PubMed]

*α*= 1.0, the AWF treats all polarimetric channels as being the same. While this method can sometimes produce a reasonable {

*s*

_{0,}

*} image, the {*

_{j}*s*

_{1,}

*} and {*

_{j}*s*

_{2,}

*} Stokes images are zero or have only low spatial frequency content from the local statistics processing in Eq. (39). Note that cross-hatching artifacts can be seen in Fig. 8(b), especially near the cockpit window at the front of the aircraft. These types of artifacts tend to become significantly more pronounced with*

_{j}*α*= 1.0 when more input frames are used. The single-frame RLS algorithm operating on the polarimetric channels independently [5

5. B. M. Ratliff, J. S. Tyo, W. T. Black, and C. F. LaCasse, “Exploiting motion-based redundancy to enhance microgrid polarimeter imagery,” Proc. SPIE **7461**, 74610K (2009). [CrossRef]

*ρ*=

*α*= 0.7 and

*σ*= 9 is presented in Fig. 8(e). Finally, the 20 frame microgrid RLS output with

**p**are shown in Fig. 10. Since DoLP involves all of the polarimetric channels, these results reveal characteristics of the estimates in all channels. DoLP also tends to expose sampling errors rather dramatically because it is made up of image differences, as can be seen in Eq. (2). The DoLP images for bicubic interpolation, the Tyo method, and the LSF method are shown in Figs. 10(a)–10(c), respectively. The DoLP image for the independent channel RLS method [5

5. B. M. Ratliff, J. S. Tyo, W. T. Black, and C. F. LaCasse, “Exploiting motion-based redundancy to enhance microgrid polarimeter imagery,” Proc. SPIE **7461**, 74610K (2009). [CrossRef]

### 5.2. Visible polarimetric data with simulated microgrid sampling

*μ*m detectors. The camera is fitted with a Computar 5mm F/1.4 lens. Full frames at each of the four polarization angles are acquired of an outdoor scene containing two vehicles. The polarization filter is rotated in 45

*°*increments between acquisition to provide the four full-frame polarization images. We define these data to be the true

**p**. These data are then put through the observation model with

*L*= 4 and a noise variance of 1. We use a set of randomly generated shifts and the PSF in Fig. 4 to create 20 simulated microgrid images. The Stokes intensity image results for these data are shown in Fig. 11. In particular, one of the simulated microgrid images is shown in Fig. 11(a). The corresponding true {

*s*

_{0,}

*} Stokes image is shown in Fig. 11(b). Bicubic interpolation yields the result presented in Fig. 11(c). Note that the bicubic interpolation result looks blurred relative to the true image. Since {*

_{j}*s*

_{0,}

*} is an average of the four polarimetric channels, some of the aliasing artifacts that are more clearly seen in the individual polarimetric channels are reduced here. The output of the Tyo method is shown in Fig. 11(d) and the 20 frame microgrid AWF output with*

_{j}*ρ*=

*α*= 0.7 and

*σ*= 9 is shown in Fig. 11(e). Finally, the 20 frame microgrid RLS output with

*s*

_{0,}

*} from Fig. 11 as well as several of the corresponding estimate images along Row 234 and about Column 67 are shown in Fig. 12. These cross-sections are centered about a dark line in the sidewalk concrete, providing us with approximate line spread functions. These provide some insight into the resolution characteristics of the various image estimates. Here it appears that the microgrid RLS provides the narrowest line spread, followed closely by the microgrid AWF. The single frame methods clearly have much broader line spread functions. This is to be expected as these methods do not include any deblurring. It is interesting to note that the Tyo method does appear to have a more favorable line spread function than bicubic interpolation.*

_{j}*s*

_{0,}

*}, {*

_{j}*s*

_{1,}

*}, and {*

_{j}*s*

_{2,}

*} are shown in Figs. 14(a)–14(c), respectively. Note that the Tyo and LSF methods do outperform bicubic interpolation, as expected. Furthermore, the single-frame microgrid AWF with*

_{j}*α*= 0.7 and the microgrid RLS methods do better yet. We believe this is in part due to the fact that these methods incorporate knowledge of the PSF and perform some deblurring. As more input frames are used, the microgrid AWF and RLS SR estimates improve, with diminishing returns as the sampling grid gets well populated by numerous shifted frames. Note also that the

*α*= 0.7 result outperforms the independent channel AWF (

*α*= 0.0). The multichannel RLS also outperforms its independent channel counterpart [5

5. B. M. Ratliff, J. S. Tyo, W. T. Black, and C. F. LaCasse, “Exploiting motion-based redundancy to enhance microgrid polarimeter imagery,” Proc. SPIE **7461**, 74610K (2009). [CrossRef]

*α*= 1.0 provides a relatively low MSE on {

*s*

_{0,}

*}, but does a poor job with {*

_{j}*s*

_{1,}

*} and {*

_{j}*s*

_{2,}

*} that does not improve with input frames. This is because with*

_{j}*α*= 1.0, the {

*s*

_{1,}

*} and {*

_{j}*s*

_{2,}

*} estimates are nearly zero, save for the impact of the local statistics processing beginning with Eq. (39). These results suggest that when estimating {*

_{j}*s*

_{0,}

*} with a relatively small number of frames, it is beneficial to assume more correlation between channels (i.e., use a higher*

_{j}*α*) to more fully exploit the spatial sampling diversity offered by the different channels. However, this comes at the expense of the polarization content in {

*s*

_{1,}

*} and {*

_{j}*s*

_{2,}

*}. We also see that when enough frames are available, the benefit of multichannel processing is somewhat reduced, due to the abundance of spatial samples of each polarization.*

_{j}*α*parameter impacts the MSE performance of the microgrid AWF filter on the three Stokes images. These results are for the 20 frame microgrid AWF output with

*ρ*= 0.7 and

*σ*= 9. Note that the MSE on {

*s*

_{0,}

*} is minimized with an*

_{j}*α*close to, but distinctly less than, one. The MSE on the other Stokes images is obtained with an

*α*near 0.5. Again, this shows us that if one assumes too much correlation between channels (i.e., high

*α*), the pure polarimetric content in {

*s*

_{1,}

*} and {*

_{j}*s*

_{2,}

*} is degraded. On the other hand, assuming too little correlation predominantly hurts our estimate of {*

_{j}*s*

_{0,}

*}. We have found a good compromise in the value used in our earlier results of*

_{j}*α*= 0.7.

## 6. Conclusions

## Acknowledgments

## References and links

1. | J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. |

2. | B. M. Ratliff, C. F. LaCasse, and J. S. Tyo, “Interpolation strategies for reducing IFOV artifacts in microgrid polarimeter imagery,” Opt. Express |

3. | J. S. Tyo, C. F. LaCasse, and B. M. Ratliff, “Total elimination of sampling errors in polarization imagery obtained with integrated microgrid polarimeters,” Opt. Lett. |

4. | S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. |

5. | B. M. Ratliff, J. S. Tyo, W. T. Black, and C. F. LaCasse, “Exploiting motion-based redundancy to enhance microgrid polarimeter imagery,” Proc. SPIE |

6. | D. A. Lemaster, “Resolution enhancement by image fusion for microgrid polarization imagers,” in IEEE Aerospace Conference, Big Sky, MT (2010). |

7. | R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, “High resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system,” Opt. Eng. |

8. | R. Hardie, “A fast image super-resolution algorithm using an adaptive Wiener filter,” IEEE Trans. Image Process. |

9. | R. M. A. Azzam and N. M. Bashara, |

10. | J. Goodman, |

11. | R. R. Shannon, “Aberrations and their effect on images,” Proc. SPIE |

12. | T. Gotoh and M. Okutomi, “Direct super-resolution and registration using raw CFA images,” in |

13. | S. Farsiu, M. Elad, and P. Milanfar, “Multi-frame demosaicing and super-resolution of color images,” IEEE Trans. Image Process. |

14. | L. Condat, “A generic variational approach for demosaicking from an arbitrary color filter array,” in Proceedings of IEEE ICIP , pp. 1625–1628 (2009). |

15. | R. C. Hardie, K. J. Barnard, and E. E. Armstrong, “Joint MAP registration and high resolution image estimation using a sequence of undersampled images,” IEEE Trans. Image Process. |

16. | B. M. Ratliff, J. S. Tyo, J. K. Boger, W. T. Black, D. L. Bowers, and M. P. Fetrow, “Dead pixel replacement in LWIR microgrid polarimeters,” Opt. Express |

**OCIS Codes**

(100.6640) Image processing : Superresolution

(120.2130) Instrumentation, measurement, and metrology : Ellipsometry and polarimetry

(120.5410) Instrumentation, measurement, and metrology : Polarimetry

(110.5405) Imaging systems : Polarimetric imaging

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: May 6, 2011

Revised Manuscript: June 1, 2011

Manuscript Accepted: June 2, 2011

Published: June 20, 2011

**Citation**

Russell C. Hardie, Daniel A. LeMaster, and Bradley M. Ratliff, "Super-resolution for imagery from integrated microgrid polarimeters," Opt. Express **19**, 12937-12960 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-14-12937

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### References

- J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45(22), 5453–5469 (2006). [CrossRef] [PubMed]
- B. M. Ratliff, C. F. LaCasse, and J. S. Tyo, “Interpolation strategies for reducing IFOV artifacts in microgrid polarimeter imagery,” Opt. Express 17(11), 9112–9125 (2009). [CrossRef] [PubMed]
- J. S. Tyo, C. F. LaCasse, and B. M. Ratliff, “Total elimination of sampling errors in polarization imagery obtained with integrated microgrid polarimeters,” Opt. Lett. 34(20), 3187–3189 (2009). [CrossRef] [PubMed]
- S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20(3), 21–36 (2003). [CrossRef]
- B. M. Ratliff, J. S. Tyo, W. T. Black, and C. F. LaCasse, “Exploiting motion-based redundancy to enhance microgrid polarimeter imagery,” Proc. SPIE 7461, 74610K (2009). [CrossRef]
- D. A. Lemaster, “Resolution enhancement by image fusion for microgrid polarization imagers,” in IEEE Aerospace Conference, Big Sky, MT (2010).
- R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, “High resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system,” Opt. Eng. 37(1), 247–260 (1998). [CrossRef]
- R. Hardie, “A fast image super-resolution algorithm using an adaptive Wiener filter,” IEEE Trans. Image Process. 16(12), 2953–2964 (2007). [CrossRef] [PubMed]
- R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).
- J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
- R. R. Shannon, “Aberrations and their effect on images,” Proc. SPIE 531, 27–37 (1985).
- T. Gotoh and M. Okutomi, “Direct super-resolution and registration using raw CFA images,” in IEEE Conference on Computer Vision and Pattern Recognition , vol. 2, pp. 600–607 (Los Alamitos, CA, 2004).
- S. Farsiu, M. Elad, and P. Milanfar, “Multi-frame demosaicing and super-resolution of color images,” IEEE Trans. Image Process. 15, 141–159 (2006). [CrossRef] [PubMed]
- L. Condat, “A generic variational approach for demosaicking from an arbitrary color filter array,” in Proceedings of IEEE ICIP , pp. 1625–1628 (2009).
- R. C. Hardie, K. J. Barnard, and E. E. Armstrong, “Joint MAP registration and high resolution image estimation using a sequence of undersampled images,” IEEE Trans. Image Process. 6(12), 1621–1633 (1997). [CrossRef] [PubMed]
- B. M. Ratliff, J. S. Tyo, J. K. Boger, W. T. Black, D. L. Bowers, and M. P. Fetrow, “Dead pixel replacement in LWIR microgrid polarimeters,” Opt. Express 15(12), 7596–7609 (2007). [CrossRef] [PubMed]

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