Microgrid polarimeters are composed of an array of micro-polarizing elements overlaid upon an FPA sensor. In the past decade systems have been designed and built in all regions of the optical spectrum. These systems have rugged, compact designs and the ability to obtain a complete set of polarimetric measurements during a single image capture. However, these systems acquire the polarization measurements through spatial modulation and each measurement has a varying instantaneous field-of-view (IFOV). When these measurements are combined to estimate the polarization images, strong edge artifacts are present that severely degrade the estimated polarization imagery. These artifacts can be reduced when interpolation strategies are first applied to the intensity data prior to Stokes vector estimation. Here we formally study IFOV error and the performance of several bilinear interpolation strategies used for reducing it.
©2009 Optical Society of America
The primary physical quantities associated with an optical field are intensity, wavelength, coherence, and polarization. Conventional panchromatic cameras measure the intensity of optical radiation over some waveband of interest. Spectral imagers measure the intensity in a number of wavebands, which can range from a few to a few hundred or more, and tend to provide information about the distribution of material in a scene . Polarization information tells us about surface features, shape , shading, and roughness , and therefore tends to provide information that is largely uncorrelated with spectral and intensity data. This fact has been exploited to defeat clutter , aid in the defeat of scatterers [5, 6], and identify target composition  and orientation .
A polarization state describes the time averaged properties of the orientation of the electric field . A common description of polarized light is the intensity parametrization of the state in the form of a Stokes vector . The typical Stokes vector definition is
where PH and PV are the detected power of horizontally and vertically polarized light, P 45 and P 135 are the detected power of linearly polarized light at ±45°, and PR and PL are the detected power of right and left circularly polarized light. Secondary measures that are often useful in imaging applications include the normalized Stokes vector
degree of linear polarization (DoLP)
and angle of polarization (AoP)
Direct measurement of the polarization properties at optical wavelengths is difficult, as most optical detectors respond to intensity only. In order to measure polarization a system must be prepared that modulates the intensity in a manner that depends on the polarization state and can be demodulated to reconstruct the polarization parameters. A recent review article  provides a description of many of the available technologies for passive polarization sensing, including an overview of the prevalent strategies used to obtain the polarized intensity measurements and their associated strengths and weaknesses that contribute to the overall quality of their measurements. Here we focus our attention on a particular class of polarimeters known as division of focal plane (DoFP) devices, also known as microgrid polarimeters.
The recent advances in FPA technology have led to the integration of micro-optical polarization elements directly onto the FPA [11, 12]. Most microgrid polarimeters are only sensitive to linear polarization, i.e., s 0, s 1, and s 2, though some discussion of full-Stokes DoFP systems has been raised . These systems have been manufactured for imaging in all regions of the optical spectrum, including visible [14, 15], SWIR , and LWIR . Most microgrids are laid out in an interlaced configuration as shown in Fig. 1, where each 2×2 region of the array is called a superpixel and contains all polarized intensity measurements necessary for estimating the Stokes vector in that region. Other layouts have been made where the polarization information is sampled on a line-by-line basis .
DoFP systems have the significant advantage that all polarization measurements are simultaneously acquired during a single image capture. However, they have the distinct disadvantage that each pixel makes only one of the three necessary polarized intensity measurements required to estimate the Stokes vector at each location. Since all of a given pixel’s neighbors are of different orientation, the Stokes vectors are estimated by combining measurements within a small neighborhood of pixels. Hence, each contributing pixel has a different instantaneous field-of-view (IFOV). When these measurements are combined, the estimated Stokes vector may contain error when there is significant intensity contrast across the neighborhood. Due to the subtractive nature of the Stokes vector estimates, the resulting polarization images contain strong edge artifacts.
IFOV artifacts are most severe when the Stokes vector imagery is estimated directly from the microgrid image samples. However, these artifacts can be reduced to a large degree if the intensity samples are first interpolated prior to Stokes estimation [18, 19]. In addition, it is possible to reduce IFOV error by blurring the PSF of the optics, but this carries a corresponding loss of resolution. Our ultimate goal is to get to (or past) the resolution of the FPA. In this paper we present a set of microgrid-specific bilinear interpolators and study their ability to reduce IFOV artifacts. The interpolators can indeed reduce IFOV artifacts at the cost of smoothing the resulting images. Our goal is to find the best interpolation strategies that minimize loss in image information while simultaneously minimizing IFOV artifacts. We investigate our interpolation strategies using both simulated and real LWIR microgrid data and compare their performance to an ideal diffraction-limited polarimeter.
The remainder of this paper is structured as follows. In Sec. 2 we formally define IFOV error in the context of microgrid systems. Microgrid-specific interpolation strategies used to reduce the effects of IFOV error are presented in Sec. 3. The performance of these various interpolation strategies and their associated trade-offs are studied in Sec. 4. The interpolation schemes are then applied to LWIR data from an actual microgrid sensor in Sec. 5. Finally, conclusions are stated in Sec. 6.
2. Definition of IFOV error
IFOV error is a fundamental problem for microgrid instruments, being present even in ideal microgrid systems due to the spatial sampling nature of these devices. Here we present interpolation strategies for reducing the effects of IFOV error. In order to study the ability of each strategy to reduce IFOV error, we first formally define the meaning of IFOV error in the context of microgrid systems. To establish a basis for evaluation, we construct a theoretical imager that does not suffer from IFOV error, i.e., we assume we have an instrument that is capable of measuring the same area in space over the same interval in time for each polarizer orientation. Though such a device is clearly unrealizable, it is used as a best case in our simulation-based studies for the various interpolation strategies.
To develop this formalism we assume diffraction-limited imaging and work in object space, meaning the optical system projects the grid of pixels out to the object. This optical system has magnification factor Mo and is depicted conceptually in Fig. 2. We specify all object radiance in Stokes vector form, denoted by S(x,y), to account for fully polarized scenes, where x and y are the continuous object space coordinates. According to the described optical system, each Stokes vector traces a path from the object through the system optics to the focal plane. Both the microgrid and ideal polarimeter pixels are indicated in the figure. The system optics are described by the point spread function h(x,y). Each pixel is masked by a linear polarizer oriented at given angle θ, where θ = 0°, ±45°, and 90°. We assume each pixel’s polarizer is ideal and is described by the Mueller Matrix M(θ). In practice each detector has an associated spectral bandwidth, though we omit it from our notation for simplicity. Our model is described in greater detail elsewhere .
First, we account for the system optics by convolving the input Stokes vector with the optical point spread function, i.e., S̃(x,y) = S(x,y) *h(x,y). Next, we account for the linear polarizer by multiplying S̃(x,y) with M(θ) to obtain the modified Stokes vector at the FPA [21, 22]. Hence S(x,y)θ Det = M(θ)S̃(x,y). We only consider the s 0 component at the detector since optical detectors only sense intensity, which is
where q and r are the polarizer’s transmission coefficients for co- and cross-polarized light, respectively. To obtain the FPA detector response we next integrate s 0(x,y)θ Det over pixel area. The detector response of the theoretical polarimeter pixel for orientation angle θ is
where τt is the optical power transmission coefficient, Ω is the solid angle of the collection, m and n are the FPA detector indices assumed to be at the center of each pixel, and m′ = m + k and n′ = n + k. When the theoretical pixel is in line with the FPA detectors k = 0; when this pixel is offset from the FPA detectors k = 1/2. Placement of the ideal polarimeter detector is determined by the reconstruction points for a selected interpolation scheme and will be discussed thoroughly in Sec. 3.
The detector response for each microgrid detector is given by
IFOV error will be present in any Stokes vector estimated from the microgrid measurements since each pixel has a different IFOV and only one orientation is sampled per pixel. The Stokes vector is estimated from these intensity measurements or interpolations of them at selected reconstruction points, denoted by P̂θ μgrid (m′,n′). We therefore define IFOV error at each reconstruction point as
Thus, IFOV error is the deviation between the ideal polarimeter and interpolated microgrid pixel responses at each point. The goal of our study is to determine how well a given interpolation strategy can approximate the measurements obtained from the ideal sensor.
3. Interpolation strategies
IFOV artifacts can be reduced if the direct microgrid measurements from a defined neighborhood of pixels are first interpolated to a common reconstruction point. However, here we do not combine measurements from adjacent pixels since each is masked with a polarizer of different orientation. Instead, under the schemes presented here, we only interpolate pixels of like orientation. For the purposes of this paper we limit ourselves to a family of modified bilinear interpolators due to their computational simplicity and widespread use, though other strategies could be pursued . For each scheme we choose reconstruction points that are either aligned to each microgrid pixel or on the grid between them, as shown in Fig. 1. We define an interpolation neighborhood as all pixels whose centers are within a given radius (in pixels) of a chosen reconstruction point. Here we consider the five interpolation methods illustrated in Fig. 3.
For each interpolator we depict the pixel neighborhood defined by the given radius about the reconstruction point, indicated in yellow. Pixels of different orientation are colored in different shades of blue. When the reconstruction point is at a pixel center (Methods 1,2) there are a nonequal number of pixels of each orientation, whereas the distribution is equal for on-grid reconstructions (Methods 0,3,4). Also displayed for each method are a set of four convolution kernels, with the exception of Method 1 (discussed below). The multiple kernels result from the fact that we do mix pixels of different orientations. We denote each kernel H ij, where i = 0,…, 4 indicates the interpolation method from Fig. 3 and j = 1,…, 4 denotes one of the four kernels for the respective method, numbered from left to right in the figure.
To perform the interpolation, we convolve the M × N microgrid image Pμgrid with each kernel for a particular method, i.e.,
The resulting four images are modulated much like Pμgrid, only the modulation pattern is shifted in each image relative to the input and is determined by the particular kernel used. For example, if we align these images according to reconstruction point, each point contains one interpolated intensity value of each orientation. However, the order of orientation of the four interpolated values will vary at each location. Also, due to the convolution, there will be border effects in the resulting images . The size of the border varies depending on the size of the kernel employed. The images should thus be appropriately cropped to size M′ × N′ to account for this, though we leave the details of this step to the reader. The Stokes images can be estimated directly from the four Iij images, but is a tedious task due to their modulated form. Instead, we demodulate them into an M′ × N′ × 4 image cube I i such that each image contains intensity values of the same orientation. We populate each element of the image cube using the following demodulation scheme:
for m = 1,…, M′, n = 1,…, N′, and j = 1,…, 4 where
Both the interpolation kernels and demodulation strategy are independent of microgrid layout. This method will always demodulate the interpolated images as desired; however, their order of orientation in I i will vary depending upon the layout of the microgrid, the cropping scheme used, and the order that the interpolation kernels are applied. Our convention is to perform the demodulation so that I i has its images arranged by increasing θ. The Stokes vector images are next computed in an M′ × N′ × 3 image cube Si, ordered according to s 0, s 1 and s 2. Making use of Eq. (1) we have
The above method yields a Stokes vector cube for a particular interpolator. In the next section we compute S i for each interpolation method and compare the quality of their estimates.
A closer examination of the interpolation schemes of Fig. 3 will prove useful in our later discussions. Observe that the neighborhoods defined by Method 0 consist of a single super-pixel, resulting in each respective kernel containing a single nonzero element with weight 1. Thus, these kernels extract a single microgrid value from each 2 × 2 neighborhood. In this case no interpolation occurs among pixels of like orientation. Instead the result is similar to a nearest-neighbor interpolator, only at each reconstruction point the nearest pixel of a particular orientation is selected. Hence, we call Method 0 the nearest like-polarization neighbor interpolator.
Method 1 defines neighborhoods that only contain pixels of three orientations. Since only three measurements of different orientation are needed to estimate the linear Stokes vector this is not problematic. However, in order to estimate the linear Stokes vector image cube of Eq. (12), we must first estimate the missing orientation at a given reconstruction point from the three present ones. Since this strategy relies on the redundancy inherent in the microgrid measurements, i.e., four measurements are used to estimate the three Stokes parameters, we refer to this strategy as the redundancy estimation interpolator. The idea of redundancy estimation is the foundation of our microgrid-specific dead pixel replacement (DPR) strategy . The mathematical relationships needed to estimate the missing intensity measurements are fully derived in  and are not repeated here.
4. Interpolator performance
The five interpolation methods presented in Sec. 3 yield several options for estimating the Stokes vector imagery. Here we examine the interpolators by applying each method, estimating the Stokes vector images and studying the resulting IFOV artifacts. We do this by applying the interpolators to simulated data generated from the physical model described in Sec. 2. The imagery is created by first generating Stokes vectors at each point in a high resolution image (approximating the analog scene) and passing it through our microgrid system model. This involves accounting for the optics and microgrid polarizers before the signal is captured by the detector. The resulting microgrid image is interpolated with a selected method from Sec. 3 and the Stokes images are estimated. We then examine these reconstructed Stokes images by looking at recovered frequency content of both the original signal and IFOV terms in addition to error in the estimated DoLP images.
In our study we generate Stokes vectors in the analog scene that are spatial sinusoids of the form
where fx and fy are specified in cycles per detector pixel. We use sinusoids so that we can compute the modulation transfer function (MTF) and something we term the intermodulation transfer function (IMTF) (described below) for results from each interpolation scheme.
An illustrative example is shown in Fig. 4. We choose fully polarized Stokes vectors of s 0 (x,y) = s 2 (x,y) = t(x,y) and s 1 (x,y) = 0 with cycles/pixel and A = 1. Rather than work in continuous space, we instead generate a high resolution digital approximation to the analog scene. The resulting s 0 image for our example and its frequency domain representation are shown in Fig. 4(a). In this case the frequency domain signal consists of the two delta functions δ(u + fx,ν + fy) and δ(u − fx,ν − fy). The high resolution Stokes vector images are then convolved with h(x,y), the PSF for an ideal circular aperture, and each resulting Stokes vector is multiplied by M(θ) with θ set according to the microgrid polarizer at that location. This high resolution image is then downsampled to the detector resolution (128 × 128) and the resulting simulated microgrid image and its corresponding frequency domain representation are shown in Fig. 4(b).
The microgrid image shows significant modulation (often referred to as checkerboarding) due to the fully polarized signal passing through the mosaic of micro-polarizers. The modulation is also apparent in the corresponding frequency domain plot. This has introduced six new sinusoidal terms in addition to the two corresponding to our input signal: two at the highest possible frequencies due to the microgrid modulation, δ(0.0,0.5) and δ(0.5,0.0); two we call the horizontal intermodulation terms, δ(u + fx −0.5,ν + fy) and δ(u − fx + 0.5,ν − fy); and two we call the vertical intermodulation terms, δ(u + fx,ν+ fy − 0.5) and δ(u − fx,ν − fy + 0.5). We then apply interpolation Method 0, demodulate each resulting image, and estimate the Stokes images as described in Sec. 3. The resulting s 0 and s 1 images and their corresponding frequency domain representations are shown in Figs. 4(c) (Media 1) and 4(d), respectively. We do not show the reconstructed s 2 results because they are similar to s 0 in this case since the input s 0 and s 2 images are equal by construction. The frequency domain plots for the reconstructed s 0 and s 1 images show that the intermodulation terms remain and thus are not removed by interpolation or demodulation. These terms correspond directly to IFOV artifacts in the reconstructed images. The artifacts are especially apparent in the reconstructed s 1 image since this image should be zero.
We find that the intermodulation terms always appear at locations that are 1/2 cycle per pixel away from the input sinusoid location, which corresponds to the modulation frequency of the microgrid. Figure 4(c) (Media 1) shows an animation of this phenomena for the reconstructed s 0 image and its corresponding frequency domain representation as the input frequency is varied. As is observed in the video, each sinusoidal term has both a horizontal and vertical intermodulation term associated with it, each respectively shifted in u or ν by 1/2 cycle per pixel. When u = ν the magnitudes of the horizontal and vertical IMTF terms are equal. However, when u ≠ ν the magnitudes differ in a predicable fashion, discussed more below.
We clearly would like these intermodulation terms to be zero and next investigate the effect each interpolation method has on them. To do so, we pass sinusoidal Stokes vectors at many spatial frequencies through our model and measure the output magnitudes of the sinusoidal and intermodulation terms in the reconstructed s 0, s 1 and s 2 images under each interpolation strategy. We then use these output magnitudes to generate MTF and IMTF surfaces as a function of sinusoidal input frequency. At each frequency we also compute the mean-square error (MSE) of the DoLP image as another measure of system performance. We do this for both unpolarized (0% DoLP) and fully polarized (100%) inputs. In practice DoLP signatures greater than 20% are rarely observed and hence these cases were chosen to illustrate the two theoretical extremes. Figure 5 shows selected resulting surfaces for the unpolarized case. Here s 0(x,y) = t(x,y), s 1(x,y) = s 2(x,y) = 0, and A = 1 with fx, fy ∈ [0,0.5] cycles per detector pixel. Shown are the results from each reconstruction for the s 0 MTF, s 0 horizontal IMTF (HIMTF), s 1 HIMTF and DoLP MSE. The vertical IMTF (VIMTF) surfaces are not shown due to their similarity to the HIMTF surfaces. As was observed in Media 1, the HIMTF and VIMTF surfaces are equal along the line u = ν. When u ≡ ν the VIMTF surface is related to the HIMTF plot by mirroring the HIMTF surface about the line u = ν.
We observation that the intermodulation terms are zero in the reconstructed s 0 image when the scene is unpolarized regardless of interpolation method. However, this is not true for the reconstructed s 1 HIMTF terms as observed in the third column. As the interpolation method number increases we see the magnitude of the intermodulation terms decrease. We would thus expect a decrease in IFOV artifacts present in the reconstructed images and expect a decrease in high frequency image content according to the MTF plots in the first column, which also show a decrease in magnitude with increasing method number. The reduction in IFOV artifacts with increasing interpolation method number is also indicated by the DoLP MSE surfaces in the fourth column. Thus, from these results we would expect larger bilinear kernels to provide better reduction of IFOV artifacts at the cost of high frequency image content being lost.
The same surfaces in Fig. 5 are shown in Fig. 6 for the fully polarized case. Here s 0(x,y) = s 2(x,y) = t(x,y), s 1(x,y) = 0, and A = 1 with fx,fy ∈ [0,0.5] cycles per detector pixel. The s 0 MTF and s 1 HIMTF surfaces in the first and third columns are identical to the results from the unpolarized case. However, the s 0 HIMTF surfaces are now nonzero. Interestingly, the DoLP MSE error is on average smaller when compared with the respective surface in Fig. 5. Overall the trends are the same for the polarized and unpolarized cases in that increasing interpolation method number decreases image frequency content while also decreasing the contributions of the intermodulation terms.
Cross-sections from the MTF and DoLP MSE surfaces are plotted along the line u = ν in Fig. 7. Figures 7(a) and 7(c) contain the MTF results for the polarized and unpolarized cases, respectively. Also plotted is the MTF for an ideal circular aperture from our theoretical polarimeter system described in Sec. 2. As expected the MTF bandwidth decreases with increasing interpolation method number. Figures 7(b) and 7(d) show the corresponding DoLP MSE projections for the polarized and unpolarized cases, respectively. The ideal polarimeter DoLP MSE is zero in both cases and therefore not plotted. Here we see the expected trend that DoLP MSE decreases with increasing method number with the exception of the cases of Methods 3 and 4. In the unpolarized case Method 4 shows significantly more error at lower frequencies than Method 3. In the polarized case the two Methods are similar. This indicates that Method 3 provides a better tradeoff between loss of image frequency content and reduction in IFOV artifacts, and will indeed be observed on real data in the next section.
Though the MTF, IMTF and DoLP MSE metrics have been useful for providing a better understanding of IFOV error and insight into the performance tradeoffs of each interpolation method presented, an improved set of analysis tools is warranted for future analysis. In future work we intend to develop such a generalized polarimetric linear systems framework that will allow for easier analysis of microgrid systems and their corresponding imagery.
5. Application to microgrid data
We apply the interpolators of Sec. 3 to data obtained from a LWIR microgrid polarimeter system manufactured by DRS Technologies. The camera system consists of an HgCdTe FPA sensor with micropolarizers oriented as shown in Fig. 1. The FPA sensor is of size 640 × 480, has a square pixel pitch of 25 μm, and is cryogenically cooled to 65K. The cold filter has a pass-band of 7.8–9.8 μm. The instrument is described in greater detail elsewhere [17, 20, 26]. The data consists of 1000 image frames acquired at 30 fps and contains frame-to-frame panning motion. As with all LWIR data, spatial nonuniformity and dead pixels are present in the video sequence. To account for this we perform a linear ten-point nonuniformity correction  and apply our redundancy estimation DPR strategy  to each image frame prior to processing. The first frame of the video sequence after application of calibration and DPR is shown in Fig. 8. The scenario is an airport runway that depicts aircraft departing. Multiple aircraft can be seen along with a significant amount of both natural and manmade clutter including various buildings, paved runways, grass and other typical terrain features. Also shown in Fig. 8 are the five DoLP images resulting from application of each interpolation method from Fig. 3 for this image frame.
Examining each DoLP image we see that IFOV artifacts are most severe for Method 0 and decrease with increasing method number until Method 4. Interestingly, once the interpolation kernel becomes large enough we see a reemergence of artifacts near objects of significant thermal contrast. However, the artifacts appear as a double-edge of weaker magnitude in comparison to the single-edge, high-magnitude artifacts observed in the DoLP image of Method 0. Also, the images become smoother as the interpolation kernel size increases. The results from Methods 1–3 each have their advantages, but in general we find that Method 3 yields results that have the least artifacts and provides the best tradeoff between reduction in IFOV artifacts and reduction in image frequency content, consistent with our findings from Sec. 4.
For closer examination of the results from each method, we generated a video that contains the Stokes parameter, DoLP and false-colored full Stokes vector image (FSVI) for each method from a small region of interest (ROI) for the airport video sequence. The FSVI is generated using a modified version of the polarimetric colorization strategy by Tyo, et al. . The ROI is demarcated by the red box in the s 0 image of Fig. 8. The video contains 100 frames (30 frames in the shorter version), the first of which is displayed in Fig. 9 (Media 2).
When viewing Fig. 9 (Media 2), the s 0 images are nearly indistinguishable. This is expected since s 0 is computed by averaging, rather than differencing, pixels within a given neighborhood. The s 1 and s 2 images contain a significant amount of IFOV and inter-modulation artifacts, especially in the cases of Methods 0 and 4. These artifacts are most visible in the elongated building roofs towards the bottom of the image. Methods 1–3 are much less noisy, with the artifacts appearing to be least severe in Method 3. The DoLP images are similar in quality to each respective s 1 and s 2 image since they are directly derived from them. The FSVI images are all similar in quality since the basis of each image is s 0, with the polarization information overlaid upon the color channels, though artifacts are more apparent in the FSVI of Method 0.
We have presented a family of bilinear interpolation methods used to reduce IFOV artifacts in Stokes vector images estimated from microgrid polarimeter systems. IFOV error was defined and modeled in the context of a conventional microgrid system. We used this model to simulate microgrid imagery and then applied the various interpolators, estimated the corresponding Stokes vector imagery and studied the results. Overall we found that one technique, namely Method 3, consistently outperformed the others by providing the highest reduction in IFOV artifacts while yielding an acceptable loss in high frequency image content. We applied each interpolation strategy to data from an actual LWIR microgrid polarimeter and made similar conclusions. In future work we plan to develop more sophisticated interpolation strategies and investigate motion-based algorithms that use multiple image frames to perform resolution enhancement. Additionally, we are working on a fully polarimetric linear systems framework based on a formalism defined by McGuire and Chipman  that will simplify the task of analyzing the performance of microgrid polarimeter systems.
This work was supported in part by the Air Force Office of Scientific Research under award FA9550-07-1-0087 and in part by the National Science Foundation under award ECCS0704114. We would like to thank Ernie Atkins at DRS Sensors and Tracking for providing the LWIR microgrid data used in this paper.
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