## Abstract

This paper presents bilinear and bicubic interpolation methods tailored for the division of focal plane polarization imaging sensor. The interpolation methods are targeted for a 1-Mega pixel polarization imaging sensor operating in the visible spectrum. The five interpolation methods considered in this paper are: bilinear, weighted bilinear, bicubic spline, an approximated bicubic spline and a bicubic interpolation method. The modulation transfer function analysis is applied to the different interpolation methods, and test images as well as numerical error analyses are also presented. Based on the comparison results, the full frame bicubic spline interpolation achieves the best performance for polarization images.

© 2011 OSA

## 1. Introduction

Polarization imaging sensors can be divided into division of time [1, 2], division of amplitude [3–6], division of aperture [7, 8] and division of focal plane polarimeters (DoFP) [9–16]. DoFP polarimeters include imaging elements, i.e. photo detectors, and micropolarization filter arrays on the same substrate, and can record the first three or four Stokes parameters at every frame [17, 18]. The micropolarization filter array is composed of pixelated polarization filters oriented at several different polarization angles in order to filter the incident optical filed. The individual pixelated polarization filters are typically composed of periodic metallic wires whose width is approximately less than or equal to 1/5^{th} of the incident wavelength [19]. Rigorous coupled-wave analysis has been used to show that extinction ratio above 100 and maximum transmission of ~80% can be achieved with aluminum metallic wires whose width is 1/5th of the incident wavelength [19]. For example, the micropolarization filter array for infrared DoFP sensors imaging at 3 μm wavelength is realized with 600 nm wide metallic wires [20]. These wires can be fabricated via a standard photolithography process and integrated with an array of infrared sensitive photo detectors. The width of the wires in the micropolarization filter array for the visible spectrum have to be ~90 nm, in order to filter optical frequencies in the range of 450 nm up to 700 nm [15]. Fabricating feature sizes bellow 200 nm is not possible via standard UV lithography and possess challenges for realizing visible spectrum polarization filters.

Recent advancements in nanofabrication techniques and technology have allowed for reliable fabrication of metallic nanowires that can act as linear polarization filters in the visible spectrum [20–23]. Furthermore, the challenge of monolithic integration of metallic nanowires with an array of photo detectors implemented in either CMOS or CCD technology has been recently overcome [15]. A high resolution DoFP polarimeter has been reported where aluminum nanowires have been monolithically integrated with an array of 1000 by 1000 CCD pixels. This sensor can compute the first three Stokes parameters, i.e. S_{0}, S_{1} and S_{2}, at every frame with maximum signal-to-noise ratio of 45 dB and dynamic range of 60 dB [15]. The pixel pitch of 7.4 μm allows for imaging a scene with high spatial fidelity.

A block diagram of a typical DoFP sensor for infrared or visible spectrum is presented in Fig. 1(a) [15]. In a neighborhood of 2 by 2 pixels, four different polarization filters, offset by 45 degrees, are deposited on individual photo receptors. This neighborhood of 4 pixels is typically referred to as a “super pixel”. An analogous concept was realized in the mid 1970’s for capturing color information via direct deposition of a color filter array, known as a Bayer pattern [24], on top of the CCD’s imaging elements. A sample of the Bayer color filter pattern is presented in Fig. 1(b). The integration of color filters with an array of imaging elements has created robust and compact color imaging sensors. However, the color information from the imaged scene is sub-sampled and can lead to inaccurate color reconstruction. For instance, blue and red color information is sub-sampled by a factor of 4 and green color is sub-sampled by a factor of 2. In order to recreate high resolution color images from low resolution prime color images, various interpolation techniques have been introduced [24–28]. Since the inception of the Bayer pattern, research in the color interpolation area has created a tremendous body of literature with the main purpose of recovering as much missing color information as possible in the imaging array with high accuracy.

A similar problem exists in division of focal plane polarization imaging sensors [29, 30]. In these sensors, the imaged scene is sampled with four different polarization filters and four sub-sampled images are produced. Due to the different instantaneous field of view of the four sub-sampled images, large errors are introduced in the computed polarization information. In order to recreate accurate high resolution polarization images, interpolation techniques have to be introduced on the four primary low resolution polarization images. Although some interpolation algorithms from the color domain can be easily ported to the polarization imaging sensors, the inherent difference of the filter arrays between the two imaging sensors mandates new research directions in polarization interpolation algorithms.

In this paper, we present three interpolation methods for images produced by DoFP sensors and they are: bicubic interpolation, bicubic spline interpolation and an approximated bicubic spine interpolation algorithm. For reference purposes in Section 2, we describe two previously published interpolation methods for polarization images: bilinear interpolation and weighted bilinear interpolation [29, 30]. In Section 3, the performance of the different interpolation methods is evaluated in terms of spatial resolution and accuracy of the reconstructed images for both synthetic and real polarization images. In Section 4, we introduce a method for evaluating polarization interpolation algorithms on set of polarization images. In this section, we first present a quantitative comparison between the different interpolation methods by presenting a set of reconstructed polarization images. Second, we present a qualitative evaluation by introducing a mean square error comparison for the accuracy of the interpolated images. Concluding remarks are presented in Section 5.

## 2. Interpolation methods

#### 2.1 Bilinear interpolation methods

The bilinear interpolation methods are among the most common techniques used in image processing due to their computational simplicity. In DoFP polarization sensors, bilinear interpolation is computed separately on the four low resolution images recorded with different polarization filters offset by 45°. For example, a 1 megapixel (Mp) DOFP polarization sensor captures 0.25 Mp for each of the 4 linear polarization filters (0°, 45°, 90°, 135°). The bilinear interpolation method is then used to up-sample these 0.25 Mp components back to the full resolution (1 Mp) of the camera.

Figure 2 presents a neighborhood of 4 by 4 pixels in a typical DoFP imager composed of four different pixilated polarization filters. For this pixel neighborhood, the pixel in location (2,2) records the intensity of the light wave after filtering it with a linear polarization filter oriented at 90°. At this pixel location, the intensity value of the filtered light wave with 0°, 45° and 135° polarization filter are estimated via bilinear interpolation from the surrounding pixels and Eqs. (1) through (3) present the formulas used to estimate these pixel values.

In Eqs. (1) through (3), the 0° interpolated intensity value at pixel location (2,2) is an average intensity value of the four adjacent diagonal pixels; the 45° interpolated intensity value is the average intensity of the two adjacent vertical pixels, and the 135° interpolated intensity value is the average intensity of the two adjacent horizontal pixels. Equations (1) through (3) are slightly modified when computing the interpolated pixel values at locations (2,3), (3,2) and (3,3) in the imaging array due to the differences in the linear polarization filters in the immediate pixel neighborhood.

#### 2.2 Weighted bilinear interpolation method

Weighted bilinear interpolation for polarization images has been introduced in [29] and has higher reconstruction accuracy when compared to the traditional bilinear interpolation method. The interpolation method can be visualized as a convolution process using the four kernels presented in Fig. 3(a)
through Fig. 3(d). In this method, all four pixel orientations are interpolated and the newly computed pixels are located at the center of the convolution kernel, i.e. the location marked with a white dot in the convolution kernel presented in Fig. 3(e). The coefficients in the kernels are estimated by the distance from the location of the pixel that is interpolated. For example, pixels convolved with the coefficient *A* are closer to the estimated pixel and have higher weighting than the pixel convolved with the coefficient B.

Equations (4) through (7) are used to compute the four different polarization angle pixel values. Weighted bilinear interpolation uses a larger interpolation kernel compared to the bilinear interpolation method. Furthermore, all angle intensity values of the reconstruction pixel are interpolated using the same number of adjacent pixels in one angle orientation, i.e. three pixels per convolution kernel.

#### 2.3 Bicubic spline interpolation method

The bicubic spline interpolation method is a common algorithm used in color images with superior improvements in its performance over bilinear interpolation. However, the cost is increased computational complexity [31]. The bicubic spline interpolation method can be easily applied to the four low resolution polarization images produced by a DoFP polarimeter. This interpolation method is first applied in the horizontal direction followed by interpolation in the vertical direction. The bicubic spline interpolation method computes new values from a low resolution image by fitting a third order polynomial, *f _{i}(x),* between two neighboring pixels (

*x*and

_{i}*x*) with polarization filters oriented at the same angle. The third order polynomial,

_{i + 1}*f*can be described by Eq. (8):

_{i}(x),*x*is the spatial location of the pixel value that is being interpolated and is typically half way between

*x*and

_{i}*x*. Coefficients

_{i + 1}*a*,

_{i}*b*,

_{i}*c*and

_{i}*d*are defined as follows:

_{i}In Eqs. (9) through (12), *I(i,j)* is the intensity value of a pixel at location *(i,j)* and *M _{i}* is the second derivative of the function

*f*, i.e.

_{i}(x)*M*. The coefficients described by Eqs. (9) through (12) are derived by placing the following constraints on the interpolated curve

_{i}= f_{i}^{”}(x)*f*:

_{i}(x)- (a) the function
*f*should be continuous between pixels_{i}(x)*x*and_{1}*x*;_{n} - (b) the first derivative of the function
*f*should be continuous between pixels_{i}(x)*x*and_{1}*x*_{n} - (c) the second derivative of the function
*f*should be continuous between pixels_{i}(x)*x*and_{1}*x*._{n}

In order to solve the coefficients presented in Eqs. (9) through (12), one additional equation is derived from the requirement that the second derivative of the function *f _{i}(x)* should be continuous and is presented by Eq. (13):

Equations (8) through (13) lead to formulating a system of *N-2* linear equations with *N* unknowns, where *N* is the number of pixels in a row. In order to solve the system of linear equations presented in Fig. 4
, two more edge constraints must be introduced. In natural splines, the edge constrains are set such that${M}_{1}={M}_{n}=0$. Once the system of equations is solved, the values of *a _{i}*,

*b*,

_{i}*c*and

_{i}*d*are uniquely determined. Finally, the interpolated value

_{i}*f*is computed using Eq. (8).

_{i}(x)Due to the edge constrains that *M _{1} = M_{N} = 0*, the system of

*N-2*linear equations need to be solved which can result in

*(N-2)*computational workload. However, since the constant matrix here is in tri-diagonal form, the Crout or Doolittle method can be applied to solve the matrix [32]. This method decreases the computational workload to

^{2}*5*N-14*multiplications and divisions and

*3*N-9*additions and subtractions. This reduction of computational cycles can lead to real-time execution of the interpolation algorithm.

The drawbacks of the above implementation are the storage of the entire frame information in memory and the computational complexity necessary to estimate new pixel values. This limitation may constrain real-time execution of the algorithm. Thus, we developed an approximated bicubic spline method which is more suitable for computational systems with limited memory, such as FPGAs and DSPs [33]. This method computes the bicubic spline interpolation on a sub window of 10 by 10 pixels from the imaging array. The edge constraints of the computational window remain the same as described above, i.e. *M _{0} = M_{10} = 0*. This method can also be executed in parallel on multiple independent windows in the imaging array and can enable real-time execution of the algorithm.

#### 2.4 Bicubic interpolation method

The bicubic interpolation method attempts to fit a surface between four corner points using a third order polynomial function [34]. In order to compute a bicubic interpolation, the intensity values and the horizontal, vertical, and diagonal derivate at the four corner points should be specified. The interpolated surface, *f _{i}(x,y)*, described by a third order polynomial is presented by Eq. (14).

There are 16 coefficients *a _{ij}* that should be determined in order to compute the function in Eq. (14). Four of the coefficients are determined directly from the intensity values in the four corners; eight of the coefficients are determined from the spatial derivate in the horizontal and vertical direction and four of the coefficients are determined from the diagonal derivatives.

Figure 5(a) presents the 0° pixels in the neighborhood of 5-by-5 pixels, where the interpolation area is in the lower right region, i.e. pixels (3,4), (4,3), (4,4), (4,5), and (5,4) are estimated. The estimation of these pixel values is based on both the intensity values of the known pixels and the three spatial derivatives in the 5 by 5 pixel region. A system of 16 equations is composed in order to solve the interpolation coefficients. Once the interpolation coefficients are computed from these 16 equations, the 5 interpolated pixel values are computed using Eq. (14). Since other angle elements have the same pattern as the 0° pixels, the same algorithm is applied to compute the other three pixels with different linear polarization filters.

## 3. Modulation transfer function evaluation

The modulation transfer function (MTF) is commonly used for evaluating the optical characteristics and performance of an imaging system. The MTF measures the contrast of sinusoids as a function of spatial frequency [35]. In order to evaluate the MTF of an imaging system, a spatially varying sinusoidal pattern at various frequencies is used as a target image. The output from the imaging system is recorded and the contrast for a given input frequency is computed, i.e. the difference between the maximum and the minimum of the output sinusoid. The MTF is computed as the ratio of the contrast of the output sinusoidal pattern over the contrast of the input sinusoidal pattern for various frequencies.

In order to evaluate the MTF response for a typical CMOS or CCD grayscale or color imaging sensors, a single target composed of spatially varying intensity patterns is imaged. This is achieved by imaging a set of calibrated sinusoidal targets provided by Kodak [36]. The MTF of an imaging sensor depends on the MTFs of individual optical elements used in the system such as lenses, filters, pixel design and others.

Division of focal plane polarization imaging sensors record both intensity and polarization information of the light wave at every frame. Hence, the standard procedure for evaluating the MTF of CMOS/CCD imaging sensor should be accordingly modified in order to encompass changes in the polarization properties of an input target. Since polarization sensors record the photo response at four different pixels with linear polarization filters offset by 45 degrees, then an input target can be defined such that any of the four pixel photo responses is spatially varying at a particular spatial frequency. For example, an input target image can be defined as spatially varying sinusoids where the responses of the four pixels are defined as follows:

In Eqs. (15) through (18), *f _{x}* and

*f*are the spatial frequency elements in the horizontal and vertical directions in cycles per pixel. The spatial pattern for

_{y}*I*,

_{0}*I*, and

_{45}*I*are modulated in both the

_{90}*x*and

*y*directions, resulting in a target image that has variable intensity, i.e. variable

*S*parameter, a constant

_{0}*S*parameter and variable

_{1}*S*parameter. Hence, the target image is composed of spatially varying intensity, angle and degree of polarization patterns. The spatial patterns for

_{2}*I*,

_{0}*I*,

_{45}*I*and

_{90}*I*are projected across the entire imaging array of a division of focal plane polarimeter. Due to the pixelated polarization pattern at the focal plane, all four spatial patterns described by Eqs. (15) through (18) are decimated in order to emulate an image recorded by a DoFP imager. Using the decimated images as inputs to the different interpolation algorithms described in Section 2, four high resolution images are generated.

_{135}The original and interpolated images are used to compute the MTF of the system. First, the first Stokes parameter, S_{0}, is computed from the data recorded by the four pixels. The Fourier transform is computed on the intensity images from the original and interpolated data set. Since the target image is a cosine function with horizontal and vertical frequency (*f _{x},f_{y}*), the Fourier image has two main frequency peaks at positions (

*f*) and (

_{x},f_{y}**-**

*f*). Since the accuracy of the interpolated images is a strong function of the input frequency of the target image, the amplitude of the main frequency peaks of the S

_{x},-f_{y}_{0}image will be a function of the input frequency as well. The MTF of the imaging sensor is computed as the ratio of the magnitude of the main frequency peak in the interpolated result,

*M*over the magnitude of the main frequency peak in the original image, i.e.

_{interpolated}(f_{x},f_{y}),*M*.

_{original}(f_{x},f_{y})Figure 6
presents the MTF of the intensity image from four different interpolation algorithms. In Fig. 6(a) through Fig. 6(d), the vertical and horizontal frequency, *f _{x}* and

*f*, are swept from 0 up to 0.5 cycles per pixel and the MTF response is computed at each frequency. Figure 6(e) presents the MTF response for all four interpolation algorithms across equal horizontal and vertical frequencies, i.e.

_{y}*f*=

_{x}*f*. In Fig. 6(e), the ideal MTF response of a full resolution sensor is plotted for comparison reasons. Due to the Nyquist sampling theorem, the ideal MTF has a response of one up to 0.5 cycles per pixel and drops to zero for higher frequencies. Due to sampling a scene with four different polarization filters in DoFP polarimeters, the MTF for these class of sensors drops to zero for frequencies higher than 0.25 cycles per pixel.

_{y}Two important observations can be made regarding bilinear and weighted bilinear interpolation. Specifically, MTF function decreases relatively quickly for low spatial frequencies and goes to zero at *f _{x}* =

*f*= 0.25 cycles per pixel. Bilinear interpolation does not perform well when the interpolated signal contains high spatial frequencies. Due to Nyquist theory, the signal cannot be reconstructed beyond spatial frequency of 0.25 cycles per pixel. Meanwhile, the two bicubic interpolation methods reconstruct high spatial frequency information better when compared to the bilinear methods. The bicubic spline interpolation method preserves low frequency features the best, while the bicubic method can recover the highest frequency components better than any other method (up to spatial frequency of 0.375 cycles per pixel).

_{y}## 4. Performance evaluation

In order to evaluate the accuracy of the different interpolation methods, we acquired a set of four images using a grayscale CCD camera and a rotating linear polarization filter i.e. a division of time polarization camera. The four images are recorded at 0°, 45°, 90° and 135° orientations of the linear polarization filter. All four images have spatial resolution of 1024 by 1024 pixels and they are co-registered with respect to each other due to the fixed position of the imaging sensor. In other words, at each pixel location, the optical field has been filtered and recorded with four different polarization filters offset by 45°. Using the intensity information of the four images, true polarization information, such as angle and degree of linear polarization, is computed at every pixel location.

Four low resolution (512 by 512 pixels) images for I_{0}, I_{45}, I_{90} and I_{135} are computed by decimating the respective high resolution images. The first pixel in all four low resolution images is different with respect to each other and offset by one pixel to the right and/or down in the imaging array in order to resemble the sampling pattern of a typical division of focal plane polarization imaging sensor [15]. For example, the first pixel in the I_{0} low resolution image corresponds to pixel (1,1) in the high resolution image; the first pixel in the I_{45} low resolution image corresponds to pixel (1,2) in the high resolution image; the first pixel in the I_{135} low resolution image corresponds to pixel (2,1) in the high resolution image and the first pixel in the I_{90} low resolution image corresponds to pixel (2,2) in the high resolution image. The different interpolation methods compute a high resolution images for I_{0}, I_{45}, I_{90} and I_{135} from its respective low resolution image and the resulting images are compared against the “true” high resolution image. Note, the accuracy of the recorded polarization information is dependent on the accuracy of the rotational stage with the linear polarization filters and the alignment of the linear polarization filter with respect to the imaging sensor. These optical misalignments will introduce errors in the recorded polarization information by the four original high resolution images. These errors are not critical for our set of experiments since the interpolation algorithms are implemented on the decimated images computed from the original high resolution images. The interpolated images are compared against the original high resolution images in order to estimate the accuracy of the method. Hence, any optical misalignment errors would equally affect the original high resolution images as well as the interpolated images.

The linear polarizing filter used in this experiment is an Edmund Optics M71-939 and has polarizing extinction ratios greater than 1,000. The linear polarization filter is mounted on a manual rotation stage from Newport (RSP-2T). Figure 7 presents the high resolution intensity, degree of linear polarization and angle of linear polarization image of a still black horse figure. The degree of linear polarization is presented as a grayscale image, where dark levels represent low levels of degree of linear polarization and bright levels represent high levels of degree of linear polarization. The angle of polarization is presented in false color representation, where 0° of linear polarization is presented in red color and 90° of linear polarization is presented in blue color. The images presented in Fig. 7 are a true measurement of the linear polarization properties of a scene and will be used for comparisons against the interpolated images.

#### 4.1 Image visual comparison

In Fig. 8 , three columns of images are presented. In the first column the intensity image is presented; in the second column the degree of linear polarization is presented and in the third column the angle of linear polarization is presented.

The first row of images, i.e. Fig. 8(a), presents the true polarization images; the second row, i.e. Fig. 8(b), presents the results from bilinear interpolation; the third row, i.e. Fig. 8(c), presents the results from weighted bilinear interpolation; the fourth row, i.e. Fig. 8(d), presents the results from bicubic interpolation; the fifth and sixth row, i.e. Figs. 8(e) and 8(f), present the results from bicubic spline interpolation. From this set of images, it can be observed that both degree and angle of linear polarization images for bilinear and weighted bilinear interpolation have stronger pixellation effects compared to the true polarization images (see region B in the AoP image). Details of the horsehair are lost, and the boundary between the horsehair and the background is discrete due to the large error introduced by the bilinear interpolation method (see region A in the AoP image). For the bicubic and bicubic spline interpolation images the details of the horsehair in both degree and angle of linear polarization are recovered with similar results and closely resemble the true polarization images. Also smooth varying regions of angle and degree of polarization are recovered visually with higher accuracy than the bilinear interpolated images.

#### 4.2 MSE comparison

To quantitatively compare the different interpolation methods, we utilized the concept of mean square error (MSE). The MSE is computed as shown by Eq. (19):

*I*is the true value of the target pixel (i.e. Fig. 8(a)),

_{true}(i,j)*I*is the interpolated intensity value of the target pixel (i.e. Figs. 8(b) through 8(f)), M is the number of rows in the image array and N is the number of columns in the image array. For the toy horse test images, the MSE comparison results are shown in Table 1 . Based on the evaluating results presented in Table 1, the minimum interpolation error for all images, i.e. I

_{interpolated}(i,j)_{0}, I

_{45}, I

_{90}, I

_{135}, S

_{1}, S

_{2}, intensity, degree and angle of linear polarization images, is obtained via the bicubic spline interpolation method. Similar results are obtained for the approximated 10 by 10 bicubic spline method, which might be more suitable for real-time implementation. The bilinear and weighted bilinear interpolation methods introduce larger errors in the computed higher resolution images when compared to the three bicubic interpolation methods.

#### 4.3 Results analysis

Based on both visual comparison and numerical error analysis results, three main observations can be made. First, bicubic spline interpolation shows the best results in the visual comparison, and the numerical error comparison via MSE confirms this observation. The algorithm complexity, as well as the large window area used to compute the interpolated images, allow for higher accuracy in the recovered images. The algorithm complexity introduces higher computational workload compared to the two bilinear interpolation methods. However, the bicubic algorithms are very modular and can be executed in parallel if implemented in an FPGA, GPU or a multi core processor.

Second, the approximated bicubic spline interpolation on 10 by 10 pixel window computes images with similar accuracy as the bicubic interpolation method which uses an entire row of pixels. The difference in the MSE between the full frame bicubic spline interpolation and the approximated bicubic spline interpolation are less than 4.8% for DoLP and 0.9% for AoP. Therefore, the approximated bicubic spline interpolation could be an efficient way to achieve high precision for the portable hardware implementation of a DoFP sensor.

Third, bilinear and weighted bilinear interpolation methods show relatively poor performance. A remarkable “gap” exists between the linear-based interpolation methods and the cubic-based interpolation methods for both visual results and the MSE numerical comparison. This is because the linear-based interpolation methods have relatively simple linear interpolation kernel and less original pixels involved in interpolating an unknown pixel, which could not allow them to handle the edges and abrupt variations. The weighted bilinear interpolation method produces images that look smoother when visually compared to the bilinear method but larger error in the MSE comparison. The cubic-based algorithms will always outperform the linear-based methods regardless of the extinction ratios of the polarization filters in division of focal plane polarimeters.

## 5. Conclusion

In this paper, we present five interpolation methods for four angle division of focal plane polarization imagers. The interpolation methods discussed in this paper are: bilinear, weighted bilinear, bicubic, bicubic spline and approximated bicubic spline method. A theoretical framework for all five interpolation methods is presented in the paper. The interpolation methods are first analyzed using the MTF for their frequency performance, followed by numerical analysis on a set of images obtained via a CCD camera and a linear polarization filter rotated in front of the sensor. The presented method allows for reconstruction of a true polarization image as well as constructs lower resolution images by appropriately decimating the high resolution images.

The interpolation methods are implemented on the low resolution images and compared numerically against the true high resolution polarization images. According to the results, though the bicubic based interpolation methods have more computational complexities, they show higher MTF gain and wider validation frequency bandwidth than bilinear based interpolation methods, and also outperform the bilinear based interpolation methods both visually and numerically. Furthermore, an approximated bicubic spline interpolation on 10 by 10 pixels could largely ease the portable hardware implementation while maintaining the high performance.

## Acknowledgments

The work described in this paper was supported by National Science Foundation grant number OCE-1130897, and Air Force Office of Scientific Research grant number FA9550-10-1-0121.

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