## Bilinear and bicubic interpolation methods for division of focal plane polarimeters |

Optics Express, Vol. 19, Issue 27, pp. 26161-26173 (2011)

http://dx.doi.org/10.1364/OE.19.026161

Acrobat PDF (1226 KB)

### 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

2. J. E. Solomon, “Polarization imaging,” Appl. Opt. **20**(9), 1537–1544 (1981). [CrossRef] [PubMed]

3. R. M. A. Azzam, “Arrangement of four photodetectors for measuring the state of polarization of light,” Opt. Lett. **10**(7), 309–311 (1985). [CrossRef] [PubMed]

6. M. W. Kudenov, L. J. Pezzaniti, and G. R. Gerhart, “Microbolometer-infrared imaging Stokes polarimeter,” Opt. Eng. **48**(6), 063201 (2009). [CrossRef]

7. M. E. Roche, D. B. Chenault, J. P. Vaden, A. Lompado, D. Voelz, T. J. Schulz, R. N. Givens, and V. L. Gamiz, “Synthetic aperture imaging polarimeter,” Proc. SPIE **7672**, 767206, 767206-12 (2010). [CrossRef]

8. J. L. Pezzaniti and D. B. Chenault, “A division of aperture MWIR imaging polarimeter,” Proc. SPIE **5888**, 58880V, 58880V-12 (2005). [CrossRef]

9. C. K. Harnett and H. G. Craighead, “Liquid-crystal micropolarizer array for polarization-difference imaging,” Appl. Opt. **41**(7), 1291–1296 (2002). [CrossRef] [PubMed]

16. R. Perkins and V. Gruev, “Signal-to-noise analysis of Stokes parameters in division of focal plane polarimeters,” Opt. Express **18**(25), 25815–25824 (2010). [CrossRef] [PubMed]

17. 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]

^{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

20. A. Goldberg, T. Fischer, S. Kennerly, S. Wang, M. Sundaram, P. Uppal, M. Winn, G. Milne, and M. Stevens, “Dual band QWIP MWIR/LWIR focal plane array test results,” Proc. SPIE **4028**, 276–287 (2000). [CrossRef]

15. V. Gruev, R. Perkins, and T. York, “CCD polarization imaging sensor with aluminum nanowire optical filters,” Opt. Express **18**(18), 19087–19094 (2010). [CrossRef] [PubMed]

20. A. Goldberg, T. Fischer, S. Kennerly, S. Wang, M. Sundaram, P. Uppal, M. Winn, G. Milne, and M. Stevens, “Dual band QWIP MWIR/LWIR focal plane array test results,” Proc. SPIE **4028**, 276–287 (2000). [CrossRef]

23. V. Gruev, J. Van der Spiegel, and N. Engheta, “Dual-tier thin film polymer polarization imaging sensor,” Opt. Express **18**(18), 19292–19303 (2010). [CrossRef] [PubMed]

15. V. Gruev, R. Perkins, and T. York, “CCD polarization imaging sensor with aluminum nanowire optical filters,” Opt. Express **18**(18), 19087–19094 (2010). [CrossRef] [PubMed]

_{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

15. V. Gruev, R. Perkins, and T. York, “CCD polarization imaging sensor with aluminum nanowire optical filters,” Opt. Express **18**(18), 19087–19094 (2010). [CrossRef] [PubMed]

**18**(18), 19087–19094 (2010). [CrossRef] [PubMed]

29. 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]

29. 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]

## 2. Interpolation methods

### 2.1 Bilinear interpolation methods

### 2.2 Weighted bilinear interpolation method

29. 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]

*A*are closer to the estimated pixel and have higher weighting than the pixel convolved with the coefficient B.

### 2.3 Bicubic spline interpolation method

31. H. Hou and H. Andrews, “Cubic splines for image interpolation and digital filtering,” IEEE Trans. Acoust. Speech Signal Process. **26**(6), 508–517 (1978). [CrossRef]

*f*between two neighboring pixels (

_{i}(x),*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):where

_{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}*I(i,j)*is the intensity value of a pixel at location

*(i,j)*and

*M*is the second derivative of the function

_{i}*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}

*f*should be continuous and is presented by Eq. (13):

_{i}(x)*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

*a*,

_{i}*b*,

_{i}*c*and

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

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

_{i}(x)*M*, the system of

_{1}= M_{N}= 0*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.

33. T. York, S. Powell, and V. Gruev, “A comparison of polarization image processing across different platforms,” Proc. SPIE **8160**, 816004, 816004-7 (2011). [CrossRef]

*M*. 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.

_{0}= M_{10}= 0### 2.4 Bicubic interpolation method

34. W. S. Russell, “Polynomial interpolation schemes for internal derivative distributions on structured grids,” Appl. Numer. Math. **17**(2), 129–171 (1995). [CrossRef]

*f*, described by a third order polynomial is presented by Eq. (14).

_{i}(x,y)*a*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.

_{ij}## 3. Modulation transfer function evaluation

36. http://www.kodak.com.

*f*and

_{x}*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}_{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*), the Fourier image has two main frequency peaks at positions (

_{x},f_{y}*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})*f*and

_{x}*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}*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

_{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

**18**(18), 19087–19094 (2010). [CrossRef] [PubMed]

_{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.

### 4.1 Image visual comparison

### 4.2 MSE comparison

*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

## 5. Conclusion

## Acknowledgments

## References and links

1. | R. Walraven, “Polarization imagery,” Opt. Eng. |

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

3. | R. M. A. Azzam, “Arrangement of four photodetectors for measuring the state of polarization of light,” Opt. Lett. |

4. | C. A. Farlow, D. B. Chenault, K. D. Spradley, M. G. Gulley, M. W. Jones, and C. M. Persons, “Imaging polarimeter development and application,” Proc. SPIE |

5. | J. D. Barter, P. H. Y. Lee, and H. R. Thompson, “Stokes parameter imaging of scattering surfaces,” Proc. SPIE |

6. | M. W. Kudenov, L. J. Pezzaniti, and G. R. Gerhart, “Microbolometer-infrared imaging Stokes polarimeter,” Opt. Eng. |

7. | M. E. Roche, D. B. Chenault, J. P. Vaden, A. Lompado, D. Voelz, T. J. Schulz, R. N. Givens, and V. L. Gamiz, “Synthetic aperture imaging polarimeter,” Proc. SPIE |

8. | J. L. Pezzaniti and D. B. Chenault, “A division of aperture MWIR imaging polarimeter,” Proc. SPIE |

9. | C. K. Harnett and H. G. Craighead, “Liquid-crystal micropolarizer array for polarization-difference imaging,” Appl. Opt. |

10. | G. P. Nordin, J. T. Meier, P. C. Deguzman, and M. Jones, “Diffractive optical element for Stokes vector measurement with a focal plane array,” Proc. SPIE |

11. | M. Sarkar, D. San Segundo Bello, C. van Hoof, and A. Theuwissen, “Integrated polarization analyzing CMOS image sensor for material classification,” IEEE Sens. J. |

12. | J. S. Tyo, “Hybrid division of aperture/division of a focal-plane polarimeter for real-time polarization imagery without an instantaneous field-of-view error,” Opt. Lett. |

13. | M. Momeni and A. H. Titus, “An analog VLSI chip emulating polarization vision of Octopus retina,” IEEE Trans. Neural Netw. |

14. | V. Gruev, J. Van der Spiegel, and N. Engheta, “Dual-tier thin film polymer polarization imaging sensor,” Opt. Express |

15. | V. Gruev, R. Perkins, and T. York, “CCD polarization imaging sensor with aluminum nanowire optical filters,” Opt. Express |

16. | R. Perkins and V. Gruev, “Signal-to-noise analysis of Stokes parameters in division of focal plane polarimeters,” Opt. Express |

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

18. | D. Goldstein, |

19. | J. Wang, F. Walters, X. Liu, P. Sciortino, and X. Deng, “High-performance, large area, deep ultraviolet to infrared polarizers based on 40 nm line/78 nm space nanowire grids,” Appl. Phys. Lett. |

20. | A. Goldberg, T. Fischer, S. Kennerly, S. Wang, M. Sundaram, P. Uppal, M. Winn, G. Milne, and M. Stevens, “Dual band QWIP MWIR/LWIR focal plane array test results,” Proc. SPIE |

21. | T. Weber, T. Käsebier, E. B. Kley, and A. Tünnermann, “Broadband iridium wire grid polarizer for UV applications,” Opt. Lett. |

22. | J. G. Ok, H. J. Park, M. K. Kwak, C. A. Pina-Hernandez, S. H. Ahn, and L. J. Guo, “Continuous patterning of nanogratings by nanochannel-guided lithography on liquid resists,” Adv. Mater. (Deerfield Beach Fla.) |

23. | V. Gruev, J. Van der Spiegel, and N. Engheta, “Dual-tier thin film polymer polarization imaging sensor,” Opt. Express |

24. | B. E. Bayer, “Color imaging array,” U.S. Patent 3,971,065 (1976). |

25. | R. Kimmel, “Demosaicing: image reconstruction from color CCD samples,” IEEE Trans. Image Process. |

26. | B. K. Gunturk, Y. Altunbasak, and R. M. Mersereau, “Color plane interpolation using alternating projections,” IEEE Trans. Image Process. |

27. | B. K. Gunturk, J. Glotzbach, Y. Altunbasak, R. W. Schafer, and R. M. Mersereau, “Demosaicking: color filter array interpolation,” IEEE Signal Process. Mag. |

28. | R. C. Gonzales and R. E. Woods, |

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

30. | B. M. Ratliff, J. K. Boger, M. P. Fetrow, J. S. Tyo, and W. T. Black, “Image processing methods to compensate for IFOV errors in microgrid imaging polarimeters,” Proc. SPIE |

31. | H. Hou and H. Andrews, “Cubic splines for image interpolation and digital filtering,” IEEE Trans. Acoust. Speech Signal Process. |

32. | R. L. Burden and J. D. Faires, |

33. | T. York, S. Powell, and V. Gruev, “A comparison of polarization image processing across different platforms,” Proc. SPIE |

34. | W. S. Russell, “Polynomial interpolation schemes for internal derivative distributions on structured grids,” Appl. Numer. Math. |

35. | G. D. Boreman, |

36. |

**OCIS Codes**

(120.5410) Instrumentation, measurement, and metrology : Polarimetry

(230.5440) Optical devices : Polarization-selective devices

(260.5430) Physical optics : Polarization

(110.5405) Imaging systems : Polarimetric imaging

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: October 27, 2011

Revised Manuscript: November 20, 2011

Manuscript Accepted: November 22, 2011

Published: December 7, 2011

**Citation**

Shengkui Gao and Viktor Gruev, "Bilinear and bicubic interpolation methods for division of focal plane polarimeters," Opt. Express **19**, 26161-26173 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-27-26161

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### References

- R. Walraven, “Polarization imagery,” Opt. Eng.20, 14–18 (1981).
- J. E. Solomon, “Polarization imaging,” Appl. Opt.20(9), 1537–1544 (1981). [CrossRef] [PubMed]
- R. M. A. Azzam, “Arrangement of four photodetectors for measuring the state of polarization of light,” Opt. Lett.10(7), 309–311 (1985). [CrossRef] [PubMed]
- C. A. Farlow, D. B. Chenault, K. D. Spradley, M. G. Gulley, M. W. Jones, and C. M. Persons, “Imaging polarimeter development and application,” Proc. SPIE4819, 118–125 (2001).
- J. D. Barter, P. H. Y. Lee, and H. R. Thompson, “Stokes parameter imaging of scattering surfaces,” Proc. SPIE3121, 314–320 (1997). [CrossRef]
- M. W. Kudenov, L. J. Pezzaniti, and G. R. Gerhart, “Microbolometer-infrared imaging Stokes polarimeter,” Opt. Eng.48(6), 063201 (2009). [CrossRef]
- M. E. Roche, D. B. Chenault, J. P. Vaden, A. Lompado, D. Voelz, T. J. Schulz, R. N. Givens, and V. L. Gamiz, “Synthetic aperture imaging polarimeter,” Proc. SPIE7672, 767206, 767206-12 (2010). [CrossRef]
- J. L. Pezzaniti and D. B. Chenault, “A division of aperture MWIR imaging polarimeter,” Proc. SPIE5888, 58880V, 58880V-12 (2005). [CrossRef]
- C. K. Harnett and H. G. Craighead, “Liquid-crystal micropolarizer array for polarization-difference imaging,” Appl. Opt.41(7), 1291–1296 (2002). [CrossRef] [PubMed]
- G. P. Nordin, J. T. Meier, P. C. Deguzman, and M. Jones, “Diffractive optical element for Stokes vector measurement with a focal plane array,” Proc. SPIE3754, 169–177 (1999). [CrossRef]
- M. Sarkar, D. San Segundo Bello, C. van Hoof, and A. Theuwissen, “Integrated polarization analyzing CMOS image sensor for material classification,” IEEE Sens. J.11(8), 1692–1703 (2011). [CrossRef]
- J. S. Tyo, “Hybrid division of aperture/division of a focal-plane polarimeter for real-time polarization imagery without an instantaneous field-of-view error,” Opt. Lett.31(20), 2984–2986 (2006). [CrossRef] [PubMed]
- M. Momeni and A. H. Titus, “An analog VLSI chip emulating polarization vision of Octopus retina,” IEEE Trans. Neural Netw.17(1), 222–232 (2006). [CrossRef] [PubMed]
- V. Gruev, J. Van der Spiegel, and N. Engheta, “Dual-tier thin film polymer polarization imaging sensor,” Opt. Express18(18), 19292–19303 (2010). [CrossRef] [PubMed]
- V. Gruev, R. Perkins, and T. York, “CCD polarization imaging sensor with aluminum nanowire optical filters,” Opt. Express18(18), 19087–19094 (2010). [CrossRef] [PubMed]
- R. Perkins and V. Gruev, “Signal-to-noise analysis of Stokes parameters in division of focal plane polarimeters,” Opt. Express18(25), 25815–25824 (2010). [CrossRef] [PubMed]
- 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]
- D. Goldstein, Polarized Light (Marcel Dekker, 2003).
- J. Wang, F. Walters, X. Liu, P. Sciortino, and X. Deng, “High-performance, large area, deep ultraviolet to infrared polarizers based on 40 nm line/78 nm space nanowire grids,” Appl. Phys. Lett.90, 611041 (2007).
- A. Goldberg, T. Fischer, S. Kennerly, S. Wang, M. Sundaram, P. Uppal, M. Winn, G. Milne, and M. Stevens, “Dual band QWIP MWIR/LWIR focal plane array test results,” Proc. SPIE4028, 276–287 (2000). [CrossRef]
- T. Weber, T. Käsebier, E. B. Kley, and A. Tünnermann, “Broadband iridium wire grid polarizer for UV applications,” Opt. Lett.36(4), 445–447 (2011). [CrossRef] [PubMed]
- J. G. Ok, H. J. Park, M. K. Kwak, C. A. Pina-Hernandez, S. H. Ahn, and L. J. Guo, “Continuous patterning of nanogratings by nanochannel-guided lithography on liquid resists,” Adv. Mater. (Deerfield Beach Fla.)23(38), 4444–4448 (2011). [CrossRef] [PubMed]
- V. Gruev, J. Van der Spiegel, and N. Engheta, “Dual-tier thin film polymer polarization imaging sensor,” Opt. Express18(18), 19292–19303 (2010). [CrossRef] [PubMed]
- B. E. Bayer, “Color imaging array,” U.S. Patent 3,971,065 (1976).
- R. Kimmel, “Demosaicing: image reconstruction from color CCD samples,” IEEE Trans. Image Process.8(9), 1221–1228 (1999). [CrossRef] [PubMed]
- B. K. Gunturk, Y. Altunbasak, and R. M. Mersereau, “Color plane interpolation using alternating projections,” IEEE Trans. Image Process.11(9), 997–1013 (2002). [CrossRef] [PubMed]
- B. K. Gunturk, J. Glotzbach, Y. Altunbasak, R. W. Schafer, and R. M. Mersereau, “Demosaicking: color filter array interpolation,” IEEE Signal Process. Mag.22(1), 44–54 (2005). [CrossRef]
- R. C. Gonzales and R. E. Woods, Digital Image Processing (Prentice Hall, 2002).
- B. M. Ratliff, C. F. LaCasse, and J. S. Tyo, “Interpolation strategies for reducing IFOV artifacts in microgrid polarimeter imagery,” Opt. Express17(11), 9112–9125 (2009). [CrossRef] [PubMed]
- B. M. Ratliff, J. K. Boger, M. P. Fetrow, J. S. Tyo, and W. T. Black, “Image processing methods to compensate for IFOV errors in microgrid imaging polarimeters,” Proc. SPIE6240, 6240OE (2006).
- H. Hou and H. Andrews, “Cubic splines for image interpolation and digital filtering,” IEEE Trans. Acoust. Speech Signal Process.26(6), 508–517 (1978). [CrossRef]
- R. L. Burden and J. D. Faires, Numerical Analysis (Brooks Cole, 2010).
- T. York, S. Powell, and V. Gruev, “A comparison of polarization image processing across different platforms,” Proc. SPIE8160, 816004, 816004-7 (2011). [CrossRef]
- W. S. Russell, “Polynomial interpolation schemes for internal derivative distributions on structured grids,” Appl. Numer. Math.17(2), 129–171 (1995). [CrossRef]
- G. D. Boreman, Modulation Transfer Function in Optical and Electro-Optical Systems (SPIE, 2001).
- http://www.kodak.com .

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