Abstract

Focal planes arrays (FPA) measure values proportional to an integrated irradiance with little sensitivity to wavelength or polarization in the optical wavelength range. The measurement of spectral properties is often achieved via a spatially varying color filter array. Recently spatially varying polarization filter arrays have been used to extract polarization information. Although measurement of color and polarization utilize separate physical methods, the underlying design and engineering methodology is linked. In this communication we derive a formalism which can be used to design any type of periodic filter array on a rectangular lattice. A complete system description can be obtained from the number of unit cells, the pixel shape, and the unit cell geometry. This formalism can be used to engineer the channel structure for any type of periodic tiling of a rectangular lattice for any type of optical filter array yielding irradiance measurements.

© 2017 Optical Society of America

Full Article  |  PDF Article
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References

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  7. R. W. Schafer and R. M. Mersereau, “Demosaicking: color filter array interpolation,” Signal Processing Magazine, IEEE 22, 44–54 (2005).
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  10. J. Jia, K. J. Barnard, and K. Hirakawa, “Fourier spectral filter array for optimal multispectral imaging,” IEEE Transactions on Image Processing 25, 1530–1543 (2016).
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  11. C. F. LaCasse, T. Ririe, R. A. Chipman, and J. S. Tyo, “Spatio-temporal modulated polarimetry,” “Proc. SPIE,”  816081600K (2011).
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  12. D. A. LeMaster and K. Hirakawa, “Improved microgrid arrangement for integrated imaging polarimeters,” Optics Letters 39, 1811–1814 (2014).
    [Crossref] [PubMed]
  13. K. Hirakawa and D. A. LeMaster, “Fourier domain design of microgrid imaging polarimeters with improved spatial resolution,” “Proc. SPIE,” 909906 (2014).
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    [Crossref]
  30. M. A. Armstrong, Groups and Symmetry (Springer Science & Business Media, 2013).

2017 (1)

A. S. Alenin, I. J. Vaughn, and J. S. Tyo, “Optimal bandwidth micropolarizer arrays,” Optics Letters 42, 458–461 (2017).
[Crossref] [PubMed]

2016 (2)

J. Jia, C. Ni, A. Sarangan, and K. Hirakawa, “Guided filter demosaicking for fourier spectral filter array,” Electronic Imaging 2016, 1–5 (2016).
[Crossref]

J. Jia, K. J. Barnard, and K. Hirakawa, “Fourier spectral filter array for optimal multispectral imaging,” IEEE Transactions on Image Processing 25, 1530–1543 (2016).
[Crossref] [PubMed]

2014 (2)

D. A. LeMaster and K. Hirakawa, “Improved microgrid arrangement for integrated imaging polarimeters,” Optics Letters 39, 1811–1814 (2014).
[Crossref] [PubMed]

A. S. Alenin and J. S. Tyo, “Generalized channeled polarimetry,” J. Opt. Soc. Am. A 31, 1013–1022 (2014).
[Crossref]

2013 (2)

N. Hagen and M. W. Kudenov, “Review of snapshot spectral imaging technologies,” Optical Engineering 52, 090901 (2013).
[Crossref]

D. Diner, F. Xu, M. Garay, J. Martonchik, B. Rheingans, S. Geier, A. Davis, B. Hancock, V. Jovanovic, M. Bull, and et al., “The airborne multiangle spectropolarimetric imager (airmspi): a new tool for aerosol and cloud remote sensing,” Atmospheric Measurement Techniques 6, 2007–2025 (2013).
[Crossref]

2011 (3)

L. Condat, “A new color filter array with optimal properties for noiseless and noisy color image acquisition,” IEEE Transactions on image processing 20, 2200–2210 (2011).
[Crossref] [PubMed]

P. Hao, Y. Li, Z. Lin, and E. Dubois, “A geometric method for optimal design of color filter arrays,” Image Processing, IEEE Transactions on 20, 709–722 (2011).
[Crossref]

C. F. LaCasse, T. Ririe, R. A. Chipman, and J. S. Tyo, “Spatio-temporal modulated polarimetry,” “Proc. SPIE,”  816081600K (2011).
[Crossref]

2008 (1)

K. Hirakawa and P. J. Wolfe, “Spatio-spectral color filter array design for optimal image recovery,” Image Processing, IEEE Transactions on 17, 1876–1890 (2008).
[Crossref]

2005 (4)

D. Alleysson, S. Süsstrunk, and J. Hérault, “Linear demosaicing inspired by the human visual system,” Image Processing, IEEE Transactions on 14, 439–449 (2005).
[Crossref]

E. Dubois, “Frequency-domain methods for demosaicking of bayer-sampled color images,” IEEE Signal Processing Letters 12, 847 (2005).
[Crossref]

R. W. Schafer and R. M. Mersereau, “Demosaicking: color filter array interpolation,” Signal Processing Magazine, IEEE 22, 44–54 (2005).
[Crossref]

R. Lukac and K. N. Plataniotis, “Color filter arrays: Design and performance analysis,” IEEE Transactions on Consumer Electronics 51, 1260–1267 (2005).
[Crossref]

2000 (2)

O. Yadid-Pecht, “Geometrical modulation transfer function for different pixel active area shapes,” Optical Engineering 39, 859–865 (2000).
[Crossref]

T. Carozzi, R. Karlsson, and J. Bergman, “Parameters characterizing electromagnetic wave polarization,” Physical Review E 61, 2024 (2000).
[Crossref]

1997 (1)

J. E. Adams, “Design of practical color filter array interpolation algorithms for digital cameras,” “Electronic Imaging,  1997,117–125 (1997).

1979 (1)

R. M. Mersereau, “The processing of hexagonally sampled two-dimensional signals,” Proceedings of the IEEE 67, 930–949 (1979).
[Crossref]

Adams, J. E.

J. E. Adams, “Design of practical color filter array interpolation algorithms for digital cameras,” “Electronic Imaging,  1997,117–125 (1997).

Alenin, A. S.

A. S. Alenin, I. J. Vaughn, and J. S. Tyo, “Optimal bandwidth micropolarizer arrays,” Optics Letters 42, 458–461 (2017).
[Crossref] [PubMed]

A. S. Alenin and J. S. Tyo, “Generalized channeled polarimetry,” J. Opt. Soc. Am. A 31, 1013–1022 (2014).
[Crossref]

I. J. Vaughn, A. S. Alenin, and J. S. Tyo, “Bounds on the micropolarizer array channel assumption,” in “SPIE Commercial + Scientific Sensing and Imaging,” 83640S (2016).

A. S. Alenin and J. S. Tyo, “Task-specific snapshot mueller matrix channeled spectropolarimeter optimization,” in “Proc. SPIE,” 836402 (2012).
[Crossref]

Alleysson, D.

D. Alleysson, S. Süsstrunk, and J. Hérault, “Linear demosaicing inspired by the human visual system,” Image Processing, IEEE Transactions on 14, 439–449 (2005).
[Crossref]

D. Alleysson, S. Süsstrunk, and J. Hérault, “Color demosaicing by estimating luminance and opponent chromatic signals in the fourier domain,” in “Color and Imaging Conference,” 2002331–336 (2002).

Armstrong, M. A.

M. A. Armstrong, Groups and Symmetry (Springer Science & Business Media, 2013).

Barnard, K. J.

J. Jia, K. J. Barnard, and K. Hirakawa, “Fourier spectral filter array for optimal multispectral imaging,” IEEE Transactions on Image Processing 25, 1530–1543 (2016).
[Crossref] [PubMed]

Bergman, J.

T. Carozzi, R. Karlsson, and J. Bergman, “Parameters characterizing electromagnetic wave polarization,” Physical Review E 61, 2024 (2000).
[Crossref]

Bikov, L.

R. Priore, J. Dougherty, O. Cohen, L. Bikov, and I. Hirsh, “Design of a miniature swir hyperspectral snapshot imager utilizing multivariate optical elements,” in “SPIE Security+ Defense,” 999205 (2016).

Bull, M.

D. Diner, F. Xu, M. Garay, J. Martonchik, B. Rheingans, S. Geier, A. Davis, B. Hancock, V. Jovanovic, M. Bull, and et al., “The airborne multiangle spectropolarimetric imager (airmspi): a new tool for aerosol and cloud remote sensing,” Atmospheric Measurement Techniques 6, 2007–2025 (2013).
[Crossref]

Carozzi, T.

T. Carozzi, R. Karlsson, and J. Bergman, “Parameters characterizing electromagnetic wave polarization,” Physical Review E 61, 2024 (2000).
[Crossref]

Chipman, R. A.

C. F. LaCasse, T. Ririe, R. A. Chipman, and J. S. Tyo, “Spatio-temporal modulated polarimetry,” “Proc. SPIE,”  816081600K (2011).
[Crossref]

Cohen, O.

R. Priore, J. Dougherty, O. Cohen, L. Bikov, and I. Hirsh, “Design of a miniature swir hyperspectral snapshot imager utilizing multivariate optical elements,” in “SPIE Security+ Defense,” 999205 (2016).

Condat, L.

L. Condat, “A new color filter array with optimal properties for noiseless and noisy color image acquisition,” IEEE Transactions on image processing 20, 2200–2210 (2011).
[Crossref] [PubMed]

Davis, A.

D. Diner, F. Xu, M. Garay, J. Martonchik, B. Rheingans, S. Geier, A. Davis, B. Hancock, V. Jovanovic, M. Bull, and et al., “The airborne multiangle spectropolarimetric imager (airmspi): a new tool for aerosol and cloud remote sensing,” Atmospheric Measurement Techniques 6, 2007–2025 (2013).
[Crossref]

Diner, D.

D. Diner, F. Xu, M. Garay, J. Martonchik, B. Rheingans, S. Geier, A. Davis, B. Hancock, V. Jovanovic, M. Bull, and et al., “The airborne multiangle spectropolarimetric imager (airmspi): a new tool for aerosol and cloud remote sensing,” Atmospheric Measurement Techniques 6, 2007–2025 (2013).
[Crossref]

Dougherty, J.

R. Priore, J. Dougherty, O. Cohen, L. Bikov, and I. Hirsh, “Design of a miniature swir hyperspectral snapshot imager utilizing multivariate optical elements,” in “SPIE Security+ Defense,” 999205 (2016).

Dubois, E.

P. Hao, Y. Li, Z. Lin, and E. Dubois, “A geometric method for optimal design of color filter arrays,” Image Processing, IEEE Transactions on 20, 709–722 (2011).
[Crossref]

E. Dubois, “Frequency-domain methods for demosaicking of bayer-sampled color images,” IEEE Signal Processing Letters 12, 847 (2005).
[Crossref]

Garay, M.

D. Diner, F. Xu, M. Garay, J. Martonchik, B. Rheingans, S. Geier, A. Davis, B. Hancock, V. Jovanovic, M. Bull, and et al., “The airborne multiangle spectropolarimetric imager (airmspi): a new tool for aerosol and cloud remote sensing,” Atmospheric Measurement Techniques 6, 2007–2025 (2013).
[Crossref]

Geier, S.

D. Diner, F. Xu, M. Garay, J. Martonchik, B. Rheingans, S. Geier, A. Davis, B. Hancock, V. Jovanovic, M. Bull, and et al., “The airborne multiangle spectropolarimetric imager (airmspi): a new tool for aerosol and cloud remote sensing,” Atmospheric Measurement Techniques 6, 2007–2025 (2013).
[Crossref]

Hagen, N.

N. Hagen and M. W. Kudenov, “Review of snapshot spectral imaging technologies,” Optical Engineering 52, 090901 (2013).
[Crossref]

Hancock, B.

D. Diner, F. Xu, M. Garay, J. Martonchik, B. Rheingans, S. Geier, A. Davis, B. Hancock, V. Jovanovic, M. Bull, and et al., “The airborne multiangle spectropolarimetric imager (airmspi): a new tool for aerosol and cloud remote sensing,” Atmospheric Measurement Techniques 6, 2007–2025 (2013).
[Crossref]

Hao, P.

P. Hao, Y. Li, Z. Lin, and E. Dubois, “A geometric method for optimal design of color filter arrays,” Image Processing, IEEE Transactions on 20, 709–722 (2011).
[Crossref]

Hérault, J.

D. Alleysson, S. Süsstrunk, and J. Hérault, “Linear demosaicing inspired by the human visual system,” Image Processing, IEEE Transactions on 14, 439–449 (2005).
[Crossref]

D. Alleysson, S. Süsstrunk, and J. Hérault, “Color demosaicing by estimating luminance and opponent chromatic signals in the fourier domain,” in “Color and Imaging Conference,” 2002331–336 (2002).

Hirakawa, K.

J. Jia, K. J. Barnard, and K. Hirakawa, “Fourier spectral filter array for optimal multispectral imaging,” IEEE Transactions on Image Processing 25, 1530–1543 (2016).
[Crossref] [PubMed]

J. Jia, C. Ni, A. Sarangan, and K. Hirakawa, “Guided filter demosaicking for fourier spectral filter array,” Electronic Imaging 2016, 1–5 (2016).
[Crossref]

D. A. LeMaster and K. Hirakawa, “Improved microgrid arrangement for integrated imaging polarimeters,” Optics Letters 39, 1811–1814 (2014).
[Crossref] [PubMed]

K. Hirakawa and P. J. Wolfe, “Spatio-spectral color filter array design for optimal image recovery,” Image Processing, IEEE Transactions on 17, 1876–1890 (2008).
[Crossref]

K. Hirakawa and D. A. LeMaster, “Fourier domain design of microgrid imaging polarimeters with improved spatial resolution,” “Proc. SPIE,” 909906 (2014).

Hirsh, I.

R. Priore, J. Dougherty, O. Cohen, L. Bikov, and I. Hirsh, “Design of a miniature swir hyperspectral snapshot imager utilizing multivariate optical elements,” in “SPIE Security+ Defense,” 999205 (2016).

Jia, J.

J. Jia, K. J. Barnard, and K. Hirakawa, “Fourier spectral filter array for optimal multispectral imaging,” IEEE Transactions on Image Processing 25, 1530–1543 (2016).
[Crossref] [PubMed]

J. Jia, C. Ni, A. Sarangan, and K. Hirakawa, “Guided filter demosaicking for fourier spectral filter array,” Electronic Imaging 2016, 1–5 (2016).
[Crossref]

Jovanovic, V.

D. Diner, F. Xu, M. Garay, J. Martonchik, B. Rheingans, S. Geier, A. Davis, B. Hancock, V. Jovanovic, M. Bull, and et al., “The airborne multiangle spectropolarimetric imager (airmspi): a new tool for aerosol and cloud remote sensing,” Atmospheric Measurement Techniques 6, 2007–2025 (2013).
[Crossref]

Karlsson, R.

T. Carozzi, R. Karlsson, and J. Bergman, “Parameters characterizing electromagnetic wave polarization,” Physical Review E 61, 2024 (2000).
[Crossref]

Kokhanovsky, A. A.

A. A. Kokhanovsky and G. H. Leeuw, Satellite Aerosol Remote Sensing Over Land (Springer, 2009).
[Crossref]

Kudenov, M. W.

N. Hagen and M. W. Kudenov, “Review of snapshot spectral imaging technologies,” Optical Engineering 52, 090901 (2013).
[Crossref]

LaCasse, C. F.

C. F. LaCasse, T. Ririe, R. A. Chipman, and J. S. Tyo, “Spatio-temporal modulated polarimetry,” “Proc. SPIE,”  816081600K (2011).
[Crossref]

Lanczos, C.

C. Lanczos, Discourse on Fourier series, vol. 255 (Oliver & BoydLondon, 1966).

Leeuw, G. H.

A. A. Kokhanovsky and G. H. Leeuw, Satellite Aerosol Remote Sensing Over Land (Springer, 2009).
[Crossref]

LeMaster, D. A.

D. A. LeMaster and K. Hirakawa, “Improved microgrid arrangement for integrated imaging polarimeters,” Optics Letters 39, 1811–1814 (2014).
[Crossref] [PubMed]

K. Hirakawa and D. A. LeMaster, “Fourier domain design of microgrid imaging polarimeters with improved spatial resolution,” “Proc. SPIE,” 909906 (2014).

Li, Y.

P. Hao, Y. Li, Z. Lin, and E. Dubois, “A geometric method for optimal design of color filter arrays,” Image Processing, IEEE Transactions on 20, 709–722 (2011).
[Crossref]

Lin, Z.

P. Hao, Y. Li, Z. Lin, and E. Dubois, “A geometric method for optimal design of color filter arrays,” Image Processing, IEEE Transactions on 20, 709–722 (2011).
[Crossref]

Lukac, R.

R. Lukac and K. N. Plataniotis, “Color filter arrays: Design and performance analysis,” IEEE Transactions on Consumer Electronics 51, 1260–1267 (2005).
[Crossref]

Martonchik, J.

D. Diner, F. Xu, M. Garay, J. Martonchik, B. Rheingans, S. Geier, A. Davis, B. Hancock, V. Jovanovic, M. Bull, and et al., “The airborne multiangle spectropolarimetric imager (airmspi): a new tool for aerosol and cloud remote sensing,” Atmospheric Measurement Techniques 6, 2007–2025 (2013).
[Crossref]

Mersereau, R. M.

R. W. Schafer and R. M. Mersereau, “Demosaicking: color filter array interpolation,” Signal Processing Magazine, IEEE 22, 44–54 (2005).
[Crossref]

R. M. Mersereau, “The processing of hexagonally sampled two-dimensional signals,” Proceedings of the IEEE 67, 930–949 (1979).
[Crossref]

Ni, C.

J. Jia, C. Ni, A. Sarangan, and K. Hirakawa, “Guided filter demosaicking for fourier spectral filter array,” Electronic Imaging 2016, 1–5 (2016).
[Crossref]

Plataniotis, K. N.

R. Lukac and K. N. Plataniotis, “Color filter arrays: Design and performance analysis,” IEEE Transactions on Consumer Electronics 51, 1260–1267 (2005).
[Crossref]

Priore, R.

R. Priore, J. Dougherty, O. Cohen, L. Bikov, and I. Hirsh, “Design of a miniature swir hyperspectral snapshot imager utilizing multivariate optical elements,” in “SPIE Security+ Defense,” 999205 (2016).

Rheingans, B.

D. Diner, F. Xu, M. Garay, J. Martonchik, B. Rheingans, S. Geier, A. Davis, B. Hancock, V. Jovanovic, M. Bull, and et al., “The airborne multiangle spectropolarimetric imager (airmspi): a new tool for aerosol and cloud remote sensing,” Atmospheric Measurement Techniques 6, 2007–2025 (2013).
[Crossref]

Ririe, T.

C. F. LaCasse, T. Ririe, R. A. Chipman, and J. S. Tyo, “Spatio-temporal modulated polarimetry,” “Proc. SPIE,”  816081600K (2011).
[Crossref]

Rodríguez-Herrera, O. G.

I. J. Vaughn, O. G. Rodríguez-Herrera, M. Xu, and J. S. Tyo, “Bandwidth and crosstalk considerations in a spatio-temporally modulated polarimeter,” in “Proc. SPIE,” 9613961305 (2015).
[Crossref]

I. J. Vaughn, O. G. Rodríguez-Herrera, M. Xu, and J. S. Tyo, “A portable imaging mueller matrix polarimeter based on a spatio-temporal modulation approach: theory and implementation,” in “Proc. SPIE,” 9613961312 (2015), vol. 9613.
[Crossref]

Sarangan, A.

J. Jia, C. Ni, A. Sarangan, and K. Hirakawa, “Guided filter demosaicking for fourier spectral filter array,” Electronic Imaging 2016, 1–5 (2016).
[Crossref]

Schafer, R. W.

R. W. Schafer and R. M. Mersereau, “Demosaicking: color filter array interpolation,” Signal Processing Magazine, IEEE 22, 44–54 (2005).
[Crossref]

Süsstrunk, S.

D. Alleysson, S. Süsstrunk, and J. Hérault, “Linear demosaicing inspired by the human visual system,” Image Processing, IEEE Transactions on 14, 439–449 (2005).
[Crossref]

D. Alleysson, S. Süsstrunk, and J. Hérault, “Color demosaicing by estimating luminance and opponent chromatic signals in the fourier domain,” in “Color and Imaging Conference,” 2002331–336 (2002).

Tyo, J. S.

A. S. Alenin, I. J. Vaughn, and J. S. Tyo, “Optimal bandwidth micropolarizer arrays,” Optics Letters 42, 458–461 (2017).
[Crossref] [PubMed]

A. S. Alenin and J. S. Tyo, “Generalized channeled polarimetry,” J. Opt. Soc. Am. A 31, 1013–1022 (2014).
[Crossref]

C. F. LaCasse, T. Ririe, R. A. Chipman, and J. S. Tyo, “Spatio-temporal modulated polarimetry,” “Proc. SPIE,”  816081600K (2011).
[Crossref]

I. J. Vaughn, O. G. Rodríguez-Herrera, M. Xu, and J. S. Tyo, “A portable imaging mueller matrix polarimeter based on a spatio-temporal modulation approach: theory and implementation,” in “Proc. SPIE,” 9613961312 (2015), vol. 9613.
[Crossref]

A. S. Alenin and J. S. Tyo, “Task-specific snapshot mueller matrix channeled spectropolarimeter optimization,” in “Proc. SPIE,” 836402 (2012).
[Crossref]

I. J. Vaughn, A. S. Alenin, and J. S. Tyo, “Bounds on the micropolarizer array channel assumption,” in “SPIE Commercial + Scientific Sensing and Imaging,” 83640S (2016).

I. J. Vaughn, O. G. Rodríguez-Herrera, M. Xu, and J. S. Tyo, “Bandwidth and crosstalk considerations in a spatio-temporally modulated polarimeter,” in “Proc. SPIE,” 9613961305 (2015).
[Crossref]

Vaughn, I. J.

A. S. Alenin, I. J. Vaughn, and J. S. Tyo, “Optimal bandwidth micropolarizer arrays,” Optics Letters 42, 458–461 (2017).
[Crossref] [PubMed]

I. J. Vaughn, A. S. Alenin, and J. S. Tyo, “Bounds on the micropolarizer array channel assumption,” in “SPIE Commercial + Scientific Sensing and Imaging,” 83640S (2016).

I. J. Vaughn, O. G. Rodríguez-Herrera, M. Xu, and J. S. Tyo, “A portable imaging mueller matrix polarimeter based on a spatio-temporal modulation approach: theory and implementation,” in “Proc. SPIE,” 9613961312 (2015), vol. 9613.
[Crossref]

I. J. Vaughn, O. G. Rodríguez-Herrera, M. Xu, and J. S. Tyo, “Bandwidth and crosstalk considerations in a spatio-temporally modulated polarimeter,” in “Proc. SPIE,” 9613961305 (2015).
[Crossref]

Wolfe, P. J.

K. Hirakawa and P. J. Wolfe, “Spatio-spectral color filter array design for optimal image recovery,” Image Processing, IEEE Transactions on 17, 1876–1890 (2008).
[Crossref]

Xu, F.

D. Diner, F. Xu, M. Garay, J. Martonchik, B. Rheingans, S. Geier, A. Davis, B. Hancock, V. Jovanovic, M. Bull, and et al., “The airborne multiangle spectropolarimetric imager (airmspi): a new tool for aerosol and cloud remote sensing,” Atmospheric Measurement Techniques 6, 2007–2025 (2013).
[Crossref]

Xu, M.

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Supplementary Material (1)

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Figures (11)

Fig. 1
Fig. 1

Arbitrary filter shape represented by the function h(x, y), which is assumed to be compact, but not necessarily an indicator (binary) function. Notice that it is centered on (0, 0). Adapted from [17].

Fig. 2
Fig. 2

An example of a 5-band transmission filter element for wavelengths 443nm, 520nm, 550nm, 600nm, 660nm. This 5-band transmission is for a single filter element, corrected for the spectral integration and the quantum efficiency at the detector assuming the camera is an Andor Alta F6. This type of multi-band filter is currently being manufactured by companies like PIXELTEQ and Alluxa.

Fig. 3
Fig. 3

An example of sub-sum selection from a 6-band spectral CFA. The green sub-sum is denoted by the black circles. Each element type is periodic over the array. an,p (0, 0) through an,p (2, 1) represent the 6 different periodic sets over the FPA, while a0,0(−3, 1), a0,0(0, 1), a0,0(3, 1), a0,0(0, 3), a0,0(3, 3), · · · represent the green sub-sum.

Fig. 4
Fig. 4

The set of possible channel locations and values available for a 2 × 3 unit cell. The ξ, η-axes (horizontal and vertical respectively) have ranges in [ 1 2 , 1 2 ) in cycles per pixel showing the Nyquist bandwidth square. The circles are located at the channel centers, the magnitude of each channel is 1, and the clocking angle and color indicate the angle of the complex value of each channel.

Fig. 5
Fig. 5

Examples of approximate δ-functions (channels) for different unit cell sizes for the (0, 0) element assuming maximal sampling. Note the 3 × 3 unit cell has approximate δ-functions which lie completely within the Nyquist square. This representation conveys the physical structure available in the Fourier domain, whereas the notation used in Fig. 4 is conceptual.

Fig. 6
Fig. 6

Human perception of the unit cells for the various CFA designs. Generated via our software (Ref. [28])

Fig. 7
Fig. 7

Red (top row), green (center row), and blue (bottom row) channel descriptions for the Hirakawa, Condat, Lukac, and Vaughn 4 × 2 and 8 × 2 color filter array designs.

Fig. 8
Fig. 8

Synthetic resolution target truth image. This image varies between 0 and 1 for each color and is processed as if being incident on an ideal sensor. The white square indicates the region of interest.

Fig. 9
Fig. 9

Fourier domain filters used for the synthetic image reconstruction. Data shown is for the synthetic image and is log scaled on the magnitude of the Fourier domain data. Note that the Hirakawa filters overlap, resulting in the high filter display values there, however, Fourier domain filter sets are used individually for the reconstruction.

Fig. 10
Fig. 10

Reconstructions of the lighthouse image.

Fig. 11
Fig. 11

Reconstructions of the ROI of the synthetic image.

Tables (1)

Tables Icon

Table 1 Table of total MSE values for various unit cells. Note that the lighthouse and synthetic MSEs are not comparable due to the original images having different image values for each color, from 0 to 255 and from 0 to 1 respectively.

Equations (23)

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I ( x , y ) = h ( x x j , y y ) [ a 0 ( x , y ) + a 1 ( x , y ) + + a n ( x , y ) ] 𝔡 ( x , y ) = h ( x x j , y y ) 𝔪 ( x , y ) 𝔡 ( x , y ) ,
a 0 = a 1 + a 0 , a 1 = a 2 + a 0 , a 2 = a 3 + a 0 ,
I ( x , y ) = h ( x x j , y y ) [ a 0 ( x j , y ) 𝔡 ( x , y ) + a 1 ( x j , y ) 𝔡 ( x , y ) + + a n ( x j , y ) 𝔡 ( x , y ) ]
h ( x x j , y y ) [ a 0 ( x j , y ) + a n ( x j , y ) ] 𝔡 ^
e 2 π i ( x j ξ + y η ) H ( ξ , η ) [ a 0 ( x j , y ) + a n ( x j , y ) ] 𝔡 ^
a 0 = [ a 0 , 0 0 0 0 0 ] T , a 1 = [ 0 a 1 , 1 0 0 0 ] T , a 2 = [ 0 0 , a 2 , 2 , 0 0 ] T , a 3 = [ 0 0 0 a 3 , 3 0 ] T , a 4 = [ 0 0 0 0 a 4 , 4 ] T ,
H ( ξ , η ) j e 2 π i ( ξ x j + η y ) [ a 0 ( x j , y ) + a n ( x j , y ) ] 𝔡 ^
p = 0 k H ( ξ , η ) n ˜ = N a 0 N a 0 m ˜ = M a 0 M a 0 e 2 π i ( n ˜ P x ξ + m ˜ P y η ) a 0 , p ( n ˜ P x , m ˜ P y ) + p = 0 k H ( ξ , η ) n ˜ = N a 1 N a 1 m ˜ = M a 1 M a 1 e 2 π i ( n ˜ P x ξ + m ˜ P y η ) a 1 , p ( n ˜ P x , m ˜ P y ) + p = 0 k H ( ξ , η ) n ˜ = N a n N a n m ˜ = M a n M a n e 2 π i ( n ˜ P x ξ + m ˜ P y η ) a n , p ( n ˜ P x , m ˜ P y )
n ˜ = N N m ˜ = M M e 2 π i ( n ˜ P x ξ + m ˜ P y η ) a n , p ( n ˜ P x , m ˜ P y )
D n , p , ξ , η ( j P x , P y ) = a n , p ( 0 , 0 ) S ( 0 , 0 ) + + a n , p ( τ x P x , 0 ) S ( τ x P x , 0 ) + a n , p ( 0 , τ y P y ) S ( 0 , τ y P y ) + + a n , p ( τ x P x , τ y P y ) S ( τ x P x , τ y P y )
S ( j P x , P y ) = n ˜ = N j P x , P y N j P x , P y M j P x , P y M j P x , P y e 2 π i ( [ n ˜ T x P x + O j P x P x ] ξ + [ m ˜ T y P y + O P y P y ] η )
D n , p , ξ , η ( j P x , P y ) = a n , p ( j P x , P y ) D K ξ ( j P x , P y ) D K η ( j P x , P y )
D K ξ ( j P x , P y ) = e π i ( N j , P x , P y N j P x , P y 2 O j P x T x ) T x P x ξ { sin [ T x P x ( N j P x , P y + N j P x , P y + 1 ) π ξ ] sin ( T x P x π ξ ) }
D K η ( j P x , P y ) = e π i ( N j , P x , P y M j P x , P y 2 O P y T y ) T y P y η { sin [ T y P y ( N j P x , P y + N j P x , P y + 1 ) π η ] sin ( T y P y π η ) }
H ( ξ , η ) j = 0 T x 1 = 0 T y 1 p = 0 k D 0 , p , ξ , η ( j P x , P y ) + H ( ξ , η ) j = 0 T x 1 = 0 T y 1 p = 0 k D 1 , p , ξ , η ( j P x , P y ) + H ( ξ , η ) j = 0 T x 1 = 0 T y 1 p = 0 k D n , p , ξ , η ( j P x , P y )
M = C [ 1 C m 0 1 C m 1 1 C m k ] ,
y = Mx x = M + y ,
MSE = 1 NM g g ^ 2
0.5 ξ = k N N 0.5 , 0.5 η = k M M 0.5 , P x = P y = 1 , a n , k ( j , ) = 1 .
D n , k , ξ , η ( j , ) = exp ( π i ( N j , N j , 2 O j T x ) T x ξ ) exp ( π i ( M j , M j , 2 O T y ) T y η ) { sin [ T x ( N j , + N j , + 1 ) π ξ ] sin ( T x π ξ ) } { sin [ T y ( M j , + M j , + 1 ) π ξ ] sin ( T y π η ) }
D n , k , ξ , η ( j , ) = U x U y e ( 2 π i O j k N N ) e ( 2 π i O k M M ) ,
D n , k , ξ , η ( j , ) = U x U y e ( 2 π i O j k N N ) e ( 2 π i O k M M ) .
M = C [ 1 1 2 i cos π 6 1 2 + i cos π 6 1 1 2 + i cos π 6 1 2 i cos π 6 1 1 1 1 1 1 1 1 2 + i cos π 6 1 2 i cos π 6 1 1 2 i cos π 6 1 2 + i cos π 6 1 1 2 i cos π 6 1 2 + i cos π 6 1 1 2 i cos π 6 1 2 + i cos π 6 1 1 2 i cos π 6 1 2 + i cos π 6 1 1 2 i cos π 6 1 2 + i cos π 6 1 1 1 1 1 1 1 1 2 + i cos π 6 1 2 i cos π 6 1 1 2 + i cos π 6 1 2 i cos π 6 1 1 2 i cos π 6 1 2 + i cos π 6 1 1 2 + i cos π 6 1 2 i cos π 6 1 1 2 i cos π 6 1 2 + i cos π 6 1 1 2 + i cos π 6 1 2 i cos π 6 1 1 1 1 1 1 1 1 2 + i cos π 6 1 2 i cos π 6 1 1 2 i cos π 6 1 2 + i cos π 6 ] .

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