Abstract

Compressive sensing is a powerful sensing and reconstruction framework for recovering high dimensional signals with only a handful of observations and for spectral imaging, compressive sensing offers a novel method of multispectral imaging. Specifically, the coded aperture snapshot spectral imager (CASSI) system has been demonstrated to produce multi-spectral data cubes color images from a single snapshot taken by a monochrome image sensor. In this paper, we expand the theoretical framework of CASSI to include the spectral sensitivity of the image sensor pixels to account for color and then investigate the impact on image quality using either a traditional color image sensor that spatially multiplexes red, green, and blue light filters or a novel Foveon image sensor which stacks red, green, and blue pixels on top of one another.

© 2015 Optical Society of America

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References

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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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2015 (1)

2014 (3)

G. R. Arce, D. J. Brady, L. Carin, H. Arguello, and D. S. Kittle, “An introduction to compressive coded aperture spectral imaging,” IEEE Signal Processing Magazine,  31(1), 105–115 (2014).
[Crossref]

H. Arguello and G. R. Arce, “Colored coded aperture design by concentration of measure in compressive spectral imaging,” IEEE Transactions on Image Processing,  23(4), 1896–1908 (2014).
[Crossref] [PubMed]

C. Ma, X. Cao, R. Wu, and Q. Dai, “Content-adaptive high-resolution hyperspectral video acquisition with a hybrid camera system,” Opt. Lett. 39(4), 937–940 (2014).
[Crossref] [PubMed]

2013 (5)

2012 (2)

M. F. Duarte and R. G. Baraniuk, “Kronecker compressive sensing,” IEEE Transactions on Image Processing,  21(2), 494–504 (2012).
[Crossref]

G. Riutort-Mayol, A. Marqus-Mateu, A. Segu, and J. Lerma, “Grey level and noise evaluation of a Foveon X3 image sensor: A statistical and experimental approach”, Sensors,  12(8), 10339–10368 (2012).
[Crossref] [PubMed]

2011 (3)

Y. Wu, I. O. Mirza, G. R. Arce, and D. W. Prather, “Development of a digital-micromirror-device-based multishot snapshot spectral imaging system,” Opt. Lett 36, 2692–2694 (2011).
[Crossref] [PubMed]

J. L. Paredes and G. R. Arce, “Compressive sensing signal reconstruction by weighted median regression estimates,” IEEE Transactions on Signal Processing,  59(6), 2585–2601 (2011).
[Crossref]

M. A. T. Figueiredo, M. V. Afonso, and J. M. Bioucas-Dias, “An augmented lagrangian approach to the constrained optimization formulation of imaging inverse problems,” IEEE Transactions on Image Processing,  20(3), 681–695 (2011).
[Crossref]

2010 (2)

Z. Wang and G. R. Arce, “Variable density compressed image sampling,” IEEE Transactions on Image Processing,  19(1), 264–270 (2010).
[Crossref]

D. Kittle, K. Choi, A. A. Wagadarikar, and D. J. Brady, “Multiframe image estimation for coded aperture snapshot spectral imagers,” Appl. Opt.,  49, 6824–6833 (2010).
[Crossref] [PubMed]

2009 (1)

Y. Rivenson and A. Stern, “Compressed imaging with a separable sensing operator,” IEEE Signal Processing Letters,  16, 449–452 (2009).
[Crossref]

2007 (3)

M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE Journal of Selected Topics in Signal Processing,  1(4), 586–597 (2007).
[Crossref]

M. A. T. Figueiredo and J. M. Bioucas-Dias, “A new TwIST: Two-step Iterative Shrinkage/Thresholding algorithms for image restoration,” IEEE Transactions on Image Processing,  16(12), 2992–3004 (2007).
[Crossref] [PubMed]

A. Foi, S. Alenius, V. Katkovnik, and K. Egiazarian, “Noise measurement for raw-data of digital imaging sensors by automatic segmentation of nonuniform targets,” IEEE Sensors Journal,  7(10), 1456–1461 (2007).
[Crossref]

2006 (2)

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Transactions on Information Theory,  52(2), 489–509 (2006).
[Crossref]

D. L. Donoho, “Compressed sensing,” IEEE Transactions on Information Theory,  52(4), 1289–1306 (2006).
[Crossref]

2005 (1)

D. L. Lau and R. Yang, “Real-time multispectral color video synthesis using an array of commodity cameras,” Real-Time Imaging,  11(2), 109–116 (2005).
[Crossref]

1999 (1)

M. Dinguirard and P. Slater, “Calibration of space-multispectral imaging sensors: A review.”, Remote Sens. Environ.,  68, 194–205 (1999).
[Crossref]

Afonso, M. V.

M. A. T. Figueiredo, M. V. Afonso, and J. M. Bioucas-Dias, “An augmented lagrangian approach to the constrained optimization formulation of imaging inverse problems,” IEEE Transactions on Image Processing,  20(3), 681–695 (2011).
[Crossref]

Alenius, S.

A. Foi, S. Alenius, V. Katkovnik, and K. Egiazarian, “Noise measurement for raw-data of digital imaging sensors by automatic segmentation of nonuniform targets,” IEEE Sensors Journal,  7(10), 1456–1461 (2007).
[Crossref]

Arce, G. R.

H. Rueda, H. Arguello, and G. R. Arce, “DMD-based implementation of patterned optical filter arrays for compressive spectral imaging,” J. Opt. Soc. Am. A,  32, 80–89 (2015).
[Crossref]

H. Arguello and G. R. Arce, “Colored coded aperture design by concentration of measure in compressive spectral imaging,” IEEE Transactions on Image Processing,  23(4), 1896–1908 (2014).
[Crossref] [PubMed]

G. R. Arce, D. J. Brady, L. Carin, H. Arguello, and D. S. Kittle, “An introduction to compressive coded aperture spectral imaging,” IEEE Signal Processing Magazine,  31(1), 105–115 (2014).
[Crossref]

H. Arguello and G. R. Arce, “Rank minimization code aperture design for spectrally selective compressive imaging,” IEEE Transactions on Image Processing,  22(3), 941–954 (2013).
[Crossref]

H. Arguello, H. Rueda, Y. Wu, D. W. Prather, and G. R. Arce, “Higher-order computational model for coded aperture spectral imaging,” Appl. Opt.,  52(10), D12–D21 (2013).
[Crossref] [PubMed]

H. Arguello, C. Correa, and G. R. Arce, “Fast lapped block reconstructions in compressive spectral imaging,” Appl. Opt.,  52(10), D32–D45 (2013).
[Crossref] [PubMed]

Y. Wu, I. O. Mirza, G. R. Arce, and D. W. Prather, “Development of a digital-micromirror-device-based multishot snapshot spectral imaging system,” Opt. Lett 36, 2692–2694 (2011).
[Crossref] [PubMed]

J. L. Paredes and G. R. Arce, “Compressive sensing signal reconstruction by weighted median regression estimates,” IEEE Transactions on Signal Processing,  59(6), 2585–2601 (2011).
[Crossref]

Z. Wang and G. R. Arce, “Variable density compressed image sampling,” IEEE Transactions on Image Processing,  19(1), 264–270 (2010).
[Crossref]

Arguello, H.

H. Rueda, H. Arguello, and G. R. Arce, “DMD-based implementation of patterned optical filter arrays for compressive spectral imaging,” J. Opt. Soc. Am. A,  32, 80–89 (2015).
[Crossref]

H. Arguello and G. R. Arce, “Colored coded aperture design by concentration of measure in compressive spectral imaging,” IEEE Transactions on Image Processing,  23(4), 1896–1908 (2014).
[Crossref] [PubMed]

G. R. Arce, D. J. Brady, L. Carin, H. Arguello, and D. S. Kittle, “An introduction to compressive coded aperture spectral imaging,” IEEE Signal Processing Magazine,  31(1), 105–115 (2014).
[Crossref]

H. Arguello and G. R. Arce, “Rank minimization code aperture design for spectrally selective compressive imaging,” IEEE Transactions on Image Processing,  22(3), 941–954 (2013).
[Crossref]

H. Arguello, C. Correa, and G. R. Arce, “Fast lapped block reconstructions in compressive spectral imaging,” Appl. Opt.,  52(10), D32–D45 (2013).
[Crossref] [PubMed]

H. Arguello, H. Rueda, Y. Wu, D. W. Prather, and G. R. Arce, “Higher-order computational model for coded aperture spectral imaging,” Appl. Opt.,  52(10), D12–D21 (2013).
[Crossref] [PubMed]

August, Y.

Baraniuk, R. G.

M. F. Duarte and R. G. Baraniuk, “Kronecker compressive sensing,” IEEE Transactions on Image Processing,  21(2), 494–504 (2012).
[Crossref]

Bayer, B. E.

B. E. Bayer, “Color imaging array,” U.S. Patent3,971,065 (1976).

Bioucas-Dias, J. M.

M. A. T. Figueiredo, M. V. Afonso, and J. M. Bioucas-Dias, “An augmented lagrangian approach to the constrained optimization formulation of imaging inverse problems,” IEEE Transactions on Image Processing,  20(3), 681–695 (2011).
[Crossref]

M. A. T. Figueiredo and J. M. Bioucas-Dias, “A new TwIST: Two-step Iterative Shrinkage/Thresholding algorithms for image restoration,” IEEE Transactions on Image Processing,  16(12), 2992–3004 (2007).
[Crossref] [PubMed]

Brady, D.

X. Yuan, T. Tsai, R. Zhu, P. Llull, D. Brady, and L. Carin, “Compressive Hyperspectral Imaging with Side Information,” arXiv:1502.06260, (2015).

Brady, D. J.

G. R. Arce, D. J. Brady, L. Carin, H. Arguello, and D. S. Kittle, “An introduction to compressive coded aperture spectral imaging,” IEEE Signal Processing Magazine,  31(1), 105–115 (2014).
[Crossref]

D. Kittle, K. Choi, A. A. Wagadarikar, and D. J. Brady, “Multiframe image estimation for coded aperture snapshot spectral imagers,” Appl. Opt.,  49, 6824–6833 (2010).
[Crossref] [PubMed]

X. Yuan, P. Llull, D. J. Brady, and L. Carin, “Tree-structure bayesian compressive sensing for video,” arXiv:1410.3080 (2014).

Candes, E. J.

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Transactions on Information Theory,  52(2), 489–509 (2006).
[Crossref]

Cao, X.

Carin, L.

G. R. Arce, D. J. Brady, L. Carin, H. Arguello, and D. S. Kittle, “An introduction to compressive coded aperture spectral imaging,” IEEE Signal Processing Magazine,  31(1), 105–115 (2014).
[Crossref]

X. Yuan, P. Llull, D. J. Brady, and L. Carin, “Tree-structure bayesian compressive sensing for video,” arXiv:1410.3080 (2014).

X. Yuan, T. Tsai, R. Zhu, P. Llull, D. Brady, and L. Carin, “Compressive Hyperspectral Imaging with Side Information,” arXiv:1502.06260, (2015).

Choi, K.

Correa, C.

Dai, Q.

Dinguirard, M.

M. Dinguirard and P. Slater, “Calibration of space-multispectral imaging sensors: A review.”, Remote Sens. Environ.,  68, 194–205 (1999).
[Crossref]

Donoho, D. L.

D. L. Donoho, “Compressed sensing,” IEEE Transactions on Information Theory,  52(4), 1289–1306 (2006).
[Crossref]

Duarte, M. F.

M. F. Duarte and R. G. Baraniuk, “Kronecker compressive sensing,” IEEE Transactions on Image Processing,  21(2), 494–504 (2012).
[Crossref]

Egiazarian, K.

A. Foi, S. Alenius, V. Katkovnik, and K. Egiazarian, “Noise measurement for raw-data of digital imaging sensors by automatic segmentation of nonuniform targets,” IEEE Sensors Journal,  7(10), 1456–1461 (2007).
[Crossref]

Figueiredo, M. A. T.

M. A. T. Figueiredo, M. V. Afonso, and J. M. Bioucas-Dias, “An augmented lagrangian approach to the constrained optimization formulation of imaging inverse problems,” IEEE Transactions on Image Processing,  20(3), 681–695 (2011).
[Crossref]

M. A. T. Figueiredo and J. M. Bioucas-Dias, “A new TwIST: Two-step Iterative Shrinkage/Thresholding algorithms for image restoration,” IEEE Transactions on Image Processing,  16(12), 2992–3004 (2007).
[Crossref] [PubMed]

M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE Journal of Selected Topics in Signal Processing,  1(4), 586–597 (2007).
[Crossref]

Foi, A.

A. Foi, S. Alenius, V. Katkovnik, and K. Egiazarian, “Noise measurement for raw-data of digital imaging sensors by automatic segmentation of nonuniform targets,” IEEE Sensors Journal,  7(10), 1456–1461 (2007).
[Crossref]

Guttosch, R.

P. Hubel, J. Liu, and R. Guttosch, “Spatial frequency response of color image sensors: Bayer color filters and Foveon X3,” in Proceedings of SPIE, 5301, 402–407, EI’04, San Jose, CA, (2004).

Hubel, P.

P. Hubel, J. Liu, and R. Guttosch, “Spatial frequency response of color image sensors: Bayer color filters and Foveon X3,” in Proceedings of SPIE, 5301, 402–407, EI’04, San Jose, CA, (2004).

Katkovnik, V.

A. Foi, S. Alenius, V. Katkovnik, and K. Egiazarian, “Noise measurement for raw-data of digital imaging sensors by automatic segmentation of nonuniform targets,” IEEE Sensors Journal,  7(10), 1456–1461 (2007).
[Crossref]

Kittle, D.

Kittle, D. S.

G. R. Arce, D. J. Brady, L. Carin, H. Arguello, and D. S. Kittle, “An introduction to compressive coded aperture spectral imaging,” IEEE Signal Processing Magazine,  31(1), 105–115 (2014).
[Crossref]

Lau, D. L.

D. L. Lau and R. Yang, “Real-time multispectral color video synthesis using an array of commodity cameras,” Real-Time Imaging,  11(2), 109–116 (2005).
[Crossref]

Lerma, J.

A. Marqus-Mateu, J. Lerma, and G. Riutort-Mayol, “Statistical grey level and noise evaluation of Foveon X3 and CFA image sensors,” Optics & Laser Technology,  48, pp. 1–15, (2013).
[Crossref]

G. Riutort-Mayol, A. Marqus-Mateu, A. Segu, and J. Lerma, “Grey level and noise evaluation of a Foveon X3 image sensor: A statistical and experimental approach”, Sensors,  12(8), 10339–10368 (2012).
[Crossref] [PubMed]

Liu, J.

P. Hubel, J. Liu, and R. Guttosch, “Spatial frequency response of color image sensors: Bayer color filters and Foveon X3,” in Proceedings of SPIE, 5301, 402–407, EI’04, San Jose, CA, (2004).

Llull, P.

X. Yuan, P. Llull, D. J. Brady, and L. Carin, “Tree-structure bayesian compressive sensing for video,” arXiv:1410.3080 (2014).

X. Yuan, T. Tsai, R. Zhu, P. Llull, D. Brady, and L. Carin, “Compressive Hyperspectral Imaging with Side Information,” arXiv:1502.06260, (2015).

Ma, C.

Marqus-Mateu, A.

A. Marqus-Mateu, J. Lerma, and G. Riutort-Mayol, “Statistical grey level and noise evaluation of Foveon X3 and CFA image sensors,” Optics & Laser Technology,  48, pp. 1–15, (2013).
[Crossref]

G. Riutort-Mayol, A. Marqus-Mateu, A. Segu, and J. Lerma, “Grey level and noise evaluation of a Foveon X3 image sensor: A statistical and experimental approach”, Sensors,  12(8), 10339–10368 (2012).
[Crossref] [PubMed]

Merrill, R. B.

R. B. Merrill, “Color separation in an active pixel cell imaging array using a triple-well structure,” U.S. Patent5,965,875 (1999).

Mirza, I. O.

Y. Wu, I. O. Mirza, G. R. Arce, and D. W. Prather, “Development of a digital-micromirror-device-based multishot snapshot spectral imaging system,” Opt. Lett 36, 2692–2694 (2011).
[Crossref] [PubMed]

Nowak, R. D.

M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE Journal of Selected Topics in Signal Processing,  1(4), 586–597 (2007).
[Crossref]

Paredes, J. L.

J. L. Paredes and G. R. Arce, “Compressive sensing signal reconstruction by weighted median regression estimates,” IEEE Transactions on Signal Processing,  59(6), 2585–2601 (2011).
[Crossref]

Prather, D. W.

H. Arguello, H. Rueda, Y. Wu, D. W. Prather, and G. R. Arce, “Higher-order computational model for coded aperture spectral imaging,” Appl. Opt.,  52(10), D12–D21 (2013).
[Crossref] [PubMed]

Y. Wu, I. O. Mirza, G. R. Arce, and D. W. Prather, “Development of a digital-micromirror-device-based multishot snapshot spectral imaging system,” Opt. Lett 36, 2692–2694 (2011).
[Crossref] [PubMed]

Riutort-Mayol, G.

A. Marqus-Mateu, J. Lerma, and G. Riutort-Mayol, “Statistical grey level and noise evaluation of Foveon X3 and CFA image sensors,” Optics & Laser Technology,  48, pp. 1–15, (2013).
[Crossref]

G. Riutort-Mayol, A. Marqus-Mateu, A. Segu, and J. Lerma, “Grey level and noise evaluation of a Foveon X3 image sensor: A statistical and experimental approach”, Sensors,  12(8), 10339–10368 (2012).
[Crossref] [PubMed]

Rivenson, Y.

Romberg, J.

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Transactions on Information Theory,  52(2), 489–509 (2006).
[Crossref]

Rueda, H.

Segu, A.

G. Riutort-Mayol, A. Marqus-Mateu, A. Segu, and J. Lerma, “Grey level and noise evaluation of a Foveon X3 image sensor: A statistical and experimental approach”, Sensors,  12(8), 10339–10368 (2012).
[Crossref] [PubMed]

Slater, P.

M. Dinguirard and P. Slater, “Calibration of space-multispectral imaging sensors: A review.”, Remote Sens. Environ.,  68, 194–205 (1999).
[Crossref]

Stern, A.

Tao, T.

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Transactions on Information Theory,  52(2), 489–509 (2006).
[Crossref]

Tsai, T.

X. Yuan, T. Tsai, R. Zhu, P. Llull, D. Brady, and L. Carin, “Compressive Hyperspectral Imaging with Side Information,” arXiv:1502.06260, (2015).

Vachman, C.

Wagadarikar, A. A.

Wang, Z.

Z. Wang and G. R. Arce, “Variable density compressed image sampling,” IEEE Transactions on Image Processing,  19(1), 264–270 (2010).
[Crossref]

Wright, S. J.

M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE Journal of Selected Topics in Signal Processing,  1(4), 586–597 (2007).
[Crossref]

Wu, R.

Wu, Y.

H. Arguello, H. Rueda, Y. Wu, D. W. Prather, and G. R. Arce, “Higher-order computational model for coded aperture spectral imaging,” Appl. Opt.,  52(10), D12–D21 (2013).
[Crossref] [PubMed]

Y. Wu, I. O. Mirza, G. R. Arce, and D. W. Prather, “Development of a digital-micromirror-device-based multishot snapshot spectral imaging system,” Opt. Lett 36, 2692–2694 (2011).
[Crossref] [PubMed]

Yang, R.

D. L. Lau and R. Yang, “Real-time multispectral color video synthesis using an array of commodity cameras,” Real-Time Imaging,  11(2), 109–116 (2005).
[Crossref]

Yuan, X.

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Figures (11)

Fig. 1
Fig. 1 Illustration of the multispectral imaging principles used by CASSI.
Fig. 2
Fig. 2 Structure of the system transfer functions H for a 4 × 4 × 6 data cube, using a monochromatic sensor, where (top) shows the transfer function for a typical monochrome camera, (center) shows the same camera but with the incoming light divided into spectral bins, and (bottom) models the effects of the dispersive element by offsetting the diagonal structure from band to band. Note that the non-zero diagonals correspond to the entries of the M × N coded aperture set to be all ones.
Fig. 3
Fig. 3 FPA detector architectures along with their traditional spectral responses. (left) Monochromatic (Source: Stingray F-033B), (center) RGB-Bayer (Source: TMC) and (right) Foveon (Source: Foveon Quattro sensor).
Fig. 4
Fig. 4 Non-zero elements of the system transfer functions, H, for a 4×4×6 data cube, using (left) RGB-Bayer and (right) Foveon sensors.
Fig. 5
Fig. 5 Simulation results with first target scene. (a) Target scene for the simulations. (b) Averaged PSNR of the reconstructed data cubes as function of the compression ratio (R). (c) Reconstructed datacubes, using (first row) R = 1/8 and (second row) R = 1/2.
Fig. 6
Fig. 6 Simulation results with second target scene. (a) Target scene for the simulations. (b) Averaged PSNR of the reconstructed data cubes as function of the compression ratio (R). (c) Reconstructed datacubes, using (first row) R = 1/8 and (second row) R = 1/2. Note the color aliasing in the zoomed version of the RGB-Bayer reconstruction.
Fig. 7
Fig. 7 Absolute errors of the reconstructions of the 4th,8th,12th,16th,20th,24th spectral bands, for R = 1/8, using (First row) Monochrome sensor, (Second row) Bayer sensor, (Third row) Foveon sensor. Notice that the Foveon sensor attains the best reconstructions between the three sensors, despite the high frequencies (borders) are almost unrecoverable.
Fig. 8
Fig. 8 DMD-based CASSI testbed setup used in the experiments. The illuminated target scene is imaged onto the image plane of the DMD which plays the role of the coded aperture. Subsequently, the relay lens transmits the coded light through the Amici prism which disperses it onto the image plane of the CCD array.
Fig. 9
Fig. 9 Real measurements for a single snapshot of the monochrome, the Bayer and the Foveon sensor. Zoomed versions highlight the sensors differences.
Fig. 10
Fig. 10 Experimental reconstruction results. (a) Target scene for the experiments. (b) Real reconstructed datacubes using R = 1/8 and R = 1/2. (c) Spectral signatures of two reference points measured from the R = 1/8 reconstructions.
Fig. 11
Fig. 11 Real reconstruction of the 6th, 12th, 18th and 24th spectral bands using just a single snapshot of the (First row) Monochrome sensor, (Second row) Bayer sensor, (Third row) Foveon sensor.

Equations (5)

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g [ m , n ] = λ χ m , n ( λ ) f m , n ( λ ) d λ .
g [ m , n ] = k χ m , n , k T m , n + k f m , n + k , k ,
g = XP T f = H f ,
f ˜ = Ψ T ( arg min θ g H Ψ θ 2 + τ θ 1 ) ,
Bayer : χ n , m ( λ ) = { b B ( λ ) , n , m Ω b g B ( λ ) , n , m Ω g r B ( λ ) , n , m Ω r , Foveon : χ n , m ( λ ) { b F ( λ ) , n , m g F ( λ ) , n , m r F ( λ ) , n , m ,

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