Abstract

Compressive Spectral Imaging (CSI) is an emerging technology that aims at reconstructing a spectral image from a limited set of two-dimensional projections. To capture these projections, CSI architectures often combine light dispersive elements with coded apertures or programmable spatial light modulators. This work introduces a novel and compact CSI architecture based on a deformable mirror and a colored-filter detector to produce compressive spatio-spectral projections without the need of a grating or prism. Alongside, we propose a tensor-based reconstruction algorithm to recover the spatial-spectral information from the compressed measurements. Experimental results on both simulated and real datasets demonstrate efficacy of the proposed acquisition architecture and the especially crafted inversion algorithms.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]

2018 (2)

K. Degraux, V. Cambareri, B. Geelen, L. Jacques, and G. Lafruit, “Multispectral compressive imaging strategies using fabry–pérot filtered sensors,” IEEE Transactions on Computational Imaging 4(4), 661–673 (2018).
[Crossref]

E. Vera and P. Meza, “Snapshot compressive imaging using aberrations,” Opt. Express 26(2), 1206–1218 (2018).
[Crossref] [PubMed]

2017 (2)

S. Baek, I. Kim, D. Gutierrez, and M. Kim, “Compact single-shot hyperspectral imaging using a prism,” ACM Transactions on Graphics 36(6), 217 (2017).
[Crossref]

N. Sidiropoulos, L. De Lathauwer, X. Fu, K. Huang, E. Papalexakis, and C. Faloutsos, “Tensor decomposition for signal processing and machine learning,” IEEE Transactions on Signal Processing 65(13), 3551–3582 (2017).
[Crossref]

2016 (3)

C. Correa, C. Hinojosa, G. Arce, and H. Arguello, “Multiple snapshot colored compressive spectral imager,” Optical Engineering 56(4), 041309 (2016).
[Crossref]

C. Nansen, K. Singh, A. Mian, B. J. Allison, and C. Simmons, “Using hyperspectral imaging to characterize consistency of coffee brands and their respective roasting classes,” Journal of Food Engineering 190, 34–39 (2016).
[Crossref]

X. Cao, T. Yue, X. Lin, S. Lin, X. Yuan, Q. Dai, L. Carin, and D. J. Brady, “Computational snapshot multispectral cameras: Toward dynamic capture of the spectral world,” IEEE Signal Processing Magazine 33(5), 95–108 (2016).
[Crossref]

2015 (2)

X. Yuan, T. H. Tsai, R. Zhu, P. Llull, D. Brady, and L. Carin, “Compressive hyperspectral imaging with side information,” IEEE Journal of Selected Topics in Signal Processing 9(6), 964–976 (2015).
[Crossref]

C. V. Correa, H. Arguello, and G. R. Arce, “Snapshot colored compressive spectral imager,” J. Opt. Soc. Am. A 32(10), 1754–1763 (2015).
[Crossref]

2014 (5)

H. Arguello and G. 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]

X. Lin, Y. Liu, J. Wu, and Q. Dai, “Spatial-spectral encoded compressive hyperspectral imaging,” ACM Transactions on Graphics 33(6), 233 (2014).
[Crossref]

X. Lin, G. Wetzstein, Y. Liu, and Q. Dai, “Dual-coded compressive hyperspectral imaging,” Opt. Lett. 39(7), 2044–2047 (2014).
[Crossref] [PubMed]

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

G. Lu and B. Fei, “Medical hyperspectral imaging: a review,” Journal of Biomedical Optics 19(1), 010901 (2014).
[Crossref]

2012 (2)

Y. Murakami, M. Yamaguchi, and N. Ohyama, “Hybrid-resolution multispectral imaging using color filter array,” Opt. Express 20(7), 7173–7183 (2012).
[Crossref] [PubMed]

P. Madec, “Overview of deformable mirror technologies for adaptive optics and astronomy,” Proc. SPIE 8447, 844705 (2012).
[Crossref]

2011 (1)

M. Afonso, J. Bioucas-Dias, and M. Figueiredo, “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 (1)

F. Yasuma, T. Mitsunaga, D. Iso, and S. K. Nayar, “Generalized assorted pixel camera: postcapture control of resolution, dynamic range, and spectrum,” IEEE Transactions on Image Processing 19(9), 2241–2253 (2010).
[Crossref] [PubMed]

2009 (1)

M. D. Robinson, G. Feng, and D. G. Stork, “Spherical coded imagers: improving lens speed, depth-of-field, and manufacturing yield through enhanced spherical aberration and compensating image processing,” Proc. SPIE 7429, 74290M (2009).
[Crossref]

2008 (2)

2007 (1)

2006 (1)

2003 (1)

2001 (2)

M. Akgün, J. Garcelon, and R. Haftka, “Fast exact linear and non-linear structural reanalysis and the sherman–morrison–woodbury formulas,” International Journal for Numerical Methods in Engineering 50(7), 1587–1606 (2001).
[Crossref]

M. Kim, Y. Chen, and P. Mehl, “Hyperspectral reflectance and fluorescence imaging system for food quality and safety,” Transactions of the ASAE 44(3), 721–729 (2001).

2000 (1)

N. Gat, “Imaging spectroscopy using tunable filters: a review,” Proc. SPIE 4056, 50–64 (2000).
[Crossref]

1998 (1)

R. Green, M. Eastwood, C. Sarture, T. Chrien, M. Aronsson, B. Chippendale, J. Faust, B. Pavri, C. Chovit, M. Solis, and W. Orlesa, “Imaging spectroscopy and the airborne visible/infrared imaging spectrometer (aviris),” Remote Sensing of Environment 65(3), 227–248 (1998).
[Crossref]

1976 (1)

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66(3), 207–211 (1976).
[Crossref]

Afonso, M.

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

Akgün, M.

M. Akgün, J. Garcelon, and R. Haftka, “Fast exact linear and non-linear structural reanalysis and the sherman–morrison–woodbury formulas,” International Journal for Numerical Methods in Engineering 50(7), 1587–1606 (2001).
[Crossref]

Allison, B. J.

C. Nansen, K. Singh, A. Mian, B. J. Allison, and C. Simmons, “Using hyperspectral imaging to characterize consistency of coffee brands and their respective roasting classes,” Journal of Food Engineering 190, 34–39 (2016).
[Crossref]

Amano, K.

Arce, G.

C. Correa, C. Hinojosa, G. Arce, and H. Arguello, “Multiple snapshot colored compressive spectral imager,” Optical Engineering 56(4), 041309 (2016).
[Crossref]

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

Arce, G. R.

Arguello, H.

C. Correa, C. Hinojosa, G. Arce, and H. Arguello, “Multiple snapshot colored compressive spectral imager,” Optical Engineering 56(4), 041309 (2016).
[Crossref]

C. V. Correa, H. Arguello, and G. R. Arce, “Snapshot colored compressive spectral imager,” J. Opt. Soc. Am. A 32(10), 1754–1763 (2015).
[Crossref]

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

Aronsson, M.

R. Green, M. Eastwood, C. Sarture, T. Chrien, M. Aronsson, B. Chippendale, J. Faust, B. Pavri, C. Chovit, M. Solis, and W. Orlesa, “Imaging spectroscopy and the airborne visible/infrared imaging spectrometer (aviris),” Remote Sensing of Environment 65(3), 227–248 (1998).
[Crossref]

Artal, P.

Baek, S.

S. Baek, I. Kim, D. Gutierrez, and M. Kim, “Compact single-shot hyperspectral imaging using a prism,” ACM Transactions on Graphics 36(6), 217 (2017).
[Crossref]

Bioucas-Dias, J.

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

Brady, D.

X. Yuan, T. H. Tsai, R. Zhu, P. Llull, D. Brady, and L. Carin, “Compressive hyperspectral imaging with side information,” IEEE Journal of Selected Topics in Signal Processing 9(6), 964–976 (2015).
[Crossref]

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

A. Wagadarikar, R. John, R. Willett, and D. Brady, “Single disperser design for coded aperture snapshot spectral imaging,” Appl. Opt. 47(10), B44–B51 (2008).
[Crossref] [PubMed]

M. Gehm, R. John, D. Brady, R. Willett, and T. Schulz, “Single-shot compressive spectral imaging with a dual-disperser architecture,” Opt. Express 15(21), 14013–14027 (2007).
[Crossref] [PubMed]

Brady, D. J.

X. Cao, T. Yue, X. Lin, S. Lin, X. Yuan, Q. Dai, L. Carin, and D. J. Brady, “Computational snapshot multispectral cameras: Toward dynamic capture of the spectral world,” IEEE Signal Processing Magazine 33(5), 95–108 (2016).
[Crossref]

Cambareri, V.

K. Degraux, V. Cambareri, B. Geelen, L. Jacques, and G. Lafruit, “Multispectral compressive imaging strategies using fabry–pérot filtered sensors,” IEEE Transactions on Computational Imaging 4(4), 661–673 (2018).
[Crossref]

Cao, X.

X. Cao, T. Yue, X. Lin, S. Lin, X. Yuan, Q. Dai, L. Carin, and D. J. Brady, “Computational snapshot multispectral cameras: Toward dynamic capture of the spectral world,” IEEE Signal Processing Magazine 33(5), 95–108 (2016).
[Crossref]

Carin, L.

X. Cao, T. Yue, X. Lin, S. Lin, X. Yuan, Q. Dai, L. Carin, and D. J. Brady, “Computational snapshot multispectral cameras: Toward dynamic capture of the spectral world,” IEEE Signal Processing Magazine 33(5), 95–108 (2016).
[Crossref]

X. Yuan, T. H. Tsai, R. Zhu, P. Llull, D. Brady, and L. Carin, “Compressive hyperspectral imaging with side information,” IEEE Journal of Selected Topics in Signal Processing 9(6), 964–976 (2015).
[Crossref]

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

S. Ji, Y. Xue, and L. Carin, “Bayesian compressive sensing,” IEEE Transactions on Signal Processing 56(6), 2346 (2008).
[Crossref]

Chen, Y.

M. Kim, Y. Chen, and P. Mehl, “Hyperspectral reflectance and fluorescence imaging system for food quality and safety,” Transactions of the ASAE 44(3), 721–729 (2001).

Chippendale, B.

R. Green, M. Eastwood, C. Sarture, T. Chrien, M. Aronsson, B. Chippendale, J. Faust, B. Pavri, C. Chovit, M. Solis, and W. Orlesa, “Imaging spectroscopy and the airborne visible/infrared imaging spectrometer (aviris),” Remote Sensing of Environment 65(3), 227–248 (1998).
[Crossref]

Chovit, C.

R. Green, M. Eastwood, C. Sarture, T. Chrien, M. Aronsson, B. Chippendale, J. Faust, B. Pavri, C. Chovit, M. Solis, and W. Orlesa, “Imaging spectroscopy and the airborne visible/infrared imaging spectrometer (aviris),” Remote Sensing of Environment 65(3), 227–248 (1998).
[Crossref]

Chrien, T.

R. Green, M. Eastwood, C. Sarture, T. Chrien, M. Aronsson, B. Chippendale, J. Faust, B. Pavri, C. Chovit, M. Solis, and W. Orlesa, “Imaging spectroscopy and the airborne visible/infrared imaging spectrometer (aviris),” Remote Sensing of Environment 65(3), 227–248 (1998).
[Crossref]

Correa, C.

C. Correa, C. Hinojosa, G. Arce, and H. Arguello, “Multiple snapshot colored compressive spectral imager,” Optical Engineering 56(4), 041309 (2016).
[Crossref]

Correa, C. V.

Dai, Q.

X. Cao, T. Yue, X. Lin, S. Lin, X. Yuan, Q. Dai, L. Carin, and D. J. Brady, “Computational snapshot multispectral cameras: Toward dynamic capture of the spectral world,” IEEE Signal Processing Magazine 33(5), 95–108 (2016).
[Crossref]

X. Lin, Y. Liu, J. Wu, and Q. Dai, “Spatial-spectral encoded compressive hyperspectral imaging,” ACM Transactions on Graphics 33(6), 233 (2014).
[Crossref]

X. Lin, G. Wetzstein, Y. Liu, and Q. Dai, “Dual-coded compressive hyperspectral imaging,” Opt. Lett. 39(7), 2044–2047 (2014).
[Crossref] [PubMed]

De Lathauwer, L.

N. Sidiropoulos, L. De Lathauwer, X. Fu, K. Huang, E. Papalexakis, and C. Faloutsos, “Tensor decomposition for signal processing and machine learning,” IEEE Transactions on Signal Processing 65(13), 3551–3582 (2017).
[Crossref]

Degraux, K.

K. Degraux, V. Cambareri, B. Geelen, L. Jacques, and G. Lafruit, “Multispectral compressive imaging strategies using fabry–pérot filtered sensors,” IEEE Transactions on Computational Imaging 4(4), 661–673 (2018).
[Crossref]

Eastwood, M.

R. Green, M. Eastwood, C. Sarture, T. Chrien, M. Aronsson, B. Chippendale, J. Faust, B. Pavri, C. Chovit, M. Solis, and W. Orlesa, “Imaging spectroscopy and the airborne visible/infrared imaging spectrometer (aviris),” Remote Sensing of Environment 65(3), 227–248 (1998).
[Crossref]

Faloutsos, C.

N. Sidiropoulos, L. De Lathauwer, X. Fu, K. Huang, E. Papalexakis, and C. Faloutsos, “Tensor decomposition for signal processing and machine learning,” IEEE Transactions on Signal Processing 65(13), 3551–3582 (2017).
[Crossref]

Faust, J.

R. Green, M. Eastwood, C. Sarture, T. Chrien, M. Aronsson, B. Chippendale, J. Faust, B. Pavri, C. Chovit, M. Solis, and W. Orlesa, “Imaging spectroscopy and the airborne visible/infrared imaging spectrometer (aviris),” Remote Sensing of Environment 65(3), 227–248 (1998).
[Crossref]

Fei, B.

G. Lu and B. Fei, “Medical hyperspectral imaging: a review,” Journal of Biomedical Optics 19(1), 010901 (2014).
[Crossref]

Feng, G.

M. D. Robinson, G. Feng, and D. G. Stork, “Spherical coded imagers: improving lens speed, depth-of-field, and manufacturing yield through enhanced spherical aberration and compensating image processing,” Proc. SPIE 7429, 74290M (2009).
[Crossref]

Fernández, E.

Figueiredo, M.

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

Foster, D. H.

Foster, M. J.

Fu, X.

N. Sidiropoulos, L. De Lathauwer, X. Fu, K. Huang, E. Papalexakis, and C. Faloutsos, “Tensor decomposition for signal processing and machine learning,” IEEE Transactions on Signal Processing 65(13), 3551–3582 (2017).
[Crossref]

Garcelon, J.

M. Akgün, J. Garcelon, and R. Haftka, “Fast exact linear and non-linear structural reanalysis and the sherman–morrison–woodbury formulas,” International Journal for Numerical Methods in Engineering 50(7), 1587–1606 (2001).
[Crossref]

Gat, N.

N. Gat, “Imaging spectroscopy using tunable filters: a review,” Proc. SPIE 4056, 50–64 (2000).
[Crossref]

Geelen, B.

K. Degraux, V. Cambareri, B. Geelen, L. Jacques, and G. Lafruit, “Multispectral compressive imaging strategies using fabry–pérot filtered sensors,” IEEE Transactions on Computational Imaging 4(4), 661–673 (2018).
[Crossref]

Gehm, M.

Green, R.

R. Green, M. Eastwood, C. Sarture, T. Chrien, M. Aronsson, B. Chippendale, J. Faust, B. Pavri, C. Chovit, M. Solis, and W. Orlesa, “Imaging spectroscopy and the airborne visible/infrared imaging spectrometer (aviris),” Remote Sensing of Environment 65(3), 227–248 (1998).
[Crossref]

Gutierrez, D.

S. Baek, I. Kim, D. Gutierrez, and M. Kim, “Compact single-shot hyperspectral imaging using a prism,” ACM Transactions on Graphics 36(6), 217 (2017).
[Crossref]

Haftka, R.

M. Akgün, J. Garcelon, and R. Haftka, “Fast exact linear and non-linear structural reanalysis and the sherman–morrison–woodbury formulas,” International Journal for Numerical Methods in Engineering 50(7), 1587–1606 (2001).
[Crossref]

Hinojosa, C.

C. Correa, C. Hinojosa, G. Arce, and H. Arguello, “Multiple snapshot colored compressive spectral imager,” Optical Engineering 56(4), 041309 (2016).
[Crossref]

Huang, K.

N. Sidiropoulos, L. De Lathauwer, X. Fu, K. Huang, E. Papalexakis, and C. Faloutsos, “Tensor decomposition for signal processing and machine learning,” IEEE Transactions on Signal Processing 65(13), 3551–3582 (2017).
[Crossref]

Iso, D.

F. Yasuma, T. Mitsunaga, D. Iso, and S. K. Nayar, “Generalized assorted pixel camera: postcapture control of resolution, dynamic range, and spectrum,” IEEE Transactions on Image Processing 19(9), 2241–2253 (2010).
[Crossref] [PubMed]

Jacques, L.

K. Degraux, V. Cambareri, B. Geelen, L. Jacques, and G. Lafruit, “Multispectral compressive imaging strategies using fabry–pérot filtered sensors,” IEEE Transactions on Computational Imaging 4(4), 661–673 (2018).
[Crossref]

Ji, S.

S. Ji, Y. Xue, and L. Carin, “Bayesian compressive sensing,” IEEE Transactions on Signal Processing 56(6), 2346 (2008).
[Crossref]

John, R.

Kim, I.

S. Baek, I. Kim, D. Gutierrez, and M. Kim, “Compact single-shot hyperspectral imaging using a prism,” ACM Transactions on Graphics 36(6), 217 (2017).
[Crossref]

Kim, M.

S. Baek, I. Kim, D. Gutierrez, and M. Kim, “Compact single-shot hyperspectral imaging using a prism,” ACM Transactions on Graphics 36(6), 217 (2017).
[Crossref]

M. Kim, Y. Chen, and P. Mehl, “Hyperspectral reflectance and fluorescence imaging system for food quality and safety,” Transactions of the ASAE 44(3), 721–729 (2001).

Kittle, D.

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

Lafruit, G.

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Supplementary Material (1)

NameDescription
» Visualization 1       Spectral imaging reconstructions using the proposed approach for 4 simulated datasets

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Figures (8)

Fig. 1
Fig. 1 Diagram of four multishot compressive multispectral cameras: (a) DD-CASSI, (b) SD-CASSI, (c) 3D-CASSI and (d) proposed architecture.
Fig. 2
Fig. 2 Experimental setup implemented in the laboratory.
Fig. 3
Fig. 3 Spectral image reconstruction with K = 2 and SNR = 50[dB]. Here, for the first snapshot, in the two cases (a) and (b), the measurement is generated by using W(x, y) = aiZi, where i ∈ {4, ..., 15} and ai ∈ {0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5}. For the second snapshot, the measurement is generated by using (a) W(x, y) = 0 and (b) W ( x , y ) i = 4 15 a i Z i .
Fig. 4
Fig. 4 Reconstruction quality analysis for K = {2, 3, 4} and SNR = {20, 30, 50}. Furthermore, a measurement example generated from the Zernike polynomials (b) W = 0, (c) W = Z5, (d) W = Z12 and (e) W = Z13 are illustrated.
Fig. 5
Fig. 5 ( Visualization 1) (a) Visual comparison of the spectral reconstructions, mapped to an RGB profile, (b) absolute error of the reconstructions, and (c) comparison of 8 out of the 26 spectral bands. These simulations were developed with the parameters SNR = 20 [dB] and K = 3.
Fig. 6
Fig. 6 Visual comparison of spectral signatures for reconstruction from SD-CASSI, DD-CASSI, 3D-CASSI and proposed architecture
Fig. 7
Fig. 7 Experimental results using the “Digital beads” scene, where (a) is an RGB version of the reconstruction and (b) is a visualization of 12 reconstructed spectral bands. Spectral signature reconstructions of four points of the scene, using K = 3 snapshot measurements.
Fig. 8
Fig. 8 Experimental result using the “Toys” scene, where (a) is an RGB version of the reconstruction and (b) is a visualization of 12 reconstructed spectral bands. (c) Spectral signature reconstructions of four points of the scene, using K = 3 snapshot measurements.

Tables (2)

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Algorithm 1: Spectral image reconstruction

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Table 1 Reconstruction Results Comparison for Simulations using K = 3.

Equations (17)

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f 1 ( x , y , λ ) = | h ( x x , y y , λ ) | 2 f o ( x , y , λ ) d x d y ,
h ( x , y , λ ) = 1 λ z 𝒫 ( x , y ) e i 2 π 𝒲 ( x , y ) e i 2 π λ z ( x x + y y ) d x d y ,
t ( x , y , λ ) = i 1 , i 2 , i 3 T i 1 , i 2 , i 3 rect ( x Δ c i 1 , y Δ c i 2 , λ Δ d i 3 ) ,
g ( x , y ) = λ 1 λ 2 Δ d Δ d Δ d Δ d t ( x , y , λ ) f 1 ( x , y , λ ) d x d y d λ .
G _ = [ T _ ( i = 1 3 [ W _ ( [ j = 1 3 F _ × j A j ] × 4 1 ) ] × i A i T ) ] × 3 1 T
arg min F _ G _ [ T _ ( i = 1 3 [ W _ ( [ j = 1 3 F _ × j A j ] × 4 1 ) ] × i A i T ) ] × 3 1 T F 2 ,
arg min Θ _ G _ [ T _ ( i = 1 3 [ W _ ( [ j = 1 3 Θ _ × j A j Ψ j ] × 4 1 ) ] × i A i T ) ] × 3 1 T F 2 + λ Θ _ 1 ,
Θ _ ι + 1 arg min Θ _ G _ [ T _ ( i = 1 3 [ W _ ( [ j = 1 3 Θ _ × j A j Ψ j ] × 4 1 ) ] × i A i T ) ] × 3 1 T F 2 + β Θ _ ι Θ _ F + λ 1 Θ _ 1 + λ 2 j = 1 3 Θ _ × j Ψ j ,
arg min Θ _ , Ω _ , Ξ _ Γ 1 ( Θ _ ) + Γ 3 ( Ω _ ) + Γ 4 ( Ξ _ ) s . t . Θ _ = Ω _ , Ξ _ = j = 1 3 Θ _ × j Ψ j ,
Θ _ ˜ ι + 1 = arg min Θ Y _ [ T _ ( i = 1 3 [ W _ Γ 2 ( Θ _ ) ] × i A i T ) ] × 3 1 T F 2 + μ 1 Θ _ ι Ω _ ι ν _ 1 ι F 2 + μ 2 Ξ _ ι j = 1 3 Θ _ ι × j Ψ j ν _ 2 ι F 2 ,
Ω _ ι + 1 = arg min Ω β Ω _ ι Ω _ F 2 + μ 1 Θ _ ι Ω _ ι ν _ 1 ι F 2 + λ Ω _ 1 ,
Ξ _ ι + 1 = arg min Ξ Ξ _ TV + μ 2 Ξ _ ι j = 1 3 Θ _ ι × j Ψ j ν _ 2 ι F 2 .
PSNR = 1 L i 3 = 1 L [ 10 log 10 ( max ( F ˜ : , : , i 3 ) 2 MSE ( F , F ˜ ) ) ] ,
SAM = 1 M N i 1 = 1 M i 2 = 1 N cos 1 ( i 3 = 1 L F i 1 , i 2 , i 3 F ˜ i 1 , i 2 , i 3 F : , : , i 3 2 F ˜ : , : , i 3 2 ) ,
y i 1 , , i n 1 , j , i n + 1 , , i T = i n N n x i 1 , , i n , , i T a j , i n ,
Θ _ = n = 1 T X _ × n A n ,
y = vec ( Y _ ) = vec [ n T X _ × n A n ] = A vec ( X _ ) ,