Abstract

The coded aperture snapshot spectral imager (CASSI) senses the spatial and spectral information of a scene using a set of K random projections of the scene onto focal plane array measurements. The reconstruction of the underlying three-dimensional (3D) scene is then obtained by 1 norm-based inverse optimization algorithms such as the gradient projections for sparse reconstruction (GPSR). The computational complexity of the inverse problem in this case grows with order O(KN4L) per iteration, where N2 and L are the spatial and spectral dimensions of the scene, respectively. In some applications the computational complexity becomes overwhelming since reconstructions can take up to several hours in desktop architectures. This paper presents a mathematical model for lapped block reconstructions in CASSI with O(KB4L) complexity per GPSR iteration where BN is the block size. The approach takes advantage of the structure of the sensing matrix thus allowing the independent recovery of smaller overlapping blocks spanning the measurement set. The reconstructed 3D lapped parallelepipeds are then merged to reduce the block-artifacts in the reconstructed scenes. The full data cube is reconstructed with complexity O(K(N4/(N)2)L), per iteration, where N=N/B. Simulations show the benefits of the new model as data cube reconstruction can be accelerated by an order of magnitude. Furthermore, the lapped block reconstructions lead to comparable or higher image reconstruction quality.

© 2013 Optical Society of America

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References

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  1. A. A. Wagadarikar, R. John, R. Willett, and D. Brady, “Single disperser design for coded aperture snapshot spectral imaging,” Appl. Opt. 47, B44–B51 (2008).
    [CrossRef]
  2. H. Arguello and G. R. Arce, “Code aperture optimization for spectrally agile compressive imaging,” J. Opt. Soc. Am. 28, 2400–2413 (2011).
    [CrossRef]
  3. H. Arguello and G. R. Arce, “Rank minimization code aperture design for spectrally selective compressive imaging,” IEEE Trans. Image Process. 22, 941–954 (2013).
    [CrossRef]
  4. H. Arguello and G. R. Arce, “Spectrally selective compressive imaging by matrix system analysis,” presented at OSA Optics and Photonics Congress, Monterey, California, June 2012.
  5. 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]
  6. 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]
  7. 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, D12–D21 (2013).
  8. J. Tropp and S. Wright, “Computational methods for sparse solution of linear inverse problems,” Proc. IEEE 98, 948–958 (2010).
    [CrossRef]
  9. J. Tropp and A. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53, 4655–4666 (2007).
    [CrossRef]
  10. D. Donoho, Y. Tsaig, I. Drori, and J. Starck, “Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit (StOMP),” Stat. Dept. Tech. Rep. (Stanford University, 2006).
  11. D. Needell and J. Tropp, “Iterative signal recovery from incomplete and inaccurate samples,” Appl. Comput. Harmon. Anal. 26, 301–321 (2009).
    [CrossRef]
  12. S. Wright, R. Nowak, and M. Figueiredo, “Sparse reconstruction by separable approximation,” IEEE Trans. Signal Process. 57, 2479–2493 (2009).
    [CrossRef]
  13. J. Bioucas-Dias and M. Figueiredo, “A new TwIST: two-step iterative shrinking/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
    [CrossRef]
  14. M. Figueiredo, R. Nowak, and S. Wright, “Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Signal Process. 1, 586–597 (2007).
    [CrossRef]
  15. J. Shihao, X. Ya, and L. Carin, “Bayesian compressive sensing,” IEEE Trans. Signal Process. 56, 2346–2356(2008).
    [CrossRef]
  16. R. Chartrand, “Exact reconstruction of sparse signals via nonconvex minimization,” IEEE Signal Process. Lett. 14, 707–710 (2007).
    [CrossRef]
  17. A. Wagadarikar, N. Pitsianis, X. Sun, and D. Brady, “Video rate spectral imaging using a coded aperture snapshot spectral imager,” Opt. Express 17, 6368–6388 (2009).
    [CrossRef]
  18. Y. Rivensons and A. Stern, “Compressed imaging with a separable sensing operator,” IEEE Signal Process. Lett. 16, 449–452 (2009).
    [CrossRef]
  19. M. F. Duarte and R. G. Baraniuk, “Kronecker compressive sensing,” IEEE Trans. Image Process. 21, 494–504(2012).
    [CrossRef]
  20. L. Gan, “Block compressed sensing of natural images,” Proceedings of International Conference on Digital Signal Processing, Cardiff, UK, 1–4 July2007.
  21. Y. Fang, L. Chen, J. Wu, and B. Huang, “GPU implementation of orthogonal matching pursuit for compressive sensing,” presented at IEEE 17th International Conference on Parallel and Distributed Systems (ICPADS), Tainan, Taiwan, December 2011.
  22. M. P. McLoughlin and G. R. Arce, “Deterministic properties of the recursive separable median filter,” IEEE Trans. Acoust. Speech Signal Process. 35, 98–106 (1987).
    [CrossRef]
  23. S. Kalluri and G. R. Arce, “Robust frequency-selective filtering using weighted myriad filters admitting real-valued weights,” IEEE Trans. Signal Process. 49, 2721–2733(2001).
    [CrossRef]
  24. Airborne Visible Infrared Imaging Spectrometer, http://aviris.jpl.nasa.gov/data/free_data.html .

2013

H. Arguello and G. R. Arce, “Rank minimization code aperture design for spectrally selective compressive imaging,” IEEE Trans. Image Process. 22, 941–954 (2013).
[CrossRef]

2012

M. F. Duarte and R. G. Baraniuk, “Kronecker compressive sensing,” IEEE Trans. Image Process. 21, 494–504(2012).
[CrossRef]

2011

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]

H. Arguello and G. R. Arce, “Code aperture optimization for spectrally agile compressive imaging,” J. Opt. Soc. Am. 28, 2400–2413 (2011).
[CrossRef]

2010

J. Tropp and S. Wright, “Computational methods for sparse solution of linear inverse problems,” Proc. IEEE 98, 948–958 (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]

2009

A. Wagadarikar, N. Pitsianis, X. Sun, and D. Brady, “Video rate spectral imaging using a coded aperture snapshot spectral imager,” Opt. Express 17, 6368–6388 (2009).
[CrossRef]

Y. Rivensons and A. Stern, “Compressed imaging with a separable sensing operator,” IEEE Signal Process. Lett. 16, 449–452 (2009).
[CrossRef]

D. Needell and J. Tropp, “Iterative signal recovery from incomplete and inaccurate samples,” Appl. Comput. Harmon. Anal. 26, 301–321 (2009).
[CrossRef]

S. Wright, R. Nowak, and M. Figueiredo, “Sparse reconstruction by separable approximation,” IEEE Trans. Signal Process. 57, 2479–2493 (2009).
[CrossRef]

2008

2007

R. Chartrand, “Exact reconstruction of sparse signals via nonconvex minimization,” IEEE Signal Process. Lett. 14, 707–710 (2007).
[CrossRef]

J. Tropp and A. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53, 4655–4666 (2007).
[CrossRef]

J. Bioucas-Dias and M. Figueiredo, “A new TwIST: two-step iterative shrinking/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
[CrossRef]

M. Figueiredo, R. Nowak, and S. Wright, “Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Signal Process. 1, 586–597 (2007).
[CrossRef]

2001

S. Kalluri and G. R. Arce, “Robust frequency-selective filtering using weighted myriad filters admitting real-valued weights,” IEEE Trans. Signal Process. 49, 2721–2733(2001).
[CrossRef]

1987

M. P. McLoughlin and G. R. Arce, “Deterministic properties of the recursive separable median filter,” IEEE Trans. Acoust. Speech Signal Process. 35, 98–106 (1987).
[CrossRef]

Arce, G. R.

H. Arguello and G. R. Arce, “Rank minimization code aperture design for spectrally selective compressive imaging,” IEEE Trans. Image Process. 22, 941–954 (2013).
[CrossRef]

H. Arguello and G. R. Arce, “Code aperture optimization for spectrally agile compressive imaging,” J. Opt. Soc. Am. 28, 2400–2413 (2011).
[CrossRef]

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]

S. Kalluri and G. R. Arce, “Robust frequency-selective filtering using weighted myriad filters admitting real-valued weights,” IEEE Trans. Signal Process. 49, 2721–2733(2001).
[CrossRef]

M. P. McLoughlin and G. R. Arce, “Deterministic properties of the recursive separable median filter,” IEEE Trans. Acoust. Speech Signal Process. 35, 98–106 (1987).
[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, D12–D21 (2013).

H. Arguello and G. R. Arce, “Spectrally selective compressive imaging by matrix system analysis,” presented at OSA Optics and Photonics Congress, Monterey, California, June 2012.

Arguello, H.

H. Arguello and G. R. Arce, “Rank minimization code aperture design for spectrally selective compressive imaging,” IEEE Trans. Image Process. 22, 941–954 (2013).
[CrossRef]

H. Arguello and G. R. Arce, “Code aperture optimization for spectrally agile compressive imaging,” J. Opt. Soc. Am. 28, 2400–2413 (2011).
[CrossRef]

H. Arguello and G. R. Arce, “Spectrally selective compressive imaging by matrix system analysis,” presented at OSA Optics and Photonics Congress, Monterey, California, June 2012.

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, D12–D21 (2013).

Baraniuk, R. G.

M. F. Duarte and R. G. Baraniuk, “Kronecker compressive sensing,” IEEE Trans. Image Process. 21, 494–504(2012).
[CrossRef]

Bioucas-Dias, J.

J. Bioucas-Dias and M. Figueiredo, “A new TwIST: two-step iterative shrinking/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
[CrossRef]

Brady, D.

Brady, D. J.

Carin, L.

J. Shihao, X. Ya, and L. Carin, “Bayesian compressive sensing,” IEEE Trans. Signal Process. 56, 2346–2356(2008).
[CrossRef]

Chartrand, R.

R. Chartrand, “Exact reconstruction of sparse signals via nonconvex minimization,” IEEE Signal Process. Lett. 14, 707–710 (2007).
[CrossRef]

Chen, L.

Y. Fang, L. Chen, J. Wu, and B. Huang, “GPU implementation of orthogonal matching pursuit for compressive sensing,” presented at IEEE 17th International Conference on Parallel and Distributed Systems (ICPADS), Tainan, Taiwan, December 2011.

Choi, K.

Donoho, D.

D. Donoho, Y. Tsaig, I. Drori, and J. Starck, “Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit (StOMP),” Stat. Dept. Tech. Rep. (Stanford University, 2006).

Drori, I.

D. Donoho, Y. Tsaig, I. Drori, and J. Starck, “Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit (StOMP),” Stat. Dept. Tech. Rep. (Stanford University, 2006).

Duarte, M. F.

M. F. Duarte and R. G. Baraniuk, “Kronecker compressive sensing,” IEEE Trans. Image Process. 21, 494–504(2012).
[CrossRef]

Fang, Y.

Y. Fang, L. Chen, J. Wu, and B. Huang, “GPU implementation of orthogonal matching pursuit for compressive sensing,” presented at IEEE 17th International Conference on Parallel and Distributed Systems (ICPADS), Tainan, Taiwan, December 2011.

Figueiredo, M.

S. Wright, R. Nowak, and M. Figueiredo, “Sparse reconstruction by separable approximation,” IEEE Trans. Signal Process. 57, 2479–2493 (2009).
[CrossRef]

M. Figueiredo, R. Nowak, and S. Wright, “Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Signal Process. 1, 586–597 (2007).
[CrossRef]

J. Bioucas-Dias and M. Figueiredo, “A new TwIST: two-step iterative shrinking/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
[CrossRef]

Gan, L.

L. Gan, “Block compressed sensing of natural images,” Proceedings of International Conference on Digital Signal Processing, Cardiff, UK, 1–4 July2007.

Gilbert, A.

J. Tropp and A. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53, 4655–4666 (2007).
[CrossRef]

Huang, B.

Y. Fang, L. Chen, J. Wu, and B. Huang, “GPU implementation of orthogonal matching pursuit for compressive sensing,” presented at IEEE 17th International Conference on Parallel and Distributed Systems (ICPADS), Tainan, Taiwan, December 2011.

John, R.

Kalluri, S.

S. Kalluri and G. R. Arce, “Robust frequency-selective filtering using weighted myriad filters admitting real-valued weights,” IEEE Trans. Signal Process. 49, 2721–2733(2001).
[CrossRef]

Kittle, D.

McLoughlin, M. P.

M. P. McLoughlin and G. R. Arce, “Deterministic properties of the recursive separable median filter,” IEEE Trans. Acoust. Speech Signal Process. 35, 98–106 (1987).
[CrossRef]

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]

Needell, D.

D. Needell and J. Tropp, “Iterative signal recovery from incomplete and inaccurate samples,” Appl. Comput. Harmon. Anal. 26, 301–321 (2009).
[CrossRef]

Nowak, R.

S. Wright, R. Nowak, and M. Figueiredo, “Sparse reconstruction by separable approximation,” IEEE Trans. Signal Process. 57, 2479–2493 (2009).
[CrossRef]

M. Figueiredo, R. Nowak, and S. Wright, “Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Signal Process. 1, 586–597 (2007).
[CrossRef]

Pitsianis, N.

Prather, D. W.

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]

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, D12–D21 (2013).

Rivensons, Y.

Y. Rivensons and A. Stern, “Compressed imaging with a separable sensing operator,” IEEE Signal Process. Lett. 16, 449–452 (2009).
[CrossRef]

Rueda, H.

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, D12–D21 (2013).

Shihao, J.

J. Shihao, X. Ya, and L. Carin, “Bayesian compressive sensing,” IEEE Trans. Signal Process. 56, 2346–2356(2008).
[CrossRef]

Starck, J.

D. Donoho, Y. Tsaig, I. Drori, and J. Starck, “Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit (StOMP),” Stat. Dept. Tech. Rep. (Stanford University, 2006).

Stern, A.

Y. Rivensons and A. Stern, “Compressed imaging with a separable sensing operator,” IEEE Signal Process. Lett. 16, 449–452 (2009).
[CrossRef]

Sun, X.

Tropp, J.

J. Tropp and S. Wright, “Computational methods for sparse solution of linear inverse problems,” Proc. IEEE 98, 948–958 (2010).
[CrossRef]

D. Needell and J. Tropp, “Iterative signal recovery from incomplete and inaccurate samples,” Appl. Comput. Harmon. Anal. 26, 301–321 (2009).
[CrossRef]

J. Tropp and A. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53, 4655–4666 (2007).
[CrossRef]

Tsaig, Y.

D. Donoho, Y. Tsaig, I. Drori, and J. Starck, “Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit (StOMP),” Stat. Dept. Tech. Rep. (Stanford University, 2006).

Wagadarikar, A.

Wagadarikar, A. A.

Willett, R.

Wright, S.

J. Tropp and S. Wright, “Computational methods for sparse solution of linear inverse problems,” Proc. IEEE 98, 948–958 (2010).
[CrossRef]

S. Wright, R. Nowak, and M. Figueiredo, “Sparse reconstruction by separable approximation,” IEEE Trans. Signal Process. 57, 2479–2493 (2009).
[CrossRef]

M. Figueiredo, R. Nowak, and S. Wright, “Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Signal Process. 1, 586–597 (2007).
[CrossRef]

Wu, J.

Y. Fang, L. Chen, J. Wu, and B. Huang, “GPU implementation of orthogonal matching pursuit for compressive sensing,” presented at IEEE 17th International Conference on Parallel and Distributed Systems (ICPADS), Tainan, Taiwan, December 2011.

Wu, Y.

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]

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, D12–D21 (2013).

Ya, X.

J. Shihao, X. Ya, and L. Carin, “Bayesian compressive sensing,” IEEE Trans. Signal Process. 56, 2346–2356(2008).
[CrossRef]

Appl. Comput. Harmon. Anal.

D. Needell and J. Tropp, “Iterative signal recovery from incomplete and inaccurate samples,” Appl. Comput. Harmon. Anal. 26, 301–321 (2009).
[CrossRef]

Appl. Opt.

IEEE J. Sel. Top. Signal Process.

M. Figueiredo, R. Nowak, and S. Wright, “Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Signal Process. 1, 586–597 (2007).
[CrossRef]

IEEE Signal Process. Lett.

R. Chartrand, “Exact reconstruction of sparse signals via nonconvex minimization,” IEEE Signal Process. Lett. 14, 707–710 (2007).
[CrossRef]

Y. Rivensons and A. Stern, “Compressed imaging with a separable sensing operator,” IEEE Signal Process. Lett. 16, 449–452 (2009).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process.

M. P. McLoughlin and G. R. Arce, “Deterministic properties of the recursive separable median filter,” IEEE Trans. Acoust. Speech Signal Process. 35, 98–106 (1987).
[CrossRef]

IEEE Trans. Image Process.

H. Arguello and G. R. Arce, “Rank minimization code aperture design for spectrally selective compressive imaging,” IEEE Trans. Image Process. 22, 941–954 (2013).
[CrossRef]

M. F. Duarte and R. G. Baraniuk, “Kronecker compressive sensing,” IEEE Trans. Image Process. 21, 494–504(2012).
[CrossRef]

J. Bioucas-Dias and M. Figueiredo, “A new TwIST: two-step iterative shrinking/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
[CrossRef]

IEEE Trans. Inf. Theory

J. Tropp and A. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53, 4655–4666 (2007).
[CrossRef]

IEEE Trans. Signal Process.

S. Wright, R. Nowak, and M. Figueiredo, “Sparse reconstruction by separable approximation,” IEEE Trans. Signal Process. 57, 2479–2493 (2009).
[CrossRef]

J. Shihao, X. Ya, and L. Carin, “Bayesian compressive sensing,” IEEE Trans. Signal Process. 56, 2346–2356(2008).
[CrossRef]

S. Kalluri and G. R. Arce, “Robust frequency-selective filtering using weighted myriad filters admitting real-valued weights,” IEEE Trans. Signal Process. 49, 2721–2733(2001).
[CrossRef]

J. Opt. Soc. Am.

H. Arguello and G. R. Arce, “Code aperture optimization for spectrally agile compressive imaging,” J. Opt. Soc. Am. 28, 2400–2413 (2011).
[CrossRef]

Opt. Express

Opt. Lett

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]

Proc. IEEE

J. Tropp and S. Wright, “Computational methods for sparse solution of linear inverse problems,” Proc. IEEE 98, 948–958 (2010).
[CrossRef]

Other

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, D12–D21 (2013).

D. Donoho, Y. Tsaig, I. Drori, and J. Starck, “Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit (StOMP),” Stat. Dept. Tech. Rep. (Stanford University, 2006).

L. Gan, “Block compressed sensing of natural images,” Proceedings of International Conference on Digital Signal Processing, Cardiff, UK, 1–4 July2007.

Y. Fang, L. Chen, J. Wu, and B. Huang, “GPU implementation of orthogonal matching pursuit for compressive sensing,” presented at IEEE 17th International Conference on Parallel and Distributed Systems (ICPADS), Tainan, Taiwan, December 2011.

Airborne Visible Infrared Imaging Spectrometer, http://aviris.jpl.nasa.gov/data/free_data.html .

H. Arguello and G. R. Arce, “Spectrally selective compressive imaging by matrix system analysis,” presented at OSA Optics and Photonics Congress, Monterey, California, June 2012.

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Figures (18)

Fig. 1.
Fig. 1.

CASSI architecture. The input signal f 0 ( x , y , λ ) is coded by the coded aperture T ( x , y ) and dispersed by the prism. The coded and dispersed signal f 2 ( x , y , λ ) is integrated on the FPA detector.

Fig. 2.
Fig. 2.

Process of CASSI imaging is depicted. A N × N × L spectral data cube is spatially coded by the coded aperture and dispersed by the prism. Each pixel at the detector contains the integration of the spectral information from the correspondent entries of the data cube.

Fig. 3.
Fig. 3.

Structure of the matrix H for N = 4 , L = 3 , and K = 2 .

Fig. 4.
Fig. 4.

Each B × B window at the detector results from sensing a B × B × L oblique parallelepiped block of the data cube. The CASSI sensing process is not modified.

Fig. 5.
Fig. 5.

Procedure to obtain the H m n i matrices from H i for N = 4 , L = 3 , and B = 2 . The nonzero elements are colored, where each color corresponds to a different submatrix H m n i . The resultant B 2 × B 2 L matrices are obtained by selecting the correspondent rows for each block and removing the zero-valued columns.

Fig. 6.
Fig. 6.

(a) Original spectral slice. (b) Reconstruction of a spectral band using the nonoverlapping block-model of CASSI. Horizontal and vertical artifacts can be noticed on the block boundaries.

Fig. 7.
Fig. 7.

Example of the block CASSI model using Δ overlapping rows and columns in the FPA measurement windows. Top: horizontal overlapping, bottom: vertical overlapping.

Fig. 8.
Fig. 8.

Vertically overlapped measurement windows lead to vertically overlapped oblique parallelepipeds within the data cube. Horizontally overlapped measurement windows and their reconstructions have a similar relation.

Fig. 9.
Fig. 9.

Top: subdivision of the k th spectral band of a recovered block of the data cube. Shaded regions correspond to Δ / 2 columns or rows on each side. Pixels in those regions are duplicated since they are reconstructed by the ( m , n ) th block and one of its four neighbors. Bottom: special cases for the blocks in the boundaries of the image.

Fig. 10.
Fig. 10.

Tiling in the reconstruction of a spectral band. The shaded zones show the average of the overlapped regions on consecutive parallelepipeds.

Fig. 11.
Fig. 11.

Twenty-four spectral band data cubes with wavelengths ranging from 452 to 667 nm. Each spectral slice has a spatial resolution of 256 × 256 pixels.

Fig. 12.
Fig. 12.

Reconstruction (Rec.) PSNR for the full data cube reconstruction and by the lapped block CASSI reconstruction for the (a)  256 × 256 × 24 , (b)  512 × 512 × 24 , (c)  1024 × 1024 × 24 , and (d)  512 × 512 × 32 data cubes.

Fig. 13.
Fig. 13.

(a) Original zoomed version of the 256 × 256 × 24 data cube. Zoomed versions of the reconstructions for 6 FPA measurement shots using, (b) lapped block approach with block size B = 64 and overlap Δ = 24 , 31.46 dB, and (c) traditional reconstruction approach, 28.1 dB.

Fig. 14.
Fig. 14.

(a) Original zoomed version of the 512 × 512 × 24 data cube. Reconstructions for 6 FPA measurements using, (b) lapped block reconstruction with block size B = 64 and overlap Δ = 24 , 33.45 dB, (c) traditional reconstruction, 31.09 dB, (d) original zoomed version of the 1024 × 1024 × 24 data cube, (e) lapped block reconstruction with B = 128 and Δ = 32 , 33.17 dB, and (f) traditional reconstruction, 32.99 dB.

Fig. 15.
Fig. 15.

(a) Original RGB and zoomed version of the 512 × 512 × 32 data cube. Reconstructions for 10 FPA measurement shots using, (b) lapped block approach with block size B = 64 and overlap Δ = 24 , 31.84 dB, and (c) traditional reconstruction approach, 30.99 dB.

Fig. 16.
Fig. 16.

Spectral reconstruction of the highlighted pixels in (a) for (b) pixel B, (c) pixel C, and (d) pixel D.

Fig. 17.
Fig. 17.

Reconstruction (Rec.) PSNR from noisy measurements, for the 256 × 256 × 24 full data cube reconstruction and by the lapped block reconstruction. (a)  SNR = 20 dB and (b)  SNR = 25 dB .

Fig. 18.
Fig. 18.

Reconstruction (Rec.) time as a function of the number of FPA measurements. Results for the traditional reconstruction and lapped block reconstruction approaches are shown. The average time for recovering an individual block using a single processor (one block) is also presented for (a)  256 × 256 , (b)  512 × 512 , (c)  1024 × 1024 spatial dimensions of the data cubes with L = 24 spectral bands, and (d)  512 × 512 × 32 data cube.

Tables (6)

Tables Icon

Table 1. CASSI Model Variables

Tables Icon

Table 2. CASSI Block Reconstruction Model Variablesa

Tables Icon

Table 3. CASSI Reconstruction Models Comparison

Tables Icon

Table 4. Individual Block Data Cube Reconstruction from Lapped Windows

Tables Icon

Table 5. Summary: Merging Process for Lapped Block CASSI Model

Tables Icon

Table 6. Parameters for Lapped Block Reconstructions

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

Y j i = k = 0 L 1 F j ( + k ) ( k ) T j ( + k ) i + ω j i i = 0 , , K 1 .
( y i ) = Y ( r N ) r i for = 0 , , M 1 , i = 0 , , K 1 ,
( f k ) = F ( r N ) r k for = 0 , , N 2 1 , k = 0 , , L 1 ,
H i = [ t 00 i t ( N 1 ) ( N 1 ) i 0 N ( L 1 ) × N 2 | 0 N ( 1 ) × N 2 t 00 i t ( N 1 ) ( N 1 ) i 0 N ( L 2 ) × N 2 | | 0 N ( L 1 ) × N 2 t 00 i t ( N 1 ) ( N 1 ) i ] ,
y = H f + ω ,
Y i = [ Y 0 , 0 i Y 0 , 1 i Y 0 , V 1 i Y N 1 , 0 i Y N 1 , 1 i Y N 1 , V 1 i ] ,
y m n i = H m n i f m n + ω m n i ,
( F m n ) j , , k = ( F ) m B + j + 1 , n B + + k + 1 , k ,
( H m n i ) , j = ( H i ) r , r j ,
y m n = H m n f m n + ω m n .
f ^ m n = Ψ ( argmin θ y m n H m n Ψ θ 2 + τ θ 1 ) ,
F ^ k = [ F ^ 00 k F ^ 0 ( V 1 ) k F ^ ( N 1 ) 0 k F ^ ( N 1 ) ( V 1 ) k ] ,
( H m n i ) , j = ( H i ) r , r j
F ˜ m , n , k = [ A m , n , k C m , n , k B m , n , k D m , n , k E m , n , k ] ,
E m ( m + 1 ) , n , k = E m , n , k + A ( m + 1 ) , n , k 2 , D m , n ( n + 1 ) , k = D m , n , k + C m , ( n + 1 ) , k 2
F ^ m , n , k = [ B m , n , k D m , n ( n + 1 ) , k E m ( m + 1 ) , n , k ] .
F ^ k = [ F ^ 0 , 0 , k F ^ 0 , 1 , k F ^ 0 , V 1 , k F ^ N 1 , 0 , k F ^ N 1 , 1 , k F ^ N 1 , V 1 , k ]

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