Abstract

This paper describes a generalized theoretical framework for a multiplexed spatially encoded imaging system to acquire multi-channel data. The framework is confirmed with simulations and experimental demonstrations. In the system, each channel associated with the object is spatially encoded, and the resultant signals are multiplexed onto a detector array. In the demultiplexing process, a numerical estimation algorithm with a sparsity constraint is used to solve the underdetermined reconstruction problem. The system can acquire object data in which the number of elements is larger than that of the captured data. This case includes multi-channel data acquisition by a single-shot with a detector array. In the experiments, wide field-of-view imaging and spectral imaging were demonstrated with sparse objects. A compressive sensing algorithm, called the two-step iterative shrinkage/thresholding algorithm with total variation, was adapted for object reconstruction.

© 2010 Optical Society of America

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References

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  1. A. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).
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  5. A. Levin, R. Fergus, F. Durand, and W. T. Freeman, “Image and depth from a conventional camera with a coded aperture,” in “SIGGRAPH ’07: ACM SIGGRAPH 2007 papers,” (ACM, New York, NY, USA, 2007), p. 70.
    [Crossref]
  6. A. Rajagopalan and S. Chaudhuri, “A variational approach to recovering depth from defocused images,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 1158–1164 (1997).
    [Crossref]
  7. Y. Tsaig and D. L. Donoho, “Compressed sensing,” IEEE Trans. Info. Theory 52, 1289–1306 (2006).
    [Crossref]
  8. E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
    [Crossref]
  9. R. Baraniuk, “Compressive sensing,” IEEE Signal Process. Mag. 24, 118–121 (2007).
    [Crossref]
  10. M. Wakin, J. Laska, M. Duarte, D. Baron, S. Sarvotham, D. Takhar, K. Kelly, and R. Baraniuk, “An architecture for compressive imaging,” in “ICIP06,” (2006), pp. 1273–1276.
  11. P. Ye, J. L. Paredes, G. R. Arce, Y. Wu, C. Chen, and D. W. Prather, “Compressive confocal microscopy,” in “ICASSP ’09: Proceedings of the 2009 IEEE International Conference on Acoustics, Speech and Signal Processing,” (IEEE Computer Society, Washington, DC, USA, 2009), pp. 429–432.
  12. A. Wagadarikar, R. John, R. Willett, and D. Brady, “Single disperser design for coded aperture snapshot spectral imaging,” Appl. Opt. 47, B44–BB51 (2008).
    [Crossref] [PubMed]
  13. D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17, 13040–13049 (2009).
    [Crossref] [PubMed]
  14. R. Horisaki, K. Choi, J. Hahn, J. Tanida, and D. J. Brady, “Generalized sampling using a compound-eye imaging system for multi-dimensional object acquisition,” Opt. Express 18, 19367–19378 (2010).
    [Crossref] [PubMed]
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  17. R. F. Marcia, C. Kim, C. Eldeniz, J. Kim, D. J. Brady, and R. M. Willett, “Superimposed video disambiguation for increased field of view,” Opt. Express 16, 16352–16363 (2008).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  21. R. Gribonval and M. Nielsen, “Sparse representations in unions of bases,” IEEE Trans. Info. Theory 49, 3320–3325 (2003).
    [Crossref]
  22. J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Proc. 16, 2992–3004 (2007).
    [Crossref]
  23. L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259–268 (1992).
    [Crossref]
  24. W. H. Richardson, “Bayesian-based iterative method of image restoration,” J. Opt. Soc. Am. 62, 55–59 (1972).
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2010 (1)

2009 (1)

2008 (3)

2007 (3)

R. Baraniuk, “Compressive sensing,” IEEE Signal Process. Mag. 24, 118–121 (2007).
[Crossref]

J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Proc. 16, 2992–3004 (2007).
[Crossref]

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

2006 (3)

Y. Tsaig and D. L. Donoho, “Compressed sensing,” IEEE Trans. Info. Theory 52, 1289–1306 (2006).
[Crossref]

M. Levoy, “Light fields and computational imaging,” IEEE Computer 39, 46–55 (2006).

M. Wakin, J. Laska, M. Duarte, D. Baron, S. Sarvotham, D. Takhar, K. Kelly, and R. Baraniuk, “An architecture for compressive imaging,” in “ICIP06,” (2006), pp. 1273–1276.

2005 (1)

E. J. Candes and T. Tao, “Decoding by linear programming,” IEEE Trans. Info. Theory 51, 4203–4215 (2005).
[Crossref]

2003 (1)

R. Gribonval and M. Nielsen, “Sparse representations in unions of bases,” IEEE Trans. Info. Theory 49, 3320–3325 (2003).
[Crossref]

1997 (1)

A. Rajagopalan and S. Chaudhuri, “A variational approach to recovering depth from defocused images,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 1158–1164 (1997).
[Crossref]

1995 (1)

1992 (1)

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259–268 (1992).
[Crossref]

1974 (1)

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[Crossref]

1972 (1)

Arce, G. R.

P. Ye, J. L. Paredes, G. R. Arce, Y. Wu, C. Chen, and D. W. Prather, “Compressive confocal microscopy,” in “ICASSP ’09: Proceedings of the 2009 IEEE International Conference on Acoustics, Speech and Signal Processing,” (IEEE Computer Society, Washington, DC, USA, 2009), pp. 429–432.

Ashok, A.

V. Treeaporn, A. Ashok, and M. A. Neifeld, “Increased field of view through optical multiplexing,” in “Imaging Systems,” (Optical Society of America, 2010, p. IMC4.

Baraniuk, R.

R. Baraniuk, “Compressive sensing,” IEEE Signal Process. Mag. 24, 118–121 (2007).
[Crossref]

M. Wakin, J. Laska, M. Duarte, D. Baron, S. Sarvotham, D. Takhar, K. Kelly, and R. Baraniuk, “An architecture for compressive imaging,” in “ICIP06,” (2006), pp. 1273–1276.

Baron, D.

M. Wakin, J. Laska, M. Duarte, D. Baron, S. Sarvotham, D. Takhar, K. Kelly, and R. Baraniuk, “An architecture for compressive imaging,” in “ICIP06,” (2006), pp. 1273–1276.

Bioucas-Dias, J. M.

J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Proc. 16, 2992–3004 (2007).
[Crossref]

Brady, D.

Brady, D. J.

Candes, E. J.

E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
[Crossref]

E. J. Candes and T. Tao, “Decoding by linear programming,” IEEE Trans. Info. Theory 51, 4203–4215 (2005).
[Crossref]

Cathey, W. T.

Chaudhuri, S.

A. Rajagopalan and S. Chaudhuri, “A variational approach to recovering depth from defocused images,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 1158–1164 (1997).
[Crossref]

Chen, C.

P. Ye, J. L. Paredes, G. R. Arce, Y. Wu, C. Chen, and D. W. Prather, “Compressive confocal microscopy,” in “ICASSP ’09: Proceedings of the 2009 IEEE International Conference on Acoustics, Speech and Signal Processing,” (IEEE Computer Society, Washington, DC, USA, 2009), pp. 429–432.

Choi, K.

Donoho, D. L.

Y. Tsaig and D. L. Donoho, “Compressed sensing,” IEEE Trans. Info. Theory 52, 1289–1306 (2006).
[Crossref]

Dowski, E. R.

Duarte, M.

M. Wakin, J. Laska, M. Duarte, D. Baron, S. Sarvotham, D. Takhar, K. Kelly, and R. Baraniuk, “An architecture for compressive imaging,” in “ICIP06,” (2006), pp. 1273–1276.

Durand, F.

A. Levin, R. Fergus, F. Durand, and W. T. Freeman, “Image and depth from a conventional camera with a coded aperture,” in “SIGGRAPH ’07: ACM SIGGRAPH 2007 papers,” (ACM, New York, NY, USA, 2007), p. 70.
[Crossref]

Eldeniz, C.

Fatemi, E.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259–268 (1992).
[Crossref]

Fergus, R.

A. Levin, R. Fergus, F. Durand, and W. T. Freeman, “Image and depth from a conventional camera with a coded aperture,” in “SIGGRAPH ’07: ACM SIGGRAPH 2007 papers,” (ACM, New York, NY, USA, 2007), p. 70.
[Crossref]

Figueiredo, M. A. T.

J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Proc. 16, 2992–3004 (2007).
[Crossref]

Freeman, W. T.

A. Levin, R. Fergus, F. Durand, and W. T. Freeman, “Image and depth from a conventional camera with a coded aperture,” in “SIGGRAPH ’07: ACM SIGGRAPH 2007 papers,” (ACM, New York, NY, USA, 2007), p. 70.
[Crossref]

Gehm, M. E.

Gribonval, R.

R. Gribonval and M. Nielsen, “Sparse representations in unions of bases,” IEEE Trans. Info. Theory 49, 3320–3325 (2003).
[Crossref]

Hahn, J.

Hecht, E.

E. Hecht, Optics (Addison Wesley, 2001), 4th ed.

Hiura, S.

S. Hiura and T. Matsuyama, “Depth measurement by the multi-focus camera,” in “CVPR ’98: Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition,” (IEEE Computer Society, Washington, DC, USA, 1998), p. 953.

Horisaki, R.

John, R.

Kak, A.

A. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).

Kelly, K.

M. Wakin, J. Laska, M. Duarte, D. Baron, S. Sarvotham, D. Takhar, K. Kelly, and R. Baraniuk, “An architecture for compressive imaging,” in “ICIP06,” (2006), pp. 1273–1276.

Kim, C.

R. F. Marcia, C. Kim, C. Eldeniz, J. Kim, D. J. Brady, and R. M. Willett, “Superimposed video disambiguation for increased field of view,” Opt. Express 16, 16352–16363 (2008).
[Crossref] [PubMed]

R. F. Marcia, C. Kim, J. Kim, D. J. Brady, and R. M. Willett, “Fast disambiguation of superimposed images for increased field of view,” in “Proceedings of the IEEE International Conference on Image Processing,” (San Diego, CA, 2008), pp. 2620–2623.

Kim, J.

R. F. Marcia, C. Kim, C. Eldeniz, J. Kim, D. J. Brady, and R. M. Willett, “Superimposed video disambiguation for increased field of view,” Opt. Express 16, 16352–16363 (2008).
[Crossref] [PubMed]

R. F. Marcia, C. Kim, J. Kim, D. J. Brady, and R. M. Willett, “Fast disambiguation of superimposed images for increased field of view,” in “Proceedings of the IEEE International Conference on Image Processing,” (San Diego, CA, 2008), pp. 2620–2623.

Laska, J.

M. Wakin, J. Laska, M. Duarte, D. Baron, S. Sarvotham, D. Takhar, K. Kelly, and R. Baraniuk, “An architecture for compressive imaging,” in “ICIP06,” (2006), pp. 1273–1276.

Levin, A.

A. Levin, R. Fergus, F. Durand, and W. T. Freeman, “Image and depth from a conventional camera with a coded aperture,” in “SIGGRAPH ’07: ACM SIGGRAPH 2007 papers,” (ACM, New York, NY, USA, 2007), p. 70.
[Crossref]

Levoy, M.

M. Levoy, “Light fields and computational imaging,” IEEE Computer 39, 46–55 (2006).

Lim, S.

Lucy, L. B.

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[Crossref]

Marcia, R. F.

R. F. Marcia, C. Kim, C. Eldeniz, J. Kim, D. J. Brady, and R. M. Willett, “Superimposed video disambiguation for increased field of view,” Opt. Express 16, 16352–16363 (2008).
[Crossref] [PubMed]

R. F. Marcia, C. Kim, J. Kim, D. J. Brady, and R. M. Willett, “Fast disambiguation of superimposed images for increased field of view,” in “Proceedings of the IEEE International Conference on Image Processing,” (San Diego, CA, 2008), pp. 2620–2623.

Marks, D. L.

Matsuyama, T.

S. Hiura and T. Matsuyama, “Depth measurement by the multi-focus camera,” in “CVPR ’98: Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition,” (IEEE Computer Society, Washington, DC, USA, 1998), p. 953.

Neifeld, M. A.

M. D. Stenner, P. Shankar, and M. A. Neifeld, “Wide-field feature-specific imaging,” in “Frontiers in Optics,” (Optical Society of America, 2007), p. FMJ2.

V. Treeaporn, A. Ashok, and M. A. Neifeld, “Increased field of view through optical multiplexing,” in “Imaging Systems,” (Optical Society of America, 2010, p. IMC4.

Nielsen, M.

R. Gribonval and M. Nielsen, “Sparse representations in unions of bases,” IEEE Trans. Info. Theory 49, 3320–3325 (2003).
[Crossref]

Osher, S.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259–268 (1992).
[Crossref]

Paredes, J. L.

P. Ye, J. L. Paredes, G. R. Arce, Y. Wu, C. Chen, and D. W. Prather, “Compressive confocal microscopy,” in “ICASSP ’09: Proceedings of the 2009 IEEE International Conference on Acoustics, Speech and Signal Processing,” (IEEE Computer Society, Washington, DC, USA, 2009), pp. 429–432.

Prather, D. W.

P. Ye, J. L. Paredes, G. R. Arce, Y. Wu, C. Chen, and D. W. Prather, “Compressive confocal microscopy,” in “ICASSP ’09: Proceedings of the 2009 IEEE International Conference on Acoustics, Speech and Signal Processing,” (IEEE Computer Society, Washington, DC, USA, 2009), pp. 429–432.

Rajagopalan, A.

A. Rajagopalan and S. Chaudhuri, “A variational approach to recovering depth from defocused images,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 1158–1164 (1997).
[Crossref]

Richardson, W. H.

Rudin, L. I.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259–268 (1992).
[Crossref]

Sarvotham, S.

M. Wakin, J. Laska, M. Duarte, D. Baron, S. Sarvotham, D. Takhar, K. Kelly, and R. Baraniuk, “An architecture for compressive imaging,” in “ICIP06,” (2006), pp. 1273–1276.

Schulz, T. J.

Shankar, P.

M. D. Stenner, P. Shankar, and M. A. Neifeld, “Wide-field feature-specific imaging,” in “Frontiers in Optics,” (Optical Society of America, 2007), p. FMJ2.

Slaney, M.

A. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).

Stenner, M. D.

M. D. Stenner, P. Shankar, and M. A. Neifeld, “Wide-field feature-specific imaging,” in “Frontiers in Optics,” (Optical Society of America, 2007), p. FMJ2.

Takhar, D.

M. Wakin, J. Laska, M. Duarte, D. Baron, S. Sarvotham, D. Takhar, K. Kelly, and R. Baraniuk, “An architecture for compressive imaging,” in “ICIP06,” (2006), pp. 1273–1276.

Tanida, J.

Tao, T.

E. J. Candes and T. Tao, “Decoding by linear programming,” IEEE Trans. Info. Theory 51, 4203–4215 (2005).
[Crossref]

Treeaporn, V.

V. Treeaporn, A. Ashok, and M. A. Neifeld, “Increased field of view through optical multiplexing,” in “Imaging Systems,” (Optical Society of America, 2010, p. IMC4.

Tsaig, Y.

Y. Tsaig and D. L. Donoho, “Compressed sensing,” IEEE Trans. Info. Theory 52, 1289–1306 (2006).
[Crossref]

Wagadarikar, A.

Wakin, M.

M. Wakin, J. Laska, M. Duarte, D. Baron, S. Sarvotham, D. Takhar, K. Kelly, and R. Baraniuk, “An architecture for compressive imaging,” in “ICIP06,” (2006), pp. 1273–1276.

Wakin, M. B.

E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
[Crossref]

Willett, R.

Willett, R. M.

Wu, Y.

P. Ye, J. L. Paredes, G. R. Arce, Y. Wu, C. Chen, and D. W. Prather, “Compressive confocal microscopy,” in “ICASSP ’09: Proceedings of the 2009 IEEE International Conference on Acoustics, Speech and Signal Processing,” (IEEE Computer Society, Washington, DC, USA, 2009), pp. 429–432.

Ye, P.

P. Ye, J. L. Paredes, G. R. Arce, Y. Wu, C. Chen, and D. W. Prather, “Compressive confocal microscopy,” in “ICASSP ’09: Proceedings of the 2009 IEEE International Conference on Acoustics, Speech and Signal Processing,” (IEEE Computer Society, Washington, DC, USA, 2009), pp. 429–432.

“ICIP06,” (1)

M. Wakin, J. Laska, M. Duarte, D. Baron, S. Sarvotham, D. Takhar, K. Kelly, and R. Baraniuk, “An architecture for compressive imaging,” in “ICIP06,” (2006), pp. 1273–1276.

Appl. Opt. (2)

Astron. J. (1)

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[Crossref]

IEEE Computer (1)

M. Levoy, “Light fields and computational imaging,” IEEE Computer 39, 46–55 (2006).

IEEE Signal Process. Mag. (2)

E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
[Crossref]

R. Baraniuk, “Compressive sensing,” IEEE Signal Process. Mag. 24, 118–121 (2007).
[Crossref]

IEEE Trans. Image Proc. (1)

J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Proc. 16, 2992–3004 (2007).
[Crossref]

IEEE Trans. Info. Theory (3)

E. J. Candes and T. Tao, “Decoding by linear programming,” IEEE Trans. Info. Theory 51, 4203–4215 (2005).
[Crossref]

R. Gribonval and M. Nielsen, “Sparse representations in unions of bases,” IEEE Trans. Info. Theory 49, 3320–3325 (2003).
[Crossref]

Y. Tsaig and D. L. Donoho, “Compressed sensing,” IEEE Trans. Info. Theory 52, 1289–1306 (2006).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

A. Rajagopalan and S. Chaudhuri, “A variational approach to recovering depth from defocused images,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 1158–1164 (1997).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Express (4)

Phys. D (1)

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259–268 (1992).
[Crossref]

Other (8)

V. Treeaporn, A. Ashok, and M. A. Neifeld, “Increased field of view through optical multiplexing,” in “Imaging Systems,” (Optical Society of America, 2010, p. IMC4.

E. Hecht, Optics (Addison Wesley, 2001), 4th ed.

S. Hiura and T. Matsuyama, “Depth measurement by the multi-focus camera,” in “CVPR ’98: Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition,” (IEEE Computer Society, Washington, DC, USA, 1998), p. 953.

A. Levin, R. Fergus, F. Durand, and W. T. Freeman, “Image and depth from a conventional camera with a coded aperture,” in “SIGGRAPH ’07: ACM SIGGRAPH 2007 papers,” (ACM, New York, NY, USA, 2007), p. 70.
[Crossref]

P. Ye, J. L. Paredes, G. R. Arce, Y. Wu, C. Chen, and D. W. Prather, “Compressive confocal microscopy,” in “ICASSP ’09: Proceedings of the 2009 IEEE International Conference on Acoustics, Speech and Signal Processing,” (IEEE Computer Society, Washington, DC, USA, 2009), pp. 429–432.

A. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).

M. D. Stenner, P. Shankar, and M. A. Neifeld, “Wide-field feature-specific imaging,” in “Frontiers in Optics,” (Optical Society of America, 2007), p. FMJ2.

R. F. Marcia, C. Kim, J. Kim, D. J. Brady, and R. M. Willett, “Fast disambiguation of superimposed images for increased field of view,” in “Proceedings of the IEEE International Conference on Image Processing,” (San Diego, CA, 2008), pp. 2620–2623.

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Figures (13)

Fig. 1
Fig. 1 System model of multi-channel data acquisition using multiplexed imaging with spatial encoding.

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Fig. 2
Fig. 2 Simulations with an object with a smaller sparsity s. (a) The object. The captured data and the reconstructed results in (b), (c) system A, (d), (e) system B, and (f), (g) system C. (h) The result reconstructed from Fig. (b) by the Richardson-Lucy method.

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Fig. 3
Fig. 3 Simulations with an object with larger sparsity s. (a) The object, (b) the captured data, and (c) the reconstructed result in system A.

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Fig. 4
Fig. 4 Plots of reconstruction PSNR from noisy measurements in the proposed system with selected parameters.

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Fig. 5
Fig. 5 Experimental setup of wide field-of-view imaging with phase grating.

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Fig. 6
Fig. 6 Impulse response from sub-field 1. The numbers indicate the index l of the copies of a printed dot.

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Fig. 7
Fig. 7 Experimental results of wide field-of-view imaging with phase grating. (a) Captured data and (b) the reconstructed result.

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Fig. 8
Fig. 8 Experimental setup of wide field-of-view imaging with multiple overlapping sub-fields. Dashed lines show fields-of-view.

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Fig. 9
Fig. 9 Impulse response of wide field-of-view imaging with multiple overlapping sub-fields. The numbers indicate the index l of the copies of a printed dot.

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Fig. 10
Fig. 10 Experimental results of wide field-of-view imaging with multiple overlapping sub-fields. (a) Captured data in the center sub-field, (b) captured data with all sub-fields, and (c) the reconstructed result.

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Fig. 11
Fig. 11 Schematic diagrams of spectral imaging demonstration. The solid lines and the dashed lines show diffracted rays of the 0-th and the 1-st orders in (a) red and (b) green lights, respectively.

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Fig. 12
Fig. 12 Impulse responses of (a) the red and (b) the green inks. The numbers indicate the index l of the copies of a printed dot.

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Fig. 13
Fig. 13 Experimental results of spectral imaging. (a) Original data, (b) the captured data, and (c) the reconstructed result.

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Tables (4)

Tables Icon

Table 1 Parameters of (a) system A, (b) system B, and (c) system C in the simulations

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Table 2 The parameters of the impulse responses for wide field-of-view imaging

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Table 3 The parameters of the impulse responses for wide field-of-view imaging with multiple overlapping sub-fields

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Table 4 The parameters of the impulse responses for spectral imaging

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Equations (10)

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g = Φ f = Φ Ψ β = Θ β ,
( 1 − ɛ ) ‖ β Λ ‖ 2 ≤ ‖ Θ Λ β Λ ‖ 2 ≤ ( 1 + ɛ ) ‖ β Λ ‖ 2 ,
β ^ = argmin β ‖ β ‖ 1   subject   to   g = Θ β ,
𝒢 ( x ) = ∑ c = 0 N c − 1 ∑ l = 0 ℒ ( c ) − 1 𝒲 ( l , c ) ℱ ( x − 𝒮 ( l , c ) , c ) ,
A l , c ( p , q ) = { 𝒲 ( l , c ) ( p = q + 𝒮 ( l , c ) ) , 0 ( p ≠ q + 𝒮 ( l , c ) ) ,
E c = ∑ l = 0 ℒ ( c ) − 1 A l , c .
T = [ E 0 O … O O E 1 … O ⋮ ⋮ ⋱ ⋮ O O ⋯ E N c − 1 ] ,
M = [ I   I   ⋯   I ] ,
g = Φ f = MTf = [ E 0   E 1   ⋯ E N c − 1 ] f ,
Σ c Σ x Σ y ( ℱ ( x + 1 , y , c ) − ℱ ( x , y , c ) ) 2 + ( ℱ ( x , y + 1 , c ) − ℱ ( x , y , c ) ) 2 .

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