Abstract

This paper describes a single-shot spectral imaging approach based on the concept of compressive sensing. The primary features of the system design are two dispersive elements, arranged in opposition and surrounding a binary-valued aperture code. In contrast to thin-film approaches to spectral filtering, this structure results in easily-controllable, spatially-varying, spectral filter functions with narrow features. Measurement of the input scene through these filters is equivalent to projective measurement in the spectral domain, and hence can be treated with the compressive sensing frameworks recently developed by a number of groups. We present a reconstruction framework and demonstrate its application to experimental data.

© 2007 Optical Society of America

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References

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  1. P. A. Townsend, J. R. Foster, J. R. A. Chastain, and W. S. Currie, "Application of imaging spectroscopy to mapping canopy nitrogen in the forests of the central Appalachian Mountains using Hyperion and AVIRIS," IEEE Trans. Geosci. Remote Sens. 41, 1347-1354 (2003).
    [CrossRef]
  2. W. Smith, D. Zhou, F. Harrison, H. Revercomb, A. Larar, A. Huang, and B. Huang, "Hyperspectral remote sensing of atmospheric profiles from satellites and aircraft," Proc. SPIE 4151, 94 - 102 (2001).
    [CrossRef]
  3. R. P. Lin, B. R. Dennis, and A. O. B. eds., The Reuven Ramaty High-Energy Solar Spectrscopic Imager (RHESSI) - Mission Description and Early Results (Kluwer Academic Publishers, Dordrecht, 2003).
    [PubMed]
  4. W. Denk, J. Strickler, and W. Webb, "Two-photon laser scanning fluorescence microscopy," Science 248, 73-76 (1990).
    [CrossRef] [PubMed]
  5. R. Schultz, T. Nielsen, J. Zavaleta, R. Ruch, R. Wyatt, and H. Garner, "Hyperspectral imaging: A novel approach for microscopic analysis," Cytometry 43, 239 - 247 (2001).
    [CrossRef] [PubMed]
  6. M. Hinnrichs, J. Jensen, and G. McAnally, "Handheld hyperspectral imager for standoff detection of chemical and biological aerosols," Proc. SPIE 5268, 67-78 (2003).
    [CrossRef]
  7. J. Mooney, V. Vickers, M. An, and A. Brodzik, "High-throughput hyperspectral infrared camera," J. Opt. Soc. Am. A 14, 2951-2961 (1997).
    [CrossRef]
  8. M. Descour, C. Volin, E. Dereniak, T. Gleeson, M. Hopkins, D. Wilson, and P. Maker, "Demonstration of a computed-tomography imaging spectrometer using a computer-generated hologram disperser," Appl. Opt. 36, 3694-3698 (1997).
    [CrossRef] [PubMed]
  9. M. E. Gehm and D. J. Brady, "High-throughput hyperspectral microscopy," Proc. SPIE 6090, 13-21 (2006).
  10. M. Gehm, S. McCain, N. Pitsianis, D. Brady, P. Potuluri, and M. Sullivan, "Static two-dimensional aperture coding for multimodal, multiplex spectroscopy," Appl. Opt. 45, 2965-2974 (2006).
    [CrossRef] [PubMed]
  11. S. McCain, M. Gehm, Y. Wang, N. Pitsianis, and D. Brady, "Coded Aperture Raman Spectroscopy for Quantitative Measurements of Ethanol in a Tissue Phantom," Appl. Spectrosc. 60, 663-671 (2006).
    [CrossRef] [PubMed]
  12. E. Cull, M. Gehm, D. Brady, C. Hsieh, O. Momtahan, and A. Adibi, "Dispersion multiplexing with broadband filtering for miniature spectrometers," Appl. Opt. 46, 365-374 (2007).
    [CrossRef] [PubMed]
  13. C. Fernandez, B. Guenther, M. Gehm, D. Brady, and M. Sullivan, "Longwave infrared (LWIR) coded aperture dispersive spectrometer," Opt. Express 15, 5742-5753 (2007).
    [CrossRef] [PubMed]
  14. M. E. Gehm, D. J. Brady, N. Pitsianis, and X. Sun, "Compressive sampling strategies for integrated microspectrometers," Proc. SPIE, 6232 (SPIE, 2006).
  15. D. J. Brady and M. E. Gehm, "Compressive imaging spectrometers using coded apertures," Proc. SPIE, 6246 (SPIE, 2006).
    [CrossRef]
  16. R. M. Willet, M. E. Gehm, and D. J. Brady, "Multiscale reconstruction for computational spectral imaging," Proc. SPIE, 6498 (SPIE-IS and Electronic Imaging, 2007).
    [CrossRef]
  17. E. Cand`es, J. Romberg, and T. Tao, "Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information," IEEE Trans. Inf. Theory 52, 489 - 509 (2006).
    [CrossRef]
  18. E. Candes and T. Tao, "Near Optimal Signal Recovery From Random Projections: Universal Encoding Strategies," (2006). To be published in IEEE Trans. Inf. Theory. http://www.acm.caltech.edu/ emmanuel/ papers/OptimalRecovery.pdf>.
  19. D. Donoho, "Compressed Sensing," IEEE Trans. Inf. Theory 52, 1289-1306 (2006).
    [CrossRef]
  20. M. Harwit and N. Sloane, Hadamard Transform Optics (Academic Press, New York, 1979).
  21. E. Cand`es, J. Romberg, and T. Tao, "Stable Signal Recovery from Incomplete and Inaccurate Measurements," Commun. Pure Appl. Math. 59, 1207-1223 (2006).
    [CrossRef]
  22. J. Haupt and R. Nowak, "Signal Reconstruction from Noisy Random Projections," (2006). To be published in IEEE Trans. Inf. Theory. http://www.ece.wisc.edu/ nowak/infth.pdf>.
  23. W. Richardson, "Bayesian-based iterative method of image restoration," J. Opt. Soc. of Am. 62, 55-59 (1972).
    [CrossRef]
  24. L. B. Lucy, "An iterative technique for the rectification of observed distributions," Astron. J. 79, 745-754 (1974).
    [CrossRef]
  25. R. Nowak and E. Kolaczyk, "A Multiscale Statistical Framework for Poisson Inverse Problems," IEEE Trans. Inf. Theory 46, 1811-1825 (2000).
    [CrossRef]
  26. R. Willett, "Multiscale intensity estimation for marked Poisson processes," in Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP) (2007).
  27. E. Kolaczyk and R. Nowak, "Multiscale Likelihood Analysis and Complexity Penalized Estimation," Annals of Stat. 32, 500-527 (2004).
    [CrossRef]
  28. R. Willett and R. Nowak, "Fast Multiresolution Photon-Limited Image Reconstruction," in Proc. IEEE Int. Sym. Biomedical Imaging — ISBI ’04 (15-18 April, Arlington, VA, USA, 2004).
  29. M. Lang, H. Guo, J. E. Odegard, C. S. Burrus, and R. O. Wells, "Noise reduction using an undecimated discrete wavelet transform," IEEE Signal Processing Lett. 3, 10-12 (1996).
    [CrossRef]

2007

2006

E. Cand`es, J. Romberg, and T. Tao, "Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information," IEEE Trans. Inf. Theory 52, 489 - 509 (2006).
[CrossRef]

D. Donoho, "Compressed Sensing," IEEE Trans. Inf. Theory 52, 1289-1306 (2006).
[CrossRef]

E. Cand`es, J. Romberg, and T. Tao, "Stable Signal Recovery from Incomplete and Inaccurate Measurements," Commun. Pure Appl. Math. 59, 1207-1223 (2006).
[CrossRef]

M. E. Gehm and D. J. Brady, "High-throughput hyperspectral microscopy," Proc. SPIE 6090, 13-21 (2006).

M. Gehm, S. McCain, N. Pitsianis, D. Brady, P. Potuluri, and M. Sullivan, "Static two-dimensional aperture coding for multimodal, multiplex spectroscopy," Appl. Opt. 45, 2965-2974 (2006).
[CrossRef] [PubMed]

S. McCain, M. Gehm, Y. Wang, N. Pitsianis, and D. Brady, "Coded Aperture Raman Spectroscopy for Quantitative Measurements of Ethanol in a Tissue Phantom," Appl. Spectrosc. 60, 663-671 (2006).
[CrossRef] [PubMed]

2004

E. Kolaczyk and R. Nowak, "Multiscale Likelihood Analysis and Complexity Penalized Estimation," Annals of Stat. 32, 500-527 (2004).
[CrossRef]

2003

P. A. Townsend, J. R. Foster, J. R. A. Chastain, and W. S. Currie, "Application of imaging spectroscopy to mapping canopy nitrogen in the forests of the central Appalachian Mountains using Hyperion and AVIRIS," IEEE Trans. Geosci. Remote Sens. 41, 1347-1354 (2003).
[CrossRef]

M. Hinnrichs, J. Jensen, and G. McAnally, "Handheld hyperspectral imager for standoff detection of chemical and biological aerosols," Proc. SPIE 5268, 67-78 (2003).
[CrossRef]

2001

R. Schultz, T. Nielsen, J. Zavaleta, R. Ruch, R. Wyatt, and H. Garner, "Hyperspectral imaging: A novel approach for microscopic analysis," Cytometry 43, 239 - 247 (2001).
[CrossRef] [PubMed]

2000

R. Nowak and E. Kolaczyk, "A Multiscale Statistical Framework for Poisson Inverse Problems," IEEE Trans. Inf. Theory 46, 1811-1825 (2000).
[CrossRef]

1997

1996

M. Lang, H. Guo, J. E. Odegard, C. S. Burrus, and R. O. Wells, "Noise reduction using an undecimated discrete wavelet transform," IEEE Signal Processing Lett. 3, 10-12 (1996).
[CrossRef]

1990

W. Denk, J. Strickler, and W. Webb, "Two-photon laser scanning fluorescence microscopy," Science 248, 73-76 (1990).
[CrossRef] [PubMed]

1974

L. B. Lucy, "An iterative technique for the rectification of observed distributions," Astron. J. 79, 745-754 (1974).
[CrossRef]

1972

W. Richardson, "Bayesian-based iterative method of image restoration," J. Opt. Soc. of Am. 62, 55-59 (1972).
[CrossRef]

Adibi, A.

An, M.

Brady, D.

Brady, D. J.

M. E. Gehm and D. J. Brady, "High-throughput hyperspectral microscopy," Proc. SPIE 6090, 13-21 (2006).

Brodzik, A.

Burrus, C. S.

M. Lang, H. Guo, J. E. Odegard, C. S. Burrus, and R. O. Wells, "Noise reduction using an undecimated discrete wavelet transform," IEEE Signal Processing Lett. 3, 10-12 (1996).
[CrossRef]

Cand`es, E.

E. Cand`es, J. Romberg, and T. Tao, "Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information," IEEE Trans. Inf. Theory 52, 489 - 509 (2006).
[CrossRef]

E. Cand`es, J. Romberg, and T. Tao, "Stable Signal Recovery from Incomplete and Inaccurate Measurements," Commun. Pure Appl. Math. 59, 1207-1223 (2006).
[CrossRef]

Chastain, J. R. A.

P. A. Townsend, J. R. Foster, J. R. A. Chastain, and W. S. Currie, "Application of imaging spectroscopy to mapping canopy nitrogen in the forests of the central Appalachian Mountains using Hyperion and AVIRIS," IEEE Trans. Geosci. Remote Sens. 41, 1347-1354 (2003).
[CrossRef]

Cull, E.

Currie, W. S.

P. A. Townsend, J. R. Foster, J. R. A. Chastain, and W. S. Currie, "Application of imaging spectroscopy to mapping canopy nitrogen in the forests of the central Appalachian Mountains using Hyperion and AVIRIS," IEEE Trans. Geosci. Remote Sens. 41, 1347-1354 (2003).
[CrossRef]

Denk, W.

W. Denk, J. Strickler, and W. Webb, "Two-photon laser scanning fluorescence microscopy," Science 248, 73-76 (1990).
[CrossRef] [PubMed]

Dereniak, E.

Descour, M.

Donoho, D.

D. Donoho, "Compressed Sensing," IEEE Trans. Inf. Theory 52, 1289-1306 (2006).
[CrossRef]

Fernandez, C.

Foster, J. R.

P. A. Townsend, J. R. Foster, J. R. A. Chastain, and W. S. Currie, "Application of imaging spectroscopy to mapping canopy nitrogen in the forests of the central Appalachian Mountains using Hyperion and AVIRIS," IEEE Trans. Geosci. Remote Sens. 41, 1347-1354 (2003).
[CrossRef]

Garner, H.

R. Schultz, T. Nielsen, J. Zavaleta, R. Ruch, R. Wyatt, and H. Garner, "Hyperspectral imaging: A novel approach for microscopic analysis," Cytometry 43, 239 - 247 (2001).
[CrossRef] [PubMed]

Gehm, M.

Gehm, M. E.

M. E. Gehm and D. J. Brady, "High-throughput hyperspectral microscopy," Proc. SPIE 6090, 13-21 (2006).

Gleeson, T.

Guenther, B.

Guo, H.

M. Lang, H. Guo, J. E. Odegard, C. S. Burrus, and R. O. Wells, "Noise reduction using an undecimated discrete wavelet transform," IEEE Signal Processing Lett. 3, 10-12 (1996).
[CrossRef]

Hinnrichs, M.

M. Hinnrichs, J. Jensen, and G. McAnally, "Handheld hyperspectral imager for standoff detection of chemical and biological aerosols," Proc. SPIE 5268, 67-78 (2003).
[CrossRef]

Hopkins, M.

Hsieh, C.

Jensen, J.

M. Hinnrichs, J. Jensen, and G. McAnally, "Handheld hyperspectral imager for standoff detection of chemical and biological aerosols," Proc. SPIE 5268, 67-78 (2003).
[CrossRef]

Kolaczyk, E.

E. Kolaczyk and R. Nowak, "Multiscale Likelihood Analysis and Complexity Penalized Estimation," Annals of Stat. 32, 500-527 (2004).
[CrossRef]

R. Nowak and E. Kolaczyk, "A Multiscale Statistical Framework for Poisson Inverse Problems," IEEE Trans. Inf. Theory 46, 1811-1825 (2000).
[CrossRef]

Lang, M.

M. Lang, H. Guo, J. E. Odegard, C. S. Burrus, and R. O. Wells, "Noise reduction using an undecimated discrete wavelet transform," IEEE Signal Processing Lett. 3, 10-12 (1996).
[CrossRef]

Lucy, L. B.

L. B. Lucy, "An iterative technique for the rectification of observed distributions," Astron. J. 79, 745-754 (1974).
[CrossRef]

Maker, P.

McAnally, G.

M. Hinnrichs, J. Jensen, and G. McAnally, "Handheld hyperspectral imager for standoff detection of chemical and biological aerosols," Proc. SPIE 5268, 67-78 (2003).
[CrossRef]

McCain, S.

Momtahan, O.

Mooney, J.

Nielsen, T.

R. Schultz, T. Nielsen, J. Zavaleta, R. Ruch, R. Wyatt, and H. Garner, "Hyperspectral imaging: A novel approach for microscopic analysis," Cytometry 43, 239 - 247 (2001).
[CrossRef] [PubMed]

Nowak, R.

E. Kolaczyk and R. Nowak, "Multiscale Likelihood Analysis and Complexity Penalized Estimation," Annals of Stat. 32, 500-527 (2004).
[CrossRef]

R. Nowak and E. Kolaczyk, "A Multiscale Statistical Framework for Poisson Inverse Problems," IEEE Trans. Inf. Theory 46, 1811-1825 (2000).
[CrossRef]

Odegard, J. E.

M. Lang, H. Guo, J. E. Odegard, C. S. Burrus, and R. O. Wells, "Noise reduction using an undecimated discrete wavelet transform," IEEE Signal Processing Lett. 3, 10-12 (1996).
[CrossRef]

Pitsianis, N.

Potuluri, P.

Richardson, W.

W. Richardson, "Bayesian-based iterative method of image restoration," J. Opt. Soc. of Am. 62, 55-59 (1972).
[CrossRef]

Romberg, J.

E. Cand`es, J. Romberg, and T. Tao, "Stable Signal Recovery from Incomplete and Inaccurate Measurements," Commun. Pure Appl. Math. 59, 1207-1223 (2006).
[CrossRef]

E. Cand`es, J. Romberg, and T. Tao, "Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information," IEEE Trans. Inf. Theory 52, 489 - 509 (2006).
[CrossRef]

Ruch, R.

R. Schultz, T. Nielsen, J. Zavaleta, R. Ruch, R. Wyatt, and H. Garner, "Hyperspectral imaging: A novel approach for microscopic analysis," Cytometry 43, 239 - 247 (2001).
[CrossRef] [PubMed]

Schultz, R.

R. Schultz, T. Nielsen, J. Zavaleta, R. Ruch, R. Wyatt, and H. Garner, "Hyperspectral imaging: A novel approach for microscopic analysis," Cytometry 43, 239 - 247 (2001).
[CrossRef] [PubMed]

Strickler, J.

W. Denk, J. Strickler, and W. Webb, "Two-photon laser scanning fluorescence microscopy," Science 248, 73-76 (1990).
[CrossRef] [PubMed]

Sullivan, M.

Tao, T.

E. Cand`es, J. Romberg, and T. Tao, "Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information," IEEE Trans. Inf. Theory 52, 489 - 509 (2006).
[CrossRef]

E. Cand`es, J. Romberg, and T. Tao, "Stable Signal Recovery from Incomplete and Inaccurate Measurements," Commun. Pure Appl. Math. 59, 1207-1223 (2006).
[CrossRef]

Townsend, P. A.

P. A. Townsend, J. R. Foster, J. R. A. Chastain, and W. S. Currie, "Application of imaging spectroscopy to mapping canopy nitrogen in the forests of the central Appalachian Mountains using Hyperion and AVIRIS," IEEE Trans. Geosci. Remote Sens. 41, 1347-1354 (2003).
[CrossRef]

Vickers, V.

Volin, C.

Wang, Y.

Webb, W.

W. Denk, J. Strickler, and W. Webb, "Two-photon laser scanning fluorescence microscopy," Science 248, 73-76 (1990).
[CrossRef] [PubMed]

Wells, R. O.

M. Lang, H. Guo, J. E. Odegard, C. S. Burrus, and R. O. Wells, "Noise reduction using an undecimated discrete wavelet transform," IEEE Signal Processing Lett. 3, 10-12 (1996).
[CrossRef]

Wilson, D.

Wyatt, R.

R. Schultz, T. Nielsen, J. Zavaleta, R. Ruch, R. Wyatt, and H. Garner, "Hyperspectral imaging: A novel approach for microscopic analysis," Cytometry 43, 239 - 247 (2001).
[CrossRef] [PubMed]

Zavaleta, J.

R. Schultz, T. Nielsen, J. Zavaleta, R. Ruch, R. Wyatt, and H. Garner, "Hyperspectral imaging: A novel approach for microscopic analysis," Cytometry 43, 239 - 247 (2001).
[CrossRef] [PubMed]

Annals of Stat.

E. Kolaczyk and R. Nowak, "Multiscale Likelihood Analysis and Complexity Penalized Estimation," Annals of Stat. 32, 500-527 (2004).
[CrossRef]

Appl. Opt.

Appl. Spectrosc.

Astron. J.

L. B. Lucy, "An iterative technique for the rectification of observed distributions," Astron. J. 79, 745-754 (1974).
[CrossRef]

Commun. Pure Appl. Math.

E. Cand`es, J. Romberg, and T. Tao, "Stable Signal Recovery from Incomplete and Inaccurate Measurements," Commun. Pure Appl. Math. 59, 1207-1223 (2006).
[CrossRef]

Cytometry

R. Schultz, T. Nielsen, J. Zavaleta, R. Ruch, R. Wyatt, and H. Garner, "Hyperspectral imaging: A novel approach for microscopic analysis," Cytometry 43, 239 - 247 (2001).
[CrossRef] [PubMed]

IEEE Signal Processing Lett.

M. Lang, H. Guo, J. E. Odegard, C. S. Burrus, and R. O. Wells, "Noise reduction using an undecimated discrete wavelet transform," IEEE Signal Processing Lett. 3, 10-12 (1996).
[CrossRef]

IEEE Trans. Geosci. Remote Sens.

P. A. Townsend, J. R. Foster, J. R. A. Chastain, and W. S. Currie, "Application of imaging spectroscopy to mapping canopy nitrogen in the forests of the central Appalachian Mountains using Hyperion and AVIRIS," IEEE Trans. Geosci. Remote Sens. 41, 1347-1354 (2003).
[CrossRef]

IEEE Trans. Inf. Theory

E. Cand`es, J. Romberg, and T. Tao, "Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information," IEEE Trans. Inf. Theory 52, 489 - 509 (2006).
[CrossRef]

D. Donoho, "Compressed Sensing," IEEE Trans. Inf. Theory 52, 1289-1306 (2006).
[CrossRef]

R. Nowak and E. Kolaczyk, "A Multiscale Statistical Framework for Poisson Inverse Problems," IEEE Trans. Inf. Theory 46, 1811-1825 (2000).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. of Am.

W. Richardson, "Bayesian-based iterative method of image restoration," J. Opt. Soc. of Am. 62, 55-59 (1972).
[CrossRef]

Opt. Express

Proc. SPIE

M. E. Gehm and D. J. Brady, "High-throughput hyperspectral microscopy," Proc. SPIE 6090, 13-21 (2006).

M. Hinnrichs, J. Jensen, and G. McAnally, "Handheld hyperspectral imager for standoff detection of chemical and biological aerosols," Proc. SPIE 5268, 67-78 (2003).
[CrossRef]

Science

W. Denk, J. Strickler, and W. Webb, "Two-photon laser scanning fluorescence microscopy," Science 248, 73-76 (1990).
[CrossRef] [PubMed]

Other

W. Smith, D. Zhou, F. Harrison, H. Revercomb, A. Larar, A. Huang, and B. Huang, "Hyperspectral remote sensing of atmospheric profiles from satellites and aircraft," Proc. SPIE 4151, 94 - 102 (2001).
[CrossRef]

R. P. Lin, B. R. Dennis, and A. O. B. eds., The Reuven Ramaty High-Energy Solar Spectrscopic Imager (RHESSI) - Mission Description and Early Results (Kluwer Academic Publishers, Dordrecht, 2003).
[PubMed]

M. E. Gehm, D. J. Brady, N. Pitsianis, and X. Sun, "Compressive sampling strategies for integrated microspectrometers," Proc. SPIE, 6232 (SPIE, 2006).

D. J. Brady and M. E. Gehm, "Compressive imaging spectrometers using coded apertures," Proc. SPIE, 6246 (SPIE, 2006).
[CrossRef]

R. M. Willet, M. E. Gehm, and D. J. Brady, "Multiscale reconstruction for computational spectral imaging," Proc. SPIE, 6498 (SPIE-IS and Electronic Imaging, 2007).
[CrossRef]

E. Candes and T. Tao, "Near Optimal Signal Recovery From Random Projections: Universal Encoding Strategies," (2006). To be published in IEEE Trans. Inf. Theory. http://www.acm.caltech.edu/ emmanuel/ papers/OptimalRecovery.pdf>.

R. Willett, "Multiscale intensity estimation for marked Poisson processes," in Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP) (2007).

R. Willett and R. Nowak, "Fast Multiresolution Photon-Limited Image Reconstruction," in Proc. IEEE Int. Sym. Biomedical Imaging — ISBI ’04 (15-18 April, Arlington, VA, USA, 2004).

M. Harwit and N. Sloane, Hadamard Transform Optics (Academic Press, New York, 1979).

J. Haupt and R. Nowak, "Signal Reconstruction from Noisy Random Projections," (2006). To be published in IEEE Trans. Inf. Theory. http://www.ece.wisc.edu/ nowak/infth.pdf>.

Supplementary Material (1)

» Media 1: AVI (926 KB)     

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Figures (10)

Fig. 1.
Fig. 1.

Schematic of the spectral imager.

Fig. 2.
Fig. 2.

Distribution of filter functions that arises from simple tiling of the fundamental codeword. No compact region contains all 15 filters.

Fig. 3.
Fig. 3.

Distribution of filter functions that arises from the more complicated unit cell. Any 3×5 region contains all 15 filters.

Fig. 4.
Fig. 4.

Sample partition of a spatio-spectral data cube. The spatial partition is the same at each spectral band, making it impossible for the estimation method to perform spatial smoothing at some spectral bands but not others.

Fig. 5.
Fig. 5.

The experimental prototype.

Fig. 6.
Fig. 6.

Experimental results from simple targets with monochromatic illumination. (a) Detector image recorded for 532 nm illumination. (b) Intensity image generated by summing the spectral information in the reconstruction for 532 nm illumination. (c) Spectral reconstruction at a particular spatial location for 532 nm illumination. (d) Spectral reconstruction at a particular spatial location for 543 nm illumination.

Fig. 7.
Fig. 7.

Experimental results from simple targets with narrow-band illumination. (a) Detector image recorded for illumination with a 10 nm FWHM bandpass centered at 560 nm. (b) Intensity image generated by summing the spectral information in the reconstruction for the 560 nm bandpass. (c) Spectral reconstruction at a particular spatial location for the 560 nm bandpass. (d) Spectral reconstruction at a particular spatial location for the 580 nm bandpass. The origin of the small peak near 520 nm is explained in the text.

Fig. 8.
Fig. 8.

Experimental results from real-world objects under broadband (white) illumination. (a) Detector image recorded by the system. (b) Reconstructed intensity image of the scene. (c) Spectral reconstructions for spatial locations in the three regions.

Fig. 9.
Fig. 9.

Slices through the reconstructed datacube at 8 particular spectral channels.

Fig. 10.
Fig. 10.

Animation showing all 36 reconstructed spectral channels. (Multimedia file, 1.1 MB.) [Media 1]

Equations (31)

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

S 1 ( x , y ; λ ) = dx dy δ ( x [ x + α ( λ λ c ) ] ) δ ( y y ) S 0 ( x , y ; λ )
= S 0 ( x + α ( λ λ c ) , y ; λ ) .
S 2 ( x , y ; λ ) = T ( x , y ) S 1 ( x , y ; λ ) = T ( x , y ) S 0 ( x + α ( λ λ c ) , y ; λ ) ,
S 3 ( x , y ; λ ) = dx dy δ ( x [ x α ( λ λ c ) ] ) δ ( y y ) S 2 ( x , y ; λ )
= T ( x α ( λ λ c ) , y ) S 0 ( x , y ; λ )
= H ( x , y ; λ ) S 0 ( x , y ; λ ) .
I ( x , y ) = dλH ( x , y ; λ ) S 0 ( x , y ; λ ) .
I nm = dxdydλ rect x Δ m y Δ n H ( x , y ; λ ) S 0 ( x , y ; λ ) .
T x y = m , n T n′m rect x Δ m y Δ n .
I nm = m′n′ dxdydλ rect x Δ m y Δ n rect x α ( λ λ c ) Δ m y Δ n
× T n′m′ S 0 ( x , y ; λ ) .
I nm λ = λ c = m′n′ dxdydλ rect x Δ m y Δ n rect x Δ m′ y Δ n
× T n′m′ I 0 x y δ ( λ λ c )
= m′n′ δ mm′ δ nn T n′m′ I 0 , nm
= T nm I nm ,
I nm λ = λ c + Δλ = m′n′ dxdydλ rect x Δ m y Δ m rect x Δ ( m′ + 1 ) y Δ n
× T n′m′ I 0 x y δ ( λ ( λ c + Δλ ) )
= m′n′ δ mm′ δ nn′ T n′ ( m′ + 1 ) I 0 , nm
= T n ( m 1 ) I 0 , nm .
w nmp = m′n′ dxdydλ rect x Δ m y Δ n λ λ c Δλ p
× rect x α ( λ λ c ) Δ m′ y Δ n′ T n′m′ ,
S nmp = 1 Δ 2 Δλ dxdydλ rect x Δ m y Δ n λ λ c Δλ p S 0 ( x , y ; λ ) .
I nm = p w nmp s nmp .
I = Ws ,
d ~ Poisson ( I ) = Poisson ( Ws ) ,
p ( d Ws ) = n = 1 N m = 1 M e p w nmp s nmp ( p w nmp s nmp ) d nm d nm ! .
s ̂ = arg  min s ˜ S { log p ( d | W s ˜ ) + pen ( s ˜ ) } ,
y ( t ) = s ̂ ( t ) . × W T ( d . / W s ̂ ( t ) ) ,
s ̂ ( t + 1 ) = arg  min s ˜ { log p ( y ( t ) | s ˜ ) + pen ( s ˜ ) } .
P ̂ arg  min P { log p ( y ( t ) s ˜ ( P ) ) + pen ( P ) }
s ̂ s ˜ ( P ^ ) ,

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