Abstract

Coded aperture snapshot spectral imaging (CASSI) provides a mechanism for capturing a 3D spectral cube with a single shot 2D measurement. In many applications selective spectral imaging is sought since relevant information often lies within a subset of spectral bands. Capturing and reconstructing all the spectral bands in the observed image cube, to then throw away a large portion of this data, is inefficient. To this end, this paper extends the concept of CASSI to a system admitting multiple shot measurements, which leads not only to higher quality of reconstruction but also to spectrally selective imaging when the sequence of code aperture patterns is optimized. The aperture code optimization problem is shown to be analogous to the optimization of a constrained multichannel filter bank. The optimal code apertures allow the decomposition of the CASSI measurement into several subsets, each having information from only a few selected spectral bands. The rich theory of compressive sensing is used to effectively reconstruct the spectral bands of interest from the measurements. A number of simulations are developed to illustrate the spectral imaging characteristics attained by optimal aperture codes.

© 2011 Optical Society of America

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References

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  1. A. A. Wagadarikar, N. P. Pitsianis, X. Sun, and D. J. Brady, “Spectral image estimation for coded aperture snapshot spectral imagers,” Proc. SPIE 7076, 707602 (2008).
    [CrossRef]
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    [CrossRef] [PubMed]
  3. E. Candès, 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]
  4. D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
    [CrossRef]
  5. J. L. Paredes, G. Arce, and Z. Wang, “Ultra-wideband compressed sensing: channel estimation,” IEEE J. Sel. Top. Signal Process. 1, 383–395 (2007).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  9. D. Kittle, K. Choi, A. A. Wagadarikar, and D. J. Brady, “Multiframe image estimation for coded aperture snapshot spectral imagers,” Appl. Opt. 49, 6824–6833 (2010).
    [CrossRef] [PubMed]
  10. I. Alabboud, G. Muyo, A. Gorman, D. Mordant, A. McNaught, C. Petres, Y. R. Petillot, and A. R. Harvey, “New spectral imaging techniques for blood oximetry in the retina,” Proc. SPIE 6631, 1–10 (2007).
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    [CrossRef] [PubMed]
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  15. J. Romberg, “Compressive sensing by random convolution,” SIAM J. Imaging Sci , 1098–1128 (2009).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  23. T. Shirakawa, K. L. Ishikawa, S. Suzuki, Y. Yamada, and H. Takahashi, “Design of binary diffractive microlenses with subwavelength structures using the genetic algorithm,” Opt. Express 18, 8383–8391 (2010).
    [CrossRef]

2011 (2)

J. L. Paredes and G. R. Arce, “Compressive sensing signal reconstruction by weighted median regression estimates,” IEEE Trans. Signal Process. 59, 2585–2601 (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] [PubMed]

2010 (7)

Z. Wang and G. Arce, “Variable density compressed image sampling,” IEEE Trans. Image Process. 19, 264–270 (2010).
[CrossRef]

T. Shirakawa, K. L. Ishikawa, S. Suzuki, Y. Yamada, and H. Takahashi, “Design of binary diffractive microlenses with subwavelength structures using the genetic algorithm,” Opt. Express 18, 8383–8391 (2010).
[CrossRef]

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

M. F. Duarte and R. G. Baraniuk, “Kronecker product matrices for compressive sensing,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) 2010 (IEEE, 2010), pp. 3650–3653.

D. Kittle, “Compressive spectral imaging,” Master’s thesis (Duke University, 2010).

P. Ye, H. Arguello, and G. Arce, “Spectral aperture code design for multi-shot compressive spectral imaging,” in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (CD) (Optical Society of America, 2010), paper DWA6.

H. Arguello and G. R. Arce, “Code aperture design for compressive spectral imaging,” in Proceedings of the 18th European Signal Processing Conference (EUSIPCO 2010) [European Association for Signal Processung (EURASIP), 2010], pp. 1434–1438.

2009 (3)

J. Romberg, “Compressive sensing by random convolution,” SIAM J. Imaging Sci , 1098–1128 (2009).
[CrossRef]

D. J. Brady, Optical Imaging and Spectroscopy (Wiley, 2009).

Y. Wu, C. Chen, Z. Wang, P. Ye, G. Arce, D. Prather, and G. Schneider, “Fabrication and characterization of a compressive-sampling multispectral imaging system,” Opt. Eng. 48, 123201 (2009).
[CrossRef]

2008 (2)

A. A. Wagadarikar, N. P. Pitsianis, X. Sun, and D. J. Brady, “Spectral image estimation for coded aperture snapshot spectral imagers,” Proc. SPIE 7076, 707602 (2008).
[CrossRef]

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] [PubMed]

2007 (4)

I. Alabboud, G. Muyo, A. Gorman, D. Mordant, A. McNaught, C. Petres, Y. R. Petillot, and A. R. Harvey, “New spectral imaging techniques for blood oximetry in the retina,” Proc. SPIE 6631, 1–10 (2007).

J. L. Paredes, G. Arce, and Z. Wang, “Ultra-wideband compressed sensing: channel estimation,” IEEE J. Sel. Top. Signal Process. 1, 383–395 (2007).
[CrossRef]

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE J. Sel. Top. Signal Process. , 1 606–617(2007).
[CrossRef]

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

2006 (2)

E. Candès, 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. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[CrossRef]

2004 (1)

S. Boxwell, S. G. Fox, and J. F. Román, “Design and optimization of optical components using genetic algorithms,” Opt. Eng. 43 (2004).

2002 (1)

J. L. Paredes, G. R. Arce, and L. E. Russo, “Multichannel image compression by bijection mappings onto zero-trees,” IEEE Trans. Image Process. 11, 223–233 (2002).
[CrossRef]

1989 (1)

D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning (Kluwer, 1989).
[CrossRef]

Alabboud, I.

I. Alabboud, G. Muyo, A. Gorman, D. Mordant, A. McNaught, C. Petres, Y. R. Petillot, and A. R. Harvey, “New spectral imaging techniques for blood oximetry in the retina,” Proc. SPIE 6631, 1–10 (2007).

Arce, G.

P. Ye, H. Arguello, and G. Arce, “Spectral aperture code design for multi-shot compressive spectral imaging,” in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (CD) (Optical Society of America, 2010), paper DWA6.

Z. Wang and G. Arce, “Variable density compressed image sampling,” IEEE Trans. Image Process. 19, 264–270 (2010).
[CrossRef]

Y. Wu, C. Chen, Z. Wang, P. Ye, G. Arce, D. Prather, and G. Schneider, “Fabrication and characterization of a compressive-sampling multispectral imaging system,” Opt. Eng. 48, 123201 (2009).
[CrossRef]

J. L. Paredes, G. Arce, and Z. Wang, “Ultra-wideband compressed sensing: channel estimation,” IEEE J. Sel. Top. Signal Process. 1, 383–395 (2007).
[CrossRef]

Arce, G. R.

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

J. L. Paredes and G. R. Arce, “Compressive sensing signal reconstruction by weighted median regression estimates,” IEEE Trans. Signal Process. 59, 2585–2601 (2011).
[CrossRef]

H. Arguello and G. R. Arce, “Code aperture design for compressive spectral imaging,” in Proceedings of the 18th European Signal Processing Conference (EUSIPCO 2010) [European Association for Signal Processung (EURASIP), 2010], pp. 1434–1438.

J. L. Paredes, G. R. Arce, and L. E. Russo, “Multichannel image compression by bijection mappings onto zero-trees,” IEEE Trans. Image Process. 11, 223–233 (2002).
[CrossRef]

Arguello, H.

H. Arguello and G. R. Arce, “Code aperture design for compressive spectral imaging,” in Proceedings of the 18th European Signal Processing Conference (EUSIPCO 2010) [European Association for Signal Processung (EURASIP), 2010], pp. 1434–1438.

P. Ye, H. Arguello, and G. Arce, “Spectral aperture code design for multi-shot compressive spectral imaging,” in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (CD) (Optical Society of America, 2010), paper DWA6.

Baraniuk, R. G.

M. F. Duarte and R. G. Baraniuk, “Kronecker product matrices for compressive sensing,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) 2010 (IEEE, 2010), pp. 3650–3653.

Boxwell, S.

S. Boxwell, S. G. Fox, and J. F. Román, “Design and optimization of optical components using genetic algorithms,” Opt. Eng. 43 (2004).

Boyd, S.

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE J. Sel. Top. Signal Process. , 1 606–617(2007).
[CrossRef]

Brady, D.

Brady, D. J.

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

D. J. Brady, Optical Imaging and Spectroscopy (Wiley, 2009).

A. A. Wagadarikar, N. P. Pitsianis, X. Sun, and D. J. Brady, “Spectral image estimation for coded aperture snapshot spectral imagers,” Proc. SPIE 7076, 707602 (2008).
[CrossRef]

Candès, E.

E. Candès, 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]

Chen, C.

Y. Wu, C. Chen, Z. Wang, P. Ye, G. Arce, D. Prather, and G. Schneider, “Fabrication and characterization of a compressive-sampling multispectral imaging system,” Opt. Eng. 48, 123201 (2009).
[CrossRef]

Choi, K.

Donoho, D. L.

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[CrossRef]

Duarte, M. F.

M. F. Duarte and R. G. Baraniuk, “Kronecker product matrices for compressive sensing,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) 2010 (IEEE, 2010), pp. 3650–3653.

Figueiredo, M. A. T.

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

Fox, S. G.

S. Boxwell, S. G. Fox, and J. F. Román, “Design and optimization of optical components using genetic algorithms,” Opt. Eng. 43 (2004).

Goldberg, D. E.

D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning (Kluwer, 1989).
[CrossRef]

Gorinevsky, D.

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE J. Sel. Top. Signal Process. , 1 606–617(2007).
[CrossRef]

Gorman, A.

I. Alabboud, G. Muyo, A. Gorman, D. Mordant, A. McNaught, C. Petres, Y. R. Petillot, and A. R. Harvey, “New spectral imaging techniques for blood oximetry in the retina,” Proc. SPIE 6631, 1–10 (2007).

Harvey, A. R.

I. Alabboud, G. Muyo, A. Gorman, D. Mordant, A. McNaught, C. Petres, Y. R. Petillot, and A. R. Harvey, “New spectral imaging techniques for blood oximetry in the retina,” Proc. SPIE 6631, 1–10 (2007).

Ishikawa, K. L.

John, R.

Kim, S. J.

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE J. Sel. Top. Signal Process. , 1 606–617(2007).
[CrossRef]

Kittle, D.

Koh, K.

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE J. Sel. Top. Signal Process. , 1 606–617(2007).
[CrossRef]

Lustig, M.

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE J. Sel. Top. Signal Process. , 1 606–617(2007).
[CrossRef]

McNaught, A.

I. Alabboud, G. Muyo, A. Gorman, D. Mordant, A. McNaught, C. Petres, Y. R. Petillot, and A. R. Harvey, “New spectral imaging techniques for blood oximetry in the retina,” Proc. SPIE 6631, 1–10 (2007).

Mirza, I. O.

Mordant, D.

I. Alabboud, G. Muyo, A. Gorman, D. Mordant, A. McNaught, C. Petres, Y. R. Petillot, and A. R. Harvey, “New spectral imaging techniques for blood oximetry in the retina,” Proc. SPIE 6631, 1–10 (2007).

Muyo, G.

I. Alabboud, G. Muyo, A. Gorman, D. Mordant, A. McNaught, C. Petres, Y. R. Petillot, and A. R. Harvey, “New spectral imaging techniques for blood oximetry in the retina,” Proc. SPIE 6631, 1–10 (2007).

Nowak, R. D.

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

Paredes, J. L.

J. L. Paredes and G. R. Arce, “Compressive sensing signal reconstruction by weighted median regression estimates,” IEEE Trans. Signal Process. 59, 2585–2601 (2011).
[CrossRef]

J. L. Paredes, G. Arce, and Z. Wang, “Ultra-wideband compressed sensing: channel estimation,” IEEE J. Sel. Top. Signal Process. 1, 383–395 (2007).
[CrossRef]

J. L. Paredes, G. R. Arce, and L. E. Russo, “Multichannel image compression by bijection mappings onto zero-trees,” IEEE Trans. Image Process. 11, 223–233 (2002).
[CrossRef]

Petillot, Y. R.

I. Alabboud, G. Muyo, A. Gorman, D. Mordant, A. McNaught, C. Petres, Y. R. Petillot, and A. R. Harvey, “New spectral imaging techniques for blood oximetry in the retina,” Proc. SPIE 6631, 1–10 (2007).

Petres, C.

I. Alabboud, G. Muyo, A. Gorman, D. Mordant, A. McNaught, C. Petres, Y. R. Petillot, and A. R. Harvey, “New spectral imaging techniques for blood oximetry in the retina,” Proc. SPIE 6631, 1–10 (2007).

Pitsianis, N. P.

A. A. Wagadarikar, N. P. Pitsianis, X. Sun, and D. J. Brady, “Spectral image estimation for coded aperture snapshot spectral imagers,” Proc. SPIE 7076, 707602 (2008).
[CrossRef]

Prather, D.

Y. Wu, C. Chen, Z. Wang, P. Ye, G. Arce, D. Prather, and G. Schneider, “Fabrication and characterization of a compressive-sampling multispectral imaging system,” Opt. Eng. 48, 123201 (2009).
[CrossRef]

Prather, D. W.

Román, J. F.

S. Boxwell, S. G. Fox, and J. F. Román, “Design and optimization of optical components using genetic algorithms,” Opt. Eng. 43 (2004).

Romberg, J.

J. Romberg, “Compressive sensing by random convolution,” SIAM J. Imaging Sci , 1098–1128 (2009).
[CrossRef]

E. Candès, 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]

Russo, L. E.

J. L. Paredes, G. R. Arce, and L. E. Russo, “Multichannel image compression by bijection mappings onto zero-trees,” IEEE Trans. Image Process. 11, 223–233 (2002).
[CrossRef]

Schneider, G.

Y. Wu, C. Chen, Z. Wang, P. Ye, G. Arce, D. Prather, and G. Schneider, “Fabrication and characterization of a compressive-sampling multispectral imaging system,” Opt. Eng. 48, 123201 (2009).
[CrossRef]

Shirakawa, T.

Sun, X.

A. A. Wagadarikar, N. P. Pitsianis, X. Sun, and D. J. Brady, “Spectral image estimation for coded aperture snapshot spectral imagers,” Proc. SPIE 7076, 707602 (2008).
[CrossRef]

Suzuki, S.

Takahashi, H.

Tao, T.

E. Candès, 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]

Wagadarikar, A. A.

Wang, Z.

Z. Wang and G. Arce, “Variable density compressed image sampling,” IEEE Trans. Image Process. 19, 264–270 (2010).
[CrossRef]

Y. Wu, C. Chen, Z. Wang, P. Ye, G. Arce, D. Prather, and G. Schneider, “Fabrication and characterization of a compressive-sampling multispectral imaging system,” Opt. Eng. 48, 123201 (2009).
[CrossRef]

J. L. Paredes, G. Arce, and Z. Wang, “Ultra-wideband compressed sensing: channel estimation,” IEEE J. Sel. Top. Signal Process. 1, 383–395 (2007).
[CrossRef]

Willett, R.

Wright, S. J.

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

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] [PubMed]

Y. Wu, C. Chen, Z. Wang, P. Ye, G. Arce, D. Prather, and G. Schneider, “Fabrication and characterization of a compressive-sampling multispectral imaging system,” Opt. Eng. 48, 123201 (2009).
[CrossRef]

Yamada, Y.

Ye, P.

P. Ye, H. Arguello, and G. Arce, “Spectral aperture code design for multi-shot compressive spectral imaging,” in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (CD) (Optical Society of America, 2010), paper DWA6.

Y. Wu, C. Chen, Z. Wang, P. Ye, G. Arce, D. Prather, and G. Schneider, “Fabrication and characterization of a compressive-sampling multispectral imaging system,” Opt. Eng. 48, 123201 (2009).
[CrossRef]

Appl. Opt. (2)

IEEE J. Sel. Top. Signal Process. (3)

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE J. Sel. Top. Signal Process. , 1 606–617(2007).
[CrossRef]

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

J. L. Paredes, G. Arce, and Z. Wang, “Ultra-wideband compressed sensing: channel estimation,” IEEE J. Sel. Top. Signal Process. 1, 383–395 (2007).
[CrossRef]

IEEE Trans. Image Process. (2)

J. L. Paredes, G. R. Arce, and L. E. Russo, “Multichannel image compression by bijection mappings onto zero-trees,” IEEE Trans. Image Process. 11, 223–233 (2002).
[CrossRef]

Z. Wang and G. Arce, “Variable density compressed image sampling,” IEEE Trans. Image Process. 19, 264–270 (2010).
[CrossRef]

IEEE Trans. Inf. Theory (2)

E. Candès, 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. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[CrossRef]

IEEE Trans. Signal Process. (1)

J. L. Paredes and G. R. Arce, “Compressive sensing signal reconstruction by weighted median regression estimates,” IEEE Trans. Signal Process. 59, 2585–2601 (2011).
[CrossRef]

Opt. Eng. (2)

Y. Wu, C. Chen, Z. Wang, P. Ye, G. Arce, D. Prather, and G. Schneider, “Fabrication and characterization of a compressive-sampling multispectral imaging system,” Opt. Eng. 48, 123201 (2009).
[CrossRef]

S. Boxwell, S. G. Fox, and J. F. Román, “Design and optimization of optical components using genetic algorithms,” Opt. Eng. 43 (2004).

Opt. Express (1)

Opt. Lett. (1)

Proc. SPIE (2)

A. A. Wagadarikar, N. P. Pitsianis, X. Sun, and D. J. Brady, “Spectral image estimation for coded aperture snapshot spectral imagers,” Proc. SPIE 7076, 707602 (2008).
[CrossRef]

I. Alabboud, G. Muyo, A. Gorman, D. Mordant, A. McNaught, C. Petres, Y. R. Petillot, and A. R. Harvey, “New spectral imaging techniques for blood oximetry in the retina,” Proc. SPIE 6631, 1–10 (2007).

SIAM J. Imaging Sci (1)

J. Romberg, “Compressive sensing by random convolution,” SIAM J. Imaging Sci , 1098–1128 (2009).
[CrossRef]

Other (6)

D. J. Brady, Optical Imaging and Spectroscopy (Wiley, 2009).

D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning (Kluwer, 1989).
[CrossRef]

D. Kittle, “Compressive spectral imaging,” Master’s thesis (Duke University, 2010).

P. Ye, H. Arguello, and G. Arce, “Spectral aperture code design for multi-shot compressive spectral imaging,” in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (CD) (Optical Society of America, 2010), paper DWA6.

H. Arguello and G. R. Arce, “Code aperture design for compressive spectral imaging,” in Proceedings of the 18th European Signal Processing Conference (EUSIPCO 2010) [European Association for Signal Processung (EURASIP), 2010], pp. 1434–1438.

M. F. Duarte and R. G. Baraniuk, “Kronecker product matrices for compressive sensing,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) 2010 (IEEE, 2010), pp. 3650–3653.

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Figures (17)

Fig. 1
Fig. 1

The principles behind CASSI imaging. A 6 × 6 × 8 spectral data cube with 16 nonzero spectral components is coded by the code aperture and dispersed by the prism. The detector integrates the intensity of the resulting light wave. Each pixel at the detector contains a coded linear combination of the spectral information from the respective data cube slice.

Fig. 2
Fig. 2

The CAASI system seen as a multichannel filter bank. A set of measurements { g i } are captured with a corresponding set of optimal aperture codes { w 1 , , w K } . The mapping B reorders the K sets of measurements { g i } into the V sets { ϱ i } . Each subset { ϱ i } is sufficient to independently reconstruct the desired spectral band subset { F ˜ λ i } .

Fig. 3
Fig. 3

Representation of the CASSI.

Fig. 4
Fig. 4

Model equations for a row measurement. A slice of the data cube f q impinges on a row of the code aperture t i = r w i to produce the coded slice W i R i f q . The elements of the coded slice are reordered into the matrix Γ i . The detector output g q i = Γ i u L sums the columns of Γ i .

Fig. 5
Fig. 5

(a) Traditional model where the data cube f is processed with the highly sparse matrix H i . The code aperture pattern is hidden in H i ; (b) new model of CAASI: f is first reordered and expanded into the matrix X which is then processed by the weight matrix W i whose elements are the code aperture patterns.

Fig. 6
Fig. 6

Code aperture optimization as a filter design problem. The input is a row of the data cube f s and the desired signal is f 0 . The filter coefficients (aperture codes w ¯ 1 , w ¯ 2 , , w ¯ K ) and the coefficients b i j are optimized by minimizing a cost function of the errors e 1 , e 2 , , e M + L 1 . The error at given jth position at a row at the detector is the difference between the linear combination of the measurements ( ϱ ) j and ( d ) j .

Fig. 7
Fig. 7

Slice of the data cube f s is modulated by the vector r and then reorganized into the matrix X. The output at the detector is calculated as g i = X w ¯ i .

Fig. 8
Fig. 8

Performance evolution ξ * as a function of the iterations when the vectors (a)  λ 2 and (b)  λ 4 in Eq. (36) are used as input of the GA.

Fig. 9
Fig. 9

Optimal code apertures w ¯ * i for the vectors (a)  λ 1 , (b)  λ 2 , (c)  λ 3 , and (d)  λ 4 indicated in Eq. (34).

Fig. 10
Fig. 10

128 × 128 realization of the optimal code apertures for the vectors (a)  λ 1 , (b)  λ 2 , (c)  λ 3 , (d)  λ 4 given in Eq. (34).

Fig. 11
Fig. 11

Part of the matrix B * is shown. The lth column of B * represents the optimal coefficient to construct the measurement element ( ϱ * ) l . The vector λ 4 is used as input of the optimization algorithm.

Fig. 12
Fig. 12

Reconstruction of the first spectral band of the 24 spectral band data cube. (a) Original; reconstruction using (b)  mod 4 filter bank code apertures, 4 shots; (c)  mod 12 filter bank code apertures, 12 shots; (d)  mod 24 filter bank code apertures, 24 shots.

Fig. 13
Fig. 13

Reconstruction of the first spectral band of the 24 spectral band data cube. (a) Original; (b) random code aperture, 4 shots; (c) random code aperture, 12 shots; and (d) random code aperture, 24 shots.

Fig. 14
Fig. 14

Mean PSNR for the reconstructed data cube as a function of the number of shots. The techniques of random multishot and compressive mod p filter bank are shown.

Fig. 15
Fig. 15

Reconstruction times for the full data cube as a function of the number of shots using (a) random code apertures (random multishot), (b)  mod p filter bank optimized code apertures (filter bank), and (c)  mod p filter bank code apertures using a processor for each subset of bands (filter bank parallel).

Fig. 16
Fig. 16

Reconstruction of the 1st and 18th spectral band of the 24 band data cube. (a) Original 1st band (b) original 18th band; reconstruction of the respective band using the vector λ 4 in Eq. (34). (c), (d) 4 shots; (e), (f) 8 shots.

Fig. 17
Fig. 17

Reconstruction of the 1st and 18th spectral bands indicated in Fig. 16a, 16b. Reconstruction of the respective band using the vector λ 4 in Eq. (36) for (c), (d) 12 shots; (e), (f) 16 shots.

Tables (1)

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Table 1 Iterative Optimization of Code Apertures w ¯ * 1 , , w ¯ * K and the Optimal Matrix B *

Equations (48)

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f 2 ( x , y , λ ) = T ( x , y ) f 0 ( x , y , λ ) h ( x α λ x , y y ) d x d y ,
( G ) n m = k = 1 L ( F k ) n , m + k ( T ) n , m + k + ( ω ) n , m ,
g = Hf + ω = H Ψ θ + ω ,
( f q ) ( k 1 ) · M + m = ( F k ) q m , for     m = 1 , , M     k = 1 , , L .
g q = H q f q
g [ g 1 g q g N ] = H [ H 1 0 0 0 H q 0 H N ] f [ f 1 f q f N ] ,
H q = [ diag { ( T ) q , 1 , , ( T ) q , M } 0 1 × M 0 1 × M | 0 1 × M diag { ( T ) q , 1 , , ( T ) q , M } 0 1 × M 0 1 × M | | 0 1 × M 0 1 × M diag { ( T ) q , 1 , , ( T ) q , M } ] ,
D = [ I M 0 1 × M 0 1 × M 0 1 × M I M 0 1 × M 0 1 × M 0 1 × M I M ] ,
T = I L diag { ( T ) q , 1 , ( T ) q , 2 , , ( T ) q , M } ,
g q = D T f q .
( G i ) n , m = k = 1 L ( F k ) n , m + k ( T i ) n , m + k + ( ω i ) n , m
g i = H i f + ω i i 1 , , K .
g q i = D T i f q .
T i = ( I L diag { ( w i ) 1 , , ( w i ) M } ) W i ( I L diag { ( r ) 1 , , ( r ) M } ) R ,
g q i = D W i ρ .
g q i = Γ i u L ,
Γ i = [ ρ 1 ( w i ) 1 ρ 2 ( w i ) 2 ρ L ( w i ) L ρ M ( w i ) M 0 0 0 ρ M + 1 ( w i ) 1 ρ M + L 1 ( w i ) L 1 ρ 2 M 1 ( w i ) M 1 ρ 2 M ( w i ) M 0 0 0 ρ 2 M + L 2 ( w i ) L 2 ρ 3 M 2 ( w i ) M 2 ρ 3 M 1 ( w i ) M 1 0 0 0 0 ρ ( L 1 ) M + 1 ( w i ) 1 ρ L M ( L 1 ) ( w i ) M L + 1 ρ L M ( w i ) M ] T ,
( w i ) m = ( w ¯ i ) mod ( m , L ) for     m = 1 , , M and L < M .
w i = u M w ¯ i ,
Γ ¯ i = [ ρ 1 ( w ¯ i ) 1 ρ M + 1 ( w ¯ i ) 1 ρ ( L 1 ) M + 1 ( w ¯ i ) 1 ρ L M ( L 1 ) ( w ¯ i ) 1 0 0 0 ρ 2 ( w ¯ i ) 2 ρ ( L 2 ) M + 2 ( w ¯ i ) 2 ρ ( L 1 ) M ( L 2 ) ( w ¯ i ) 2 ρ L M ( L 2 ) ( w ¯ i ) 2 0 0 0 ρ ( L 3 ) M + 3 ( w ¯ i ) 3 ρ ( L 2 ) M ( L 3 ) ( w ¯ i ) 3 ρ ( L 1 ) M ( L 3 ) ( w ¯ i ) 3 0 0 0 0 ρ L ( w ¯ i ) L ρ M ( w ¯ i ) L ρ 2 M ( w ¯ i ) L ρ L M ( w ¯ i ) L ] T .
g q i = X w ¯ i ,
X = [ ρ 1 ρ M + 1 ρ ( L 1 ) M + 1 ρ L M ( L 1 ) 0 0 0 ρ 2 ρ ( L 2 ) M + 2 ρ ( L 1 ) M ( L 2 ) ρ L M ( L 2 ) 0 0 0 ρ ( L 3 ) M + 3 ρ ( L 2 ) M ( L 3 ) ρ ( L 1 ) M ( L 3 ) 0 0 0 ρ L ρ M ρ 2 M ρ L ( M ) ] T
[ g 1 i g N i ] = [ X 1 w ¯ 1 i X N w ¯ N i ] ,
g i = χ W ¯ i ,
j = 1 j l L s j z j s l
( F k s ) n m = s k for all     m , n .
( F k d ) n m = { ( F k s ) n m if     ( λ ) k = 1 0 otherwise
( λ ) k = { 1 if the   k th spectral band is of interest 0 otherwise
d i = X d w ˜ i ,
e j = ( d ) j ( ϱ i ) j j = 1 , , M + L 1 ,
arg min W , B subject to e 0 ( W ) k i { 0 , 1 } ( B ) i j { 1 , 0 , 1 } k = 1 , , L ,     i = 1 , , K j = 1 , , M + L 1.
ϱ * = [ i = 1 K ( B * ) i 1 x 1 T w ¯ * i , i = 1 K ( B * ) i 2 x 2 T w ¯ * i , , i = 1 K ( B * ) i ( M + L 1 ) x M + L 1 T w ¯ * i ] ,
( λ n ) k = ( λ ) ( k n L ) , k = 1 , L ,
λ 1 = [ 100010100000000000110000 ] λ 2 = [ 100010001000100010001000 ] λ 3 = [ 101000101000101000101000 ] λ 4 = [ 110000001100000011000000 ] .
( λ n ) k = ( λ ) ( k n U ) L k = 1 , , L .
( λ ) k = mod ( k , p ) , k = 1 , , L ,
( λ n ) j = { 1 for     j = n , K + n , 2 K + n , , ( L K 1 ) K + n 0 otherwise . ,
( g q i ) j = { k = 1 j ( W i ) m k j , m k j ( ρ ) m k j for     j = 1 , L 1 k = 1 L ( W i ) m k l , m k j ( ρ ) m k j for     j = L , M k = j M + 1 L ( W i ) m k j , m k j ( ρ ) m k j for     j = M + 1 , M + L 1 ,
( g q i ) j = k = 1 L ( W i ) m k j , m k j ( ρ ) m k j ,
( Γ i ) j , k = ( W i ) m k j , m k j ( ρ ) m k j k = 1 , , L , and j = 1 , , M + L 1 ,
e j = x j d u L i = 1 K b i j x j T w ¯ i , j = 1 , , M + L 1 ,
k = 1 L ( x j d ) k = i = 1 K b i j k = 1 L ( x j ) k ( w ¯ i ) k .
k λ ( x j ) k = i = 1 K b i j k = 1 L ( x j ) k ( w ¯ i ) k ,
k λ s k = i = 1 K b i j k = 1 L s k ( w ¯ i ) k .
k λ s k = k = 1 L s k ( w ¯ i ) k .
7 + 11 = b 1 j ( 9 ) + b 2 j ( 11 ) ,
k λ s k = k = 1 L s k ( w ¯ i ) k .
s k = s ( k m ) L , k = 1 , , L .

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