Abstract

We present an optical imaging system based on compressive sensing (CS) along with its principal mathematical aspects. Although CS is undergoing significant advances and empowering many discussions and applications throughout various fields, this article focuses on the analysis of a single-pixel camera. This work was the core for the development of a single-pixel camera approach based on active illumination. Therefore, the active illumination concept is described along with the experimental results, which were very encouraging toward the development of compressive-sensing-based cameras for various applications, such as pixel-level programmable gain imaging.

© 2011 Optical Society of America

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  1. E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
    [CrossRef]
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    [CrossRef]
  3. J. F. Claerbout and F. Muir, “Robust modeling with erratic data,” Geophysics 38, 826–844 (1973).
    [CrossRef]
  4. S. Fadil and W. S. William, “Linear inversion of band-limited reflection seismograms,” SIAM J. Sci. Statist. Comput. 7, 1307–1330 (1986).
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    [CrossRef]
  6. E. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
    [CrossRef]
  7. D. Baron, M. B. Wakin, M. F. Duarte, S. Sarvotham, and R. G. Baraniuk, “Distributed compressive sensing,” http://arxiv.org/abs/0901.3403v1 (2005).
  8. M. Golbabaee and P. Vandergheynst, “Distributed compressed sensing for sensor networks, using thresholding,” Proc. SPIE 7446, 74461F (2009).
    [CrossRef]
  9. J. Haupt and R. Nowak, “Signal reconstruction from noisy random projections,” IEEE Trans. Inf. Theory 52, 4036–4048(2006).
    [CrossRef]
  10. M. Lustig, D. L. Donoho, and J. M. Pauly, “Rapid MR imaging with compressed sensing and randomly under-sampled 3DFT trajectories,” presented at the 14th Annual Meeting of the International Society for Magnetic Resonance in Medicine, Seattle, Washington, 6–12 May 2006.
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    [CrossRef]
  12. E. J. Candès and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies?” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
    [CrossRef]
  13. E. J. Candès and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory 51, 4203–4215 (2005).
    [CrossRef]
  14. E. J. Candès, “Compressive sampling,” in Proceedings of the International Congress of Mathematicians (ICM, 2006), pp. 1433–1452.
  15. R. G. Baraniuk, “Compressive sensing [lecture notes],” IEEE Signal Process. Mag. 24, 118–121 (2007).
    [CrossRef]
  16. E. J. Candès and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl. 23, 969–985 (2007).
    [CrossRef]
  17. S. Chen, D. Donoho, and M. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43, 129–159 (2001).
    [CrossRef]
  18. J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53, 4655–4666 (2007).
    [CrossRef]
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    [CrossRef]
  24. K. Kawase, Y. Ogawa, Y. Watanabe, and H. Inoue, “Non-destructive terahertz imaging of illicit drugs using spectral fingerprints,” Opt. Express 11, 2549–2554 (2003).
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    [CrossRef] [PubMed]
  26. W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
    [CrossRef]
  27. J. Bobin and J.-L. Starck, “Compressed sensing in astronomy and remote sensing: a data fusion perspective,” Proc. SPIE 7446, 74460I (2009).
    [CrossRef]
  28. J. Bobin, J. L. Starck, and R. Ottensamer, “Compressed sensing in astronomy,” IEEE J. Sel. Top. Signal Process. 2, 718–726 (2008).
    [CrossRef]
  29. J. H.G. Ender, “On compressive sensing applied to radar,” Signal Process. 90, 1402–1414 (2010).
    [CrossRef]

2010

J. H.G. Ender, “On compressive sensing applied to radar,” Signal Process. 90, 1402–1414 (2010).
[CrossRef]

2009

J. Bobin and J.-L. Starck, “Compressed sensing in astronomy and remote sensing: a data fusion perspective,” Proc. SPIE 7446, 74460I (2009).
[CrossRef]

L. K. Kirk, “Rethinking signal processing,” Commun. ACM 52, 13–15 (2009).
[CrossRef]

M. Golbabaee and P. Vandergheynst, “Distributed compressed sensing for sensor networks, using thresholding,” Proc. SPIE 7446, 74461F (2009).
[CrossRef]

2008

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

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, S. Ting, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

J. Bobin, J. L. Starck, and R. Ottensamer, “Compressed sensing in astronomy,” IEEE J. Sel. Top. Signal Process. 2, 718–726 (2008).
[CrossRef]

2007

R. A. Hoebe, C. H. Van Oven, T. W. J. Gadella, P. B. Dhonukshe, C. J. F. Van Noorden, and E. M. M. Manders, “Controlled light-exposure microscopy reduces photobleaching and phototoxicity in fluorescence live-cell imaging,” Nat. Biotechnol. 25, 249–253 (2007).
[CrossRef] [PubMed]

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

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

E. J. Candès and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl. 23, 969–985 (2007).
[CrossRef]

2006

J. Haupt and R. Nowak, “Signal reconstruction from noisy random projections,” IEEE Trans. Inf. Theory 52, 4036–4048(2006).
[CrossRef]

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE 6065, 606509 (2006).
[CrossRef]

E. J. Candès and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies?” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[CrossRef]

E. J. 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]

E. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

2005

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

2003

S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20, 21–36 (2003).
[CrossRef]

K. Kawase, Y. Ogawa, Y. Watanabe, and H. Inoue, “Non-destructive terahertz imaging of illicit drugs using spectral fingerprints,” Opt. Express 11, 2549–2554 (2003).
[CrossRef] [PubMed]

2001

S. Chen, D. Donoho, and M. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43, 129–159 (2001).
[CrossRef]

1973

J. F. Claerbout and F. Muir, “Robust modeling with erratic data,” Geophysics 38, 826–844 (1973).
[CrossRef]

Bagon, S.

D. Glasner, S. Bagon, and M. Irani, “Super-resolution from a single image,” in IEEE 12th International Conference on Computer Vision, 2009 (IEEE, 2009), pp. 349–356.
[CrossRef]

Baraniuk, R.

R. Baraniuk and K. Kelly, “Lecture 5: a single-pixel compressive camera,” retrieved from http://ima.umn.edu/2006-2007/ND6.4-15.07/activities/Baraniuk-Richard/baraniuk-IMA-CScamera-june07.pdf.

Baraniuk, R. G.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, S. Ting, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

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

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE 6065, 606509 (2006).
[CrossRef]

D. Baron, M. B. Wakin, M. F. Duarte, S. Sarvotham, and R. G. Baraniuk, “Distributed compressive sensing,” http://arxiv.org/abs/0901.3403v1 (2005).

Baron, D.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE 6065, 606509 (2006).
[CrossRef]

D. Baron, M. B. Wakin, M. F. Duarte, S. Sarvotham, and R. G. Baraniuk, “Distributed compressive sensing,” http://arxiv.org/abs/0901.3403v1 (2005).

Bobin, J.

J. Bobin and J.-L. Starck, “Compressed sensing in astronomy and remote sensing: a data fusion perspective,” Proc. SPIE 7446, 74460I (2009).
[CrossRef]

J. Bobin, J. L. Starck, and R. Ottensamer, “Compressed sensing in astronomy,” IEEE J. Sel. Top. Signal Process. 2, 718–726 (2008).
[CrossRef]

Candès, E.

E. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

Candès, E. J.

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

E. J. Candès and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl. 23, 969–985 (2007).
[CrossRef]

E. J. 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]

E. J. Candès and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies?” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[CrossRef]

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

E. J. Candès, “Compressive sampling,” in Proceedings of the International Congress of Mathematicians (ICM, 2006), pp. 1433–1452.

Chan, W. L.

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

Charan, K.

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

Chen, S.

S. Chen, D. Donoho, and M. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43, 129–159 (2001).
[CrossRef]

Claerbout, J. F.

J. F. Claerbout and F. Muir, “Robust modeling with erratic data,” Geophysics 38, 826–844 (1973).
[CrossRef]

Davenport, M. A.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, S. Ting, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

Dhonukshe, P. B.

R. A. Hoebe, C. H. Van Oven, T. W. J. Gadella, P. B. Dhonukshe, C. J. F. Van Noorden, and E. M. M. Manders, “Controlled light-exposure microscopy reduces photobleaching and phototoxicity in fluorescence live-cell imaging,” Nat. Biotechnol. 25, 249–253 (2007).
[CrossRef] [PubMed]

Donoho, D.

S. Chen, D. Donoho, and M. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43, 129–159 (2001).
[CrossRef]

Donoho, D. L.

M. Lustig, D. L. Donoho, and J. M. Pauly, “Rapid MR imaging with compressed sensing and randomly under-sampled 3DFT trajectories,” presented at the 14th Annual Meeting of the International Society for Magnetic Resonance in Medicine, Seattle, Washington, 6–12 May 2006.

Duarte, M. F.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, S. Ting, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE 6065, 606509 (2006).
[CrossRef]

D. Baron, M. B. Wakin, M. F. Duarte, S. Sarvotham, and R. G. Baraniuk, “Distributed compressive sensing,” http://arxiv.org/abs/0901.3403v1 (2005).

Ender, J. H.G.

J. H.G. Ender, “On compressive sensing applied to radar,” Signal Process. 90, 1402–1414 (2010).
[CrossRef]

Fadil, S.

S. Fadil and W. S. William, “Linear inversion of band-limited reflection seismograms,” SIAM J. Sci. Statist. Comput. 7, 1307–1330 (1986).

Gadella, T. W. J.

R. A. Hoebe, C. H. Van Oven, T. W. J. Gadella, P. B. Dhonukshe, C. J. F. Van Noorden, and E. M. M. Manders, “Controlled light-exposure microscopy reduces photobleaching and phototoxicity in fluorescence live-cell imaging,” Nat. Biotechnol. 25, 249–253 (2007).
[CrossRef] [PubMed]

Gilbert, A. C.

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

Glasner, D.

D. Glasner, S. Bagon, and M. Irani, “Super-resolution from a single image,” in IEEE 12th International Conference on Computer Vision, 2009 (IEEE, 2009), pp. 349–356.
[CrossRef]

Golbabaee, M.

M. Golbabaee and P. Vandergheynst, “Distributed compressed sensing for sensor networks, using thresholding,” Proc. SPIE 7446, 74461F (2009).
[CrossRef]

Haupt, J.

J. Haupt and R. Nowak, “Signal reconstruction from noisy random projections,” IEEE Trans. Inf. Theory 52, 4036–4048(2006).
[CrossRef]

Hoebe, R. A.

R. A. Hoebe, C. H. Van Oven, T. W. J. Gadella, P. B. Dhonukshe, C. J. F. Van Noorden, and E. M. M. Manders, “Controlled light-exposure microscopy reduces photobleaching and phototoxicity in fluorescence live-cell imaging,” Nat. Biotechnol. 25, 249–253 (2007).
[CrossRef] [PubMed]

Inoue, H.

Irani, M.

D. Glasner, S. Bagon, and M. Irani, “Super-resolution from a single image,” in IEEE 12th International Conference on Computer Vision, 2009 (IEEE, 2009), pp. 349–356.
[CrossRef]

Kang, M. G.

S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20, 21–36 (2003).
[CrossRef]

Kawase, K.

Kelly, K.

R. Baraniuk and K. Kelly, “Lecture 5: a single-pixel compressive camera,” retrieved from http://ima.umn.edu/2006-2007/ND6.4-15.07/activities/Baraniuk-Richard/baraniuk-IMA-CScamera-june07.pdf.

Kelly, K. F.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, S. Ting, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE 6065, 606509 (2006).
[CrossRef]

Kirk, L. K.

L. K. Kirk, “Rethinking signal processing,” Commun. ACM 52, 13–15 (2009).
[CrossRef]

Laska, J. N.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, S. Ting, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE 6065, 606509 (2006).
[CrossRef]

Lustig, M.

M. Lustig, D. L. Donoho, and J. M. Pauly, “Rapid MR imaging with compressed sensing and randomly under-sampled 3DFT trajectories,” presented at the 14th Annual Meeting of the International Society for Magnetic Resonance in Medicine, Seattle, Washington, 6–12 May 2006.

Manders, E. M. M.

R. A. Hoebe, C. H. Van Oven, T. W. J. Gadella, P. B. Dhonukshe, C. J. F. Van Noorden, and E. M. M. Manders, “Controlled light-exposure microscopy reduces photobleaching and phototoxicity in fluorescence live-cell imaging,” Nat. Biotechnol. 25, 249–253 (2007).
[CrossRef] [PubMed]

Mittleman, D. M.

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

Muir, F.

J. F. Claerbout and F. Muir, “Robust modeling with erratic data,” Geophysics 38, 826–844 (1973).
[CrossRef]

Nowak, R.

J. Haupt and R. Nowak, “Signal reconstruction from noisy random projections,” IEEE Trans. Inf. Theory 52, 4036–4048(2006).
[CrossRef]

Ogawa, Y.

Ottensamer, R.

J. Bobin, J. L. Starck, and R. Ottensamer, “Compressed sensing in astronomy,” IEEE J. Sel. Top. Signal Process. 2, 718–726 (2008).
[CrossRef]

Park, M. K.

S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20, 21–36 (2003).
[CrossRef]

Park, S. C.

S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20, 21–36 (2003).
[CrossRef]

Pauly, J. M.

M. Lustig, D. L. Donoho, and J. M. Pauly, “Rapid MR imaging with compressed sensing and randomly under-sampled 3DFT trajectories,” presented at the 14th Annual Meeting of the International Society for Magnetic Resonance in Medicine, Seattle, Washington, 6–12 May 2006.

Romberg, J.

E. J. Candès and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl. 23, 969–985 (2007).
[CrossRef]

E. J. 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]

E. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

Sarvotham, S.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE 6065, 606509 (2006).
[CrossRef]

D. Baron, M. B. Wakin, M. F. Duarte, S. Sarvotham, and R. G. Baraniuk, “Distributed compressive sensing,” http://arxiv.org/abs/0901.3403v1 (2005).

Saunders, M.

S. Chen, D. Donoho, and M. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43, 129–159 (2001).
[CrossRef]

Starck, J. L.

J. Bobin, J. L. Starck, and R. Ottensamer, “Compressed sensing in astronomy,” IEEE J. Sel. Top. Signal Process. 2, 718–726 (2008).
[CrossRef]

Starck, J.-L.

J. Bobin and J.-L. Starck, “Compressed sensing in astronomy and remote sensing: a data fusion perspective,” Proc. SPIE 7446, 74460I (2009).
[CrossRef]

Takhar, D.

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, S. Ting, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE 6065, 606509 (2006).
[CrossRef]

Tao, T.

E. J. 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]

E. J. Candès and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies?” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[CrossRef]

E. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

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

Ting, S.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, S. Ting, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

Tropp, J. A.

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

Van Noorden, C. J. F.

R. A. Hoebe, C. H. Van Oven, T. W. J. Gadella, P. B. Dhonukshe, C. J. F. Van Noorden, and E. M. M. Manders, “Controlled light-exposure microscopy reduces photobleaching and phototoxicity in fluorescence live-cell imaging,” Nat. Biotechnol. 25, 249–253 (2007).
[CrossRef] [PubMed]

Van Oven, C. H.

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W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
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E. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
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Figures (18)

Fig. 1
Fig. 1

Example of a simple recovery problem. (a) Logan–Shepp phantom test image. (b) Sampling domain in the frequency plane; the Fourier coefficients are sampled along 22 approximately radial lines. (c) Minimum energy reconstruction obtained by setting unobserved Fourier coefficients to zero. (d) Compressive sensing based reconstruction. This reconstruction is an exact replica of the image in (a) [5].

Fig. 2
Fig. 2

Geometry of 1 recovery. (a) Visualization of the 2 minimization that does not find the sparse point of contact s ^ between the 2 ball (hypersphere, in red) and the translated measurement matrix null space (in green). (b) Visualization of the 1 minimization solution that finds the sparse point of contact s ^ with high probability thanks to the pointiness of the 1 ball [15].

Fig. 3
Fig. 3

Single-pixel camera block diagram. Incident light field (corresponding to the desired image x) is reflected off a DMD array whose mirror orientations are modulated by a pseudorandom pattern. Each different mirror pattern produces a voltage at the single photodiode that corresponds to one measurement y [ m ] . From M measurements, a sparse approximation to the desired image x using CS techniques can be obtained [11].

Fig. 4
Fig. 4

Optical setup of the single-pixel camera [11].

Fig. 5
Fig. 5

Active illumination single-pixel-camera experimental setup. Following the red arrows, it can be seen that the image projected by the video projector is reflected on the wall and, by the means of a lens, is focused on the photodiode active area. The output of the photodiode amplifier circuit is connected to a data acquisition board.

Fig. 6
Fig. 6

(a) Compact active illumination single-pixel camera setup. (b) Detailed photo of the assembly comprising the lens and the photodiode circuit.

Fig. 7
Fig. 7

Example of one of the projected images, representing the product between a random measurement pattern and the image to be reconstructed.

Fig. 8
Fig. 8

Ideal image with 64 × 64 pixels ( N = 4096 ).

Fig. 9
Fig. 9

Images ( 64 × 64 pixels) reconstructed using the best K-term Haar wavelet approximation: (a)  K = 400 and (b)  K = 675 [11].

Fig. 10
Fig. 10

CS reconstruction from: (a)  M = 820 measurements 20 % and (b) M = 1600 measurements 39 % [11].

Fig. 11
Fig. 11

(a) Original object (ball), (b) 4096 pixels (800 measurements 20 % ), and (c) 4096 pixels (1600 measurements 40 % ) (pictures taken from http://dsp.rice.edu/cscamera).

Fig. 12
Fig. 12

(a) Original object (mandrill), (b) 4096 pixels (800 measurements 20 % ) and (c) 4096 pixels (1600 measurements 40 % ) (pictures taken from http://dsp.rice.edu/cscamera). This image was reconstructed using RGB color filters to separately acquire each channel and then combine them.

Fig. 13
Fig. 13

(a)  64 × 64 pixel ( N = 4096 ) raster scan image obtained with 4096 measurements. (b)  64 × 64 pixel image reconstructed via CS from 2700 measurements [20].

Fig. 14
Fig. 14

First results obtained ( 32 × 32 pixels N = 1024 ) with the active illumination single-pixel camera: (a) 205 measurements 20 % (peak signal-to-noise ratio ( PSNR ) = 11.08 dB ), (b) 410 measurements 40 % ( PSNR = 12.30 dB ), and (c) 717 measurements 70 % ( PSNR = 13.21 dB ).

Fig. 15
Fig. 15

(a) Original scene. Image reconstruction using (b) 20% of the measurements ( PSNR = 69.74 dB ), (c) 40% of the measurements ( PSNR = 75.60 dB ), and (d) 60% of the measurements. All the reconstructions are images with 64 × 64 pixels ( N = 4096 ). All the PSNR values were obtained comparing the respective image with the image reconstructed using 60% of the measurements.

Fig. 16
Fig. 16

Reconstruction of the central image of Fig. 14 after the addition of uniformly distributed noise with maximum amplitude of (a) 10% of the maximum amplitude of the measured signal ( SNR = 20.63 dB ) and (b) 20% of the maximum amplitude of the measured signal ( SNR = 14.54 dB ).

Fig. 17
Fig. 17

(a) Piece of paper with a painted red contour and green background (the coin is there only for size comparison). (b) Reconstructed color image of the painted area in (a).

Fig. 18
Fig. 18

Color reconstruction ( 64 × 64 pixels) of the real scene depicted in Fig. 15a. Forty percent of the measurements were used to reconstruct the image for each of the RGB channels.

Equations (8)

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

x = i = 1 N s i ψ i or x = Ψ s ,
( 1 δ K ) x 2 2 A x 2 2 ( 1 + δ K ) x 2 2
( 1 δ 2 s ) x 1 x 2 2 2 A x 1 A x 2 2 2 ( 1 + δ 2 s ) x 1 x 2 2 2
y = Φ x = Φ Ψ s = Θ s .
μ ( Φ , Ψ ) = n · max 1 k , j n | φ k , ψ j | ,
s ^ = arg min s 1 such that Θ s = y ,
y = Φ x + e ,
s ^ = arg min s 1 such that y Φ Ψ s 2 < ε ,

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