Abstract

Compressive sampling enables signal reconstruction using less than one measurement per reconstructed signal value. Compressive measurement is particularly useful in generating multidimensional images from lower dimensional data. We demonstrate single frame 3D tomography from 2D holographic data.

© 2009 Optical Society of America

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  1. C. E. Shannon, “Communications in the presence of noise,” in Proc.of the IREv 37, 10–21 (1949).
    [Crossref]
  2. M. Golay, “Multislit spectroscopy,” J. Opt. Soc. Am. 39, 437–444 (1949).
    [Crossref] [PubMed]
  3. E. J. Candes, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Communications on Pure and Applied Mathematics 59, 1207–1223 (2006).
    [Crossref]
  4. E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?,” IEEE Transactions on Information Theory 52(12), 5406–5425 (2006).
    [Crossref]
  5. D. L. Donoho, “Compressed sensing,” IEEE Transactions on Information Theory 52(4), 1289–1306 (2006).
    [Crossref]
  6. 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,” in Computational Imaging IV, 6065, (San Jose, CA, USA), p. 606509, SPIE, 2006.
  7. 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(21), 14013–14027 (2007).
    [Crossref]
  8. A. Wagadarikar, R. John, R. Willett, and D. J. Brady, “Single disperser design for coded aperture snapshot spectral imaging,” Appl. Opt. 47(10), B44–B51 (2008).
    [Crossref]
  9. D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
    [Crossref] [PubMed]
  10. E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. 52, 1123–1130 (1962).
    [Crossref]
  11. S. Seo, T. W. Su, D. K. Tseng, A. Erlinger, and A. Ozcan, “Lensfree holographic imaging for on-chip cytometry and diagnostics,” Lab on a Chip 9(6), 777–787 (2009).
    [Crossref]
  12. W. Jueptner and U. Schnars, Digital Holography, (New York, Springer-Verlag, Berlin Heidelberg, 2005).
  13. T. C. Poon, Digital holography and three-dimensional display, (New York; London: Springer, 2006).
    [Crossref]
  14. M. E. Brezinski, Optical coherence tomography. (Amsterdam; Boston: Academic Press, 2006).
  15. D. L. Marks, R. A. Stack, D. J. Brady, D. C. Munson, and R. B. Brady, “Visible cone-beam tomography with a lensless interferometric camera,” Science 284(5423), 2164–2166 (1999).
    [Crossref]
  16. J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3d microscopy and gene expression studies,” Science 296(5567), 541–545 (2002).
    [Crossref]
  17. A. J. Devaney, “Nonuniqueness in the inverse scattering problem,” Journal of Mathematical Physics 19(7), 1526–1531 (1978).
    [Crossref]
  18. A. J. Devaney, “Geophysical diffraction tomography,” IEEE Transactions on Geoscience and Remote Sensing GE-22, 3–13(1984).
    [Crossref]
  19. W. L. Chan, M. L. Moravec, R. G. Baraniuk, and D. M. Mittleman, “Terahertz imaging with compressed sensing and phase retrieval,” Opt. Lett. 33(9), 974–976 (2008).
    [Crossref]
  20. I. Yamaguchi, K. Yamamoto, G. A. Mills, and M. Yokota, “Image reconstruction only by phase data in phase-shifting digital holography,” Appl. Opt. 45(5), 975–983 (2006).
    [Crossref]
  21. R. Baraniuk and P. Steeghs, “Compressive radar imaging,” IEEE Radar Conference, pp. 128–133, April, (2007).
  22. L. Li, W. Zhang, and F. Li, “Compressive diffraction tomography for Weakly Scattering,” Submitted to IEEE Trans. on Geosciences and Remote Sensing (2009).
  23. D. L. Donoho and M. Elad, “Optimally sparse representation in general (nonorthogonal) dictionaries via 𡄓1 minimization,” Proceedings of the National Academy of Sciences of the United States of America,  100(5), 2197–2202 (2003).
    [Crossref]
  24. R. E. Blahut, Theory of Remote Image Formation, (Cambridge University Press, 2004).
    [Crossref]
  25. J. W. Goodman, Introduction to Fourier optics, 3rd Ed., (Roberts and Company Publishers, 2005).
  26. D. M. Paganin, Coherent X-ray Optics, (Oxford Science Publications, 2006).
    [Crossref]
  27. A. C. Kak and M. Slaney, Principle of Computerized Tomographic Imaging, (Society for Industrial and Applied Mathematics, 2001).
    [Crossref]
  28. D. J. Brady, Optical Imaging and Spectroscopy, (Wiley, 2009).
    [Crossref]
  29. L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60(1–4), 259–268 (1992).
    [Crossref]
  30. J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new twist: Two-step iterative shrinkage/thresholding algorithms for image restoration,”, IEEE Transactions on Image Processing 16, 2992–3004 (2007).
    [Crossref] [PubMed]
  31. V. Micó, J. García, Z. Zalevsky, and B. Javidi, “Phase-shifting Gabor holography,” Opt. Express 17, 1492–1494 (2009).
  32. T. Latychevskaia and H. W. Fink, “Solution to the Twin Image Problem in Holography,” Phys. Rev. Lett. 98, 233901 (2007).
    [Crossref] [PubMed]

2009 (2)

S. Seo, T. W. Su, D. K. Tseng, A. Erlinger, and A. Ozcan, “Lensfree holographic imaging for on-chip cytometry and diagnostics,” Lab on a Chip 9(6), 777–787 (2009).
[Crossref]

V. Micó, J. García, Z. Zalevsky, and B. Javidi, “Phase-shifting Gabor holography,” Opt. Express 17, 1492–1494 (2009).

2008 (2)

W. L. Chan, M. L. Moravec, R. G. Baraniuk, and D. M. Mittleman, “Terahertz imaging with compressed sensing and phase retrieval,” Opt. Lett. 33(9), 974–976 (2008).
[Crossref]

A. Wagadarikar, R. John, R. Willett, and D. J. Brady, “Single disperser design for coded aperture snapshot spectral imaging,” Appl. Opt. 47(10), B44–B51 (2008).
[Crossref]

2007 (3)

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(21), 14013–14027 (2007).
[Crossref]

T. Latychevskaia and H. W. Fink, “Solution to the Twin Image Problem in Holography,” Phys. Rev. Lett. 98, 233901 (2007).
[Crossref] [PubMed]

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

2006 (4)

E. J. Candes, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Communications on Pure and Applied Mathematics 59, 1207–1223 (2006).
[Crossref]

E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?,” IEEE Transactions on Information Theory 52(12), 5406–5425 (2006).
[Crossref]

D. L. Donoho, “Compressed sensing,” IEEE Transactions on Information Theory 52(4), 1289–1306 (2006).
[Crossref]

I. Yamaguchi, K. Yamamoto, G. A. Mills, and M. Yokota, “Image reconstruction only by phase data in phase-shifting digital holography,” Appl. Opt. 45(5), 975–983 (2006).
[Crossref]

2003 (1)

D. L. Donoho and M. Elad, “Optimally sparse representation in general (nonorthogonal) dictionaries via 𡄓1 minimization,” Proceedings of the National Academy of Sciences of the United States of America,  100(5), 2197–2202 (2003).
[Crossref]

2002 (1)

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3d microscopy and gene expression studies,” Science 296(5567), 541–545 (2002).
[Crossref]

1999 (1)

D. L. Marks, R. A. Stack, D. J. Brady, D. C. Munson, and R. B. Brady, “Visible cone-beam tomography with a lensless interferometric camera,” Science 284(5423), 2164–2166 (1999).
[Crossref]

1992 (1)

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60(1–4), 259–268 (1992).
[Crossref]

1984 (1)

A. J. Devaney, “Geophysical diffraction tomography,” IEEE Transactions on Geoscience and Remote Sensing GE-22, 3–13(1984).
[Crossref]

1978 (1)

A. J. Devaney, “Nonuniqueness in the inverse scattering problem,” Journal of Mathematical Physics 19(7), 1526–1531 (1978).
[Crossref]

1962 (1)

1949 (2)

C. E. Shannon, “Communications in the presence of noise,” in Proc.of the IREv 37, 10–21 (1949).
[Crossref]

M. Golay, “Multislit spectroscopy,” J. Opt. Soc. Am. 39, 437–444 (1949).
[Crossref] [PubMed]

1948 (1)

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[Crossref] [PubMed]

Ahlgren, U.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3d microscopy and gene expression studies,” Science 296(5567), 541–545 (2002).
[Crossref]

Baldock, R.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3d microscopy and gene expression studies,” Science 296(5567), 541–545 (2002).
[Crossref]

Baraniuk, R.

R. Baraniuk and P. Steeghs, “Compressive radar imaging,” IEEE Radar Conference, pp. 128–133, April, (2007).

Baraniuk, R. G.

W. L. Chan, M. L. Moravec, R. G. Baraniuk, and D. M. Mittleman, “Terahertz imaging with compressed sensing and phase retrieval,” Opt. Lett. 33(9), 974–976 (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,” in Computational Imaging IV, 6065, (San Jose, CA, USA), p. 606509, SPIE, 2006.

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,” in Computational Imaging IV, 6065, (San Jose, CA, USA), p. 606509, SPIE, 2006.

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 Transactions on Image Processing 16, 2992–3004 (2007).
[Crossref] [PubMed]

Blahut, R. E.

R. E. Blahut, Theory of Remote Image Formation, (Cambridge University Press, 2004).
[Crossref]

Brady, D. J.

A. Wagadarikar, R. John, R. Willett, and D. J. Brady, “Single disperser design for coded aperture snapshot spectral imaging,” Appl. Opt. 47(10), B44–B51 (2008).
[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(21), 14013–14027 (2007).
[Crossref]

D. L. Marks, R. A. Stack, D. J. Brady, D. C. Munson, and R. B. Brady, “Visible cone-beam tomography with a lensless interferometric camera,” Science 284(5423), 2164–2166 (1999).
[Crossref]

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

Brady, R. B.

D. L. Marks, R. A. Stack, D. J. Brady, D. C. Munson, and R. B. Brady, “Visible cone-beam tomography with a lensless interferometric camera,” Science 284(5423), 2164–2166 (1999).
[Crossref]

Brezinski, M. E.

M. E. Brezinski, Optical coherence tomography. (Amsterdam; Boston: Academic Press, 2006).

Candes, E. J.

E. J. Candes, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Communications on Pure and Applied Mathematics 59, 1207–1223 (2006).
[Crossref]

E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?,” IEEE Transactions on Information Theory 52(12), 5406–5425 (2006).
[Crossref]

Chan, W. L.

W. L. Chan, M. L. Moravec, R. G. Baraniuk, and D. M. Mittleman, “Terahertz imaging with compressed sensing and phase retrieval,” Opt. Lett. 33(9), 974–976 (2008).
[Crossref]

Davidson, D.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3d microscopy and gene expression studies,” Science 296(5567), 541–545 (2002).
[Crossref]

Devaney, A. J.

A. J. Devaney, “Geophysical diffraction tomography,” IEEE Transactions on Geoscience and Remote Sensing GE-22, 3–13(1984).
[Crossref]

A. J. Devaney, “Nonuniqueness in the inverse scattering problem,” Journal of Mathematical Physics 19(7), 1526–1531 (1978).
[Crossref]

Donoho, D. L.

D. L. Donoho, “Compressed sensing,” IEEE Transactions on Information Theory 52(4), 1289–1306 (2006).
[Crossref]

D. L. Donoho and M. Elad, “Optimally sparse representation in general (nonorthogonal) dictionaries via 𡄓1 minimization,” Proceedings of the National Academy of Sciences of the United States of America,  100(5), 2197–2202 (2003).
[Crossref]

Duarte, M. F.

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,” in Computational Imaging IV, 6065, (San Jose, CA, USA), p. 606509, SPIE, 2006.

Elad, M.

D. L. Donoho and M. Elad, “Optimally sparse representation in general (nonorthogonal) dictionaries via 𡄓1 minimization,” Proceedings of the National Academy of Sciences of the United States of America,  100(5), 2197–2202 (2003).
[Crossref]

Erlinger, A.

S. Seo, T. W. Su, D. K. Tseng, A. Erlinger, and A. Ozcan, “Lensfree holographic imaging for on-chip cytometry and diagnostics,” Lab on a Chip 9(6), 777–787 (2009).
[Crossref]

Fatemi, E.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60(1–4), 259–268 (1992).
[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 Transactions on Image Processing 16, 2992–3004 (2007).
[Crossref] [PubMed]

Fink, H. W.

T. Latychevskaia and H. W. Fink, “Solution to the Twin Image Problem in Holography,” Phys. Rev. Lett. 98, 233901 (2007).
[Crossref] [PubMed]

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[Crossref] [PubMed]

García, J.

V. Micó, J. García, Z. Zalevsky, and B. Javidi, “Phase-shifting Gabor holography,” Opt. Express 17, 1492–1494 (2009).

Gehm, M. E.

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(21), 14013–14027 (2007).
[Crossref]

Golay, M.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier optics, 3rd Ed., (Roberts and Company Publishers, 2005).

Hecksher-Sorensen, J.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3d microscopy and gene expression studies,” Science 296(5567), 541–545 (2002).
[Crossref]

Hill, B.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3d microscopy and gene expression studies,” Science 296(5567), 541–545 (2002).
[Crossref]

Javidi, B.

V. Micó, J. García, Z. Zalevsky, and B. Javidi, “Phase-shifting Gabor holography,” Opt. Express 17, 1492–1494 (2009).

John, R.

A. Wagadarikar, R. John, R. Willett, and D. J. Brady, “Single disperser design for coded aperture snapshot spectral imaging,” Appl. Opt. 47(10), B44–B51 (2008).
[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(21), 14013–14027 (2007).
[Crossref]

Jueptner, W.

W. Jueptner and U. Schnars, Digital Holography, (New York, Springer-Verlag, Berlin Heidelberg, 2005).

Kak, A. C.

A. C. Kak and M. Slaney, Principle of Computerized Tomographic Imaging, (Society for Industrial and Applied Mathematics, 2001).
[Crossref]

Kelly, K. F.

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,” in Computational Imaging IV, 6065, (San Jose, CA, USA), p. 606509, SPIE, 2006.

Laska, J. N.

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,” in Computational Imaging IV, 6065, (San Jose, CA, USA), p. 606509, SPIE, 2006.

Latychevskaia, T.

T. Latychevskaia and H. W. Fink, “Solution to the Twin Image Problem in Holography,” Phys. Rev. Lett. 98, 233901 (2007).
[Crossref] [PubMed]

Leith, E. N.

Li, F.

L. Li, W. Zhang, and F. Li, “Compressive diffraction tomography for Weakly Scattering,” Submitted to IEEE Trans. on Geosciences and Remote Sensing (2009).

Li, L.

L. Li, W. Zhang, and F. Li, “Compressive diffraction tomography for Weakly Scattering,” Submitted to IEEE Trans. on Geosciences and Remote Sensing (2009).

Marks, D. L.

D. L. Marks, R. A. Stack, D. J. Brady, D. C. Munson, and R. B. Brady, “Visible cone-beam tomography with a lensless interferometric camera,” Science 284(5423), 2164–2166 (1999).
[Crossref]

Micó, V.

V. Micó, J. García, Z. Zalevsky, and B. Javidi, “Phase-shifting Gabor holography,” Opt. Express 17, 1492–1494 (2009).

Mills, G. A.

I. Yamaguchi, K. Yamamoto, G. A. Mills, and M. Yokota, “Image reconstruction only by phase data in phase-shifting digital holography,” Appl. Opt. 45(5), 975–983 (2006).
[Crossref]

Mittleman, D. M.

W. L. Chan, M. L. Moravec, R. G. Baraniuk, and D. M. Mittleman, “Terahertz imaging with compressed sensing and phase retrieval,” Opt. Lett. 33(9), 974–976 (2008).
[Crossref]

Moravec, M. L.

W. L. Chan, M. L. Moravec, R. G. Baraniuk, and D. M. Mittleman, “Terahertz imaging with compressed sensing and phase retrieval,” Opt. Lett. 33(9), 974–976 (2008).
[Crossref]

Munson, D. C.

D. L. Marks, R. A. Stack, D. J. Brady, D. C. Munson, and R. B. Brady, “Visible cone-beam tomography with a lensless interferometric camera,” Science 284(5423), 2164–2166 (1999).
[Crossref]

Osher, S.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60(1–4), 259–268 (1992).
[Crossref]

Ozcan, A.

S. Seo, T. W. Su, D. K. Tseng, A. Erlinger, and A. Ozcan, “Lensfree holographic imaging for on-chip cytometry and diagnostics,” Lab on a Chip 9(6), 777–787 (2009).
[Crossref]

Paganin, D. M.

D. M. Paganin, Coherent X-ray Optics, (Oxford Science Publications, 2006).
[Crossref]

Perry, P.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3d microscopy and gene expression studies,” Science 296(5567), 541–545 (2002).
[Crossref]

Poon, T. C.

T. C. Poon, Digital holography and three-dimensional display, (New York; London: Springer, 2006).
[Crossref]

Romberg, J. K.

E. J. Candes, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Communications on Pure and Applied Mathematics 59, 1207–1223 (2006).
[Crossref]

Ross, A.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3d microscopy and gene expression studies,” Science 296(5567), 541–545 (2002).
[Crossref]

Rudin, L. I.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60(1–4), 259–268 (1992).
[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,” in Computational Imaging IV, 6065, (San Jose, CA, USA), p. 606509, SPIE, 2006.

Schnars, U.

W. Jueptner and U. Schnars, Digital Holography, (New York, Springer-Verlag, Berlin Heidelberg, 2005).

Schulz, T. J.

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(21), 14013–14027 (2007).
[Crossref]

Seo, S.

S. Seo, T. W. Su, D. K. Tseng, A. Erlinger, and A. Ozcan, “Lensfree holographic imaging for on-chip cytometry and diagnostics,” Lab on a Chip 9(6), 777–787 (2009).
[Crossref]

Shannon, C. E.

C. E. Shannon, “Communications in the presence of noise,” in Proc.of the IREv 37, 10–21 (1949).
[Crossref]

Sharpe, J.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3d microscopy and gene expression studies,” Science 296(5567), 541–545 (2002).
[Crossref]

Slaney, M.

A. C. Kak and M. Slaney, Principle of Computerized Tomographic Imaging, (Society for Industrial and Applied Mathematics, 2001).
[Crossref]

Stack, R. A.

D. L. Marks, R. A. Stack, D. J. Brady, D. C. Munson, and R. B. Brady, “Visible cone-beam tomography with a lensless interferometric camera,” Science 284(5423), 2164–2166 (1999).
[Crossref]

Steeghs, P.

R. Baraniuk and P. Steeghs, “Compressive radar imaging,” IEEE Radar Conference, pp. 128–133, April, (2007).

Su, T. W.

S. Seo, T. W. Su, D. K. Tseng, A. Erlinger, and A. Ozcan, “Lensfree holographic imaging for on-chip cytometry and diagnostics,” Lab on a Chip 9(6), 777–787 (2009).
[Crossref]

Takhar, 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,” in Computational Imaging IV, 6065, (San Jose, CA, USA), p. 606509, SPIE, 2006.

Tao, T.

E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?,” IEEE Transactions on Information Theory 52(12), 5406–5425 (2006).
[Crossref]

E. J. Candes, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Communications on Pure and Applied Mathematics 59, 1207–1223 (2006).
[Crossref]

Tseng, D. K.

S. Seo, T. W. Su, D. K. Tseng, A. Erlinger, and A. Ozcan, “Lensfree holographic imaging for on-chip cytometry and diagnostics,” Lab on a Chip 9(6), 777–787 (2009).
[Crossref]

Upatnieks, J.

Wagadarikar, A.

A. Wagadarikar, R. John, R. Willett, and D. J. Brady, “Single disperser design for coded aperture snapshot spectral imaging,” Appl. Opt. 47(10), B44–B51 (2008).
[Crossref]

Wakin, M. B.

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,” in Computational Imaging IV, 6065, (San Jose, CA, USA), p. 606509, SPIE, 2006.

Willett, R.

A. Wagadarikar, R. John, R. Willett, and D. J. Brady, “Single disperser design for coded aperture snapshot spectral imaging,” Appl. Opt. 47(10), B44–B51 (2008).
[Crossref]

Willett, R. M.

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(21), 14013–14027 (2007).
[Crossref]

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I. Yamaguchi, K. Yamamoto, G. A. Mills, and M. Yokota, “Image reconstruction only by phase data in phase-shifting digital holography,” Appl. Opt. 45(5), 975–983 (2006).
[Crossref]

Yamamoto, K.

I. Yamaguchi, K. Yamamoto, G. A. Mills, and M. Yokota, “Image reconstruction only by phase data in phase-shifting digital holography,” Appl. Opt. 45(5), 975–983 (2006).
[Crossref]

Yokota, M.

I. Yamaguchi, K. Yamamoto, G. A. Mills, and M. Yokota, “Image reconstruction only by phase data in phase-shifting digital holography,” Appl. Opt. 45(5), 975–983 (2006).
[Crossref]

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V. Micó, J. García, Z. Zalevsky, and B. Javidi, “Phase-shifting Gabor holography,” Opt. Express 17, 1492–1494 (2009).

Zhang, W.

L. Li, W. Zhang, and F. Li, “Compressive diffraction tomography for Weakly Scattering,” Submitted to IEEE Trans. on Geosciences and Remote Sensing (2009).

Appl. Opt. (2)

A. Wagadarikar, R. John, R. Willett, and D. J. Brady, “Single disperser design for coded aperture snapshot spectral imaging,” Appl. Opt. 47(10), B44–B51 (2008).
[Crossref]

I. Yamaguchi, K. Yamamoto, G. A. Mills, and M. Yokota, “Image reconstruction only by phase data in phase-shifting digital holography,” Appl. Opt. 45(5), 975–983 (2006).
[Crossref]

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S. Seo, T. W. Su, D. K. Tseng, A. Erlinger, and A. Ozcan, “Lensfree holographic imaging for on-chip cytometry and diagnostics,” Lab on a Chip 9(6), 777–787 (2009).
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Nature (1)

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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(21), 14013–14027 (2007).
[Crossref]

V. Micó, J. García, Z. Zalevsky, and B. Javidi, “Phase-shifting Gabor holography,” Opt. Express 17, 1492–1494 (2009).

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W. L. Chan, M. L. Moravec, R. G. Baraniuk, and D. M. Mittleman, “Terahertz imaging with compressed sensing and phase retrieval,” Opt. Lett. 33(9), 974–976 (2008).
[Crossref]

Phys. Rev. Lett. (1)

T. Latychevskaia and H. W. Fink, “Solution to the Twin Image Problem in Holography,” Phys. Rev. Lett. 98, 233901 (2007).
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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,” in Computational Imaging IV, 6065, (San Jose, CA, USA), p. 606509, SPIE, 2006.

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R. Baraniuk and P. Steeghs, “Compressive radar imaging,” IEEE Radar Conference, pp. 128–133, April, (2007).

L. Li, W. Zhang, and F. Li, “Compressive diffraction tomography for Weakly Scattering,” Submitted to IEEE Trans. on Geosciences and Remote Sensing (2009).

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Figures (4)

Fig. 1.
Fig. 1.

Gabor hologram geometry.

Fig. 2.
Fig. 2.

Experimental apparatus.

Fig. 3.
Fig. 3.

(a) Raw Gabor hologram for seed parachutes of taraxacum arranged as in Fig. 2, (b) and (c) photographs of the individual objects, (d) transverse slices at various ranges of the tomographic reconstruction of the 3D data volume containing both objects, and (e) transverse slices at various ranges of the backpropagated (numerical refocusing) field.

Fig. 4.
Fig. 4.

Simulations showing the effects of squared field term e: the 3D datacube estimates from the squared field with no diffraction (a) and the squared field with diffraction (b).

Equations (21)

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(1δS)fT22HTfT22(1+δS)fT22,
μ1(H,Ψ)=Nmaxhm,ψn,
1mM
1nN
MC μ12 (H,Ψ)SlogN,
θe=argminθθ1suchthatg=Hfe=HΨθe,
μ2(D)=maxij,1i,jN{diTdjdi2dj2},
S<12(1+1μ2(D)),
I(x,y)=A+E(x,y)2
=A2+E(x,y)2+A*E(x,y)+AE*(x,y),
E(x,y)=dxdydzη(x,y,z)h(xx,yy,zz),
En1n2=E (n1Δ,n2Δ)
=1(2π)2dzdxdydkxdkydxdyη(x,y,z)eikzei(kxx+kyy)Σm1Σm2δ(xm1Δ)
δ (ym2Δ)eizk2kx2ky2δ(zz)ei(kxx+kyy)δ(xn1Δ)δ(yn2Δ)Σm1Σm2δ(kxm1Δk)
δ (kym2Δk)Σlδ(zlΔz)
=1N2ΣlΣm1Σm2[Σm'2Σm2ηm1m2lei2πm1m1+m2m2N]eiklΔzeilΔzk2m12Δk2m22Δk2ei2πn1m1+n2m2N,
En1n2=𝓕2D1 {Σlη̂m1m2leiklΔzeilΔzk2m12Δk2m22Δk2} ,
gˉ=G2DQBf,
g=2Re{gˉ}=2Re{G2DQBf}=2 Re {Hf}+e+n,
f̂=argminffTVsuchthatg=Hf,
fkTV=ΣkΣn1Σn2(fk)n1,n2,

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