Abstract

This work reveals an experimental microscopy acquisition scheme successfully combining compressed sensing (CS) and digital holography in off-axis and frequency-shifting conditions. CS is a recent data acquisition theory involving signal reconstruction from randomly undersampled measurements, exploiting the fact that most images present some compact structure and redundancy. We propose a genuine CS-based imaging scheme for sparse gradient images, acquiring a diffraction map of the optical field with holographic microscopy and recovering the signal from as little as 7% of random measurements. We report experimental results demonstrating how CS can lead to an elegant and effective way to reconstruct images, opening the door for new microscopy applications.

© 2010 Optical Society of America

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  1. C. Jackson, R. F. Murphy, and J. Kovačević, IEEE Trans. Image Process. 18, 2071 (2009).
    [CrossRef]
  2. R. A. Hoebe, C. H. Van Oven, T. W. J. Gadella, Jr., P. B. Dhonukshe, C. J. F. Van Noorden, and E. M. M. Manders, Nat. Biotechnol. 25, 249 (2007).
    [CrossRef] [PubMed]
  3. M. Lustig, D. Donoho, and J. M. Pauly, Magn. Reson. Med. 58, 1182 (2007).
    [CrossRef] [PubMed]
  4. D. Takhar, J. Laska, M. Wakin, M. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. Baraniuk, Proc. SPIE 6065, 43 (2006).
  5. D. Brady, K. Choi, D. Marks, R. Horisaki, and S. Lim, Opt. Express 17, 13040 (2009).
    [CrossRef] [PubMed]
  6. E. Candès and J. Romberg, Inverse Probl. 23, 969 (2007).
    [CrossRef]
  7. D. L. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
    [CrossRef]
  8. E. Candès and J. Romberg, Proc. SPIE 5674, 76 (2005).
    [CrossRef]
  9. E. Candès, Compte Rendus de l'Academie des Sciences, Paris, Serie I 346, 589 (2008).
  10. M. Atlan and M. Gross, J. Opt. Soc. Am. A 24, 2701 (2007).
    [CrossRef]
  11. M. Gross and M. Atlan, Opt. Lett. 32, 909 (2007).
    [CrossRef] [PubMed]
  12. E. Candès, J. Romberg, and T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
    [CrossRef]
  13. M. Marim, E. Angelini, and J.-C. Olivo-Marin, Proc. SPIE 7446, 744605 (2009).
    [CrossRef]
  14. D. Donoho, M. Elad, and V. Temlyakov, IEEE Trans. Inf. Theory 52, 6 (2006).
    [CrossRef]

2009 (3)

C. Jackson, R. F. Murphy, and J. Kovačević, IEEE Trans. Image Process. 18, 2071 (2009).
[CrossRef]

D. Brady, K. Choi, D. Marks, R. Horisaki, and S. Lim, Opt. Express 17, 13040 (2009).
[CrossRef] [PubMed]

M. Marim, E. Angelini, and J.-C. Olivo-Marin, Proc. SPIE 7446, 744605 (2009).
[CrossRef]

2008 (1)

E. Candès, Compte Rendus de l'Academie des Sciences, Paris, Serie I 346, 589 (2008).

2007 (5)

M. Atlan and M. Gross, J. Opt. Soc. Am. A 24, 2701 (2007).
[CrossRef]

M. Gross and M. Atlan, Opt. Lett. 32, 909 (2007).
[CrossRef] [PubMed]

E. Candès and J. Romberg, Inverse Probl. 23, 969 (2007).
[CrossRef]

R. A. Hoebe, C. H. Van Oven, T. W. J. Gadella, Jr., P. B. Dhonukshe, C. J. F. Van Noorden, and E. M. M. Manders, Nat. Biotechnol. 25, 249 (2007).
[CrossRef] [PubMed]

M. Lustig, D. Donoho, and J. M. Pauly, Magn. Reson. Med. 58, 1182 (2007).
[CrossRef] [PubMed]

2006 (4)

D. Takhar, J. Laska, M. Wakin, M. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. Baraniuk, Proc. SPIE 6065, 43 (2006).

D. L. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
[CrossRef]

E. Candès, J. Romberg, and T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
[CrossRef]

D. Donoho, M. Elad, and V. Temlyakov, IEEE Trans. Inf. Theory 52, 6 (2006).
[CrossRef]

2005 (1)

E. Candès and J. Romberg, Proc. SPIE 5674, 76 (2005).
[CrossRef]

Angelini, E.

M. Marim, E. Angelini, and J.-C. Olivo-Marin, Proc. SPIE 7446, 744605 (2009).
[CrossRef]

Atlan, M.

Baraniuk, R.

D. Takhar, J. Laska, M. Wakin, M. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. Baraniuk, Proc. SPIE 6065, 43 (2006).

Baron, D.

D. Takhar, J. Laska, M. Wakin, M. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. Baraniuk, Proc. SPIE 6065, 43 (2006).

Brady, D.

Candès, E.

E. Candès, Compte Rendus de l'Academie des Sciences, Paris, Serie I 346, 589 (2008).

E. Candès and J. Romberg, Inverse Probl. 23, 969 (2007).
[CrossRef]

E. Candès, J. Romberg, and T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
[CrossRef]

E. Candès and J. Romberg, Proc. SPIE 5674, 76 (2005).
[CrossRef]

Choi, K.

Dhonukshe, P. B.

R. A. Hoebe, C. H. Van Oven, T. W. J. Gadella, Jr., P. B. Dhonukshe, C. J. F. Van Noorden, and E. M. M. Manders, Nat. Biotechnol. 25, 249 (2007).
[CrossRef] [PubMed]

Donoho, D.

M. Lustig, D. Donoho, and J. M. Pauly, Magn. Reson. Med. 58, 1182 (2007).
[CrossRef] [PubMed]

D. Donoho, M. Elad, and V. Temlyakov, IEEE Trans. Inf. Theory 52, 6 (2006).
[CrossRef]

Donoho, D. L.

D. L. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
[CrossRef]

Duarte, M.

D. Takhar, J. Laska, M. Wakin, M. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. Baraniuk, Proc. SPIE 6065, 43 (2006).

Elad, M.

D. Donoho, M. Elad, and V. Temlyakov, IEEE Trans. Inf. Theory 52, 6 (2006).
[CrossRef]

Gadella, T. W. J.

R. A. Hoebe, C. H. Van Oven, T. W. J. Gadella, Jr., P. B. Dhonukshe, C. J. F. Van Noorden, and E. M. M. Manders, Nat. Biotechnol. 25, 249 (2007).
[CrossRef] [PubMed]

Gross, M.

Hoebe, R. A.

R. A. Hoebe, C. H. Van Oven, T. W. J. Gadella, Jr., P. B. Dhonukshe, C. J. F. Van Noorden, and E. M. M. Manders, Nat. Biotechnol. 25, 249 (2007).
[CrossRef] [PubMed]

Horisaki, R.

Jackson, C.

C. Jackson, R. F. Murphy, and J. Kovačević, IEEE Trans. Image Process. 18, 2071 (2009).
[CrossRef]

Kelly, K.

D. Takhar, J. Laska, M. Wakin, M. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. Baraniuk, Proc. SPIE 6065, 43 (2006).

Kovacevic, J.

C. Jackson, R. F. Murphy, and J. Kovačević, IEEE Trans. Image Process. 18, 2071 (2009).
[CrossRef]

Laska, J.

D. Takhar, J. Laska, M. Wakin, M. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. Baraniuk, Proc. SPIE 6065, 43 (2006).

Lim, S.

Lustig, M.

M. Lustig, D. Donoho, and J. M. Pauly, Magn. Reson. Med. 58, 1182 (2007).
[CrossRef] [PubMed]

Manders, E. M. M.

R. A. Hoebe, C. H. Van Oven, T. W. J. Gadella, Jr., P. B. Dhonukshe, C. J. F. Van Noorden, and E. M. M. Manders, Nat. Biotechnol. 25, 249 (2007).
[CrossRef] [PubMed]

Marim, M.

M. Marim, E. Angelini, and J.-C. Olivo-Marin, Proc. SPIE 7446, 744605 (2009).
[CrossRef]

Marks, D.

Murphy, R. F.

C. Jackson, R. F. Murphy, and J. Kovačević, IEEE Trans. Image Process. 18, 2071 (2009).
[CrossRef]

Olivo-Marin, J. -C.

M. Marim, E. Angelini, and J.-C. Olivo-Marin, Proc. SPIE 7446, 744605 (2009).
[CrossRef]

Pauly, J. M.

M. Lustig, D. Donoho, and J. M. Pauly, Magn. Reson. Med. 58, 1182 (2007).
[CrossRef] [PubMed]

Romberg, J.

E. Candès and J. Romberg, Inverse Probl. 23, 969 (2007).
[CrossRef]

E. Candès, J. Romberg, and T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
[CrossRef]

E. Candès and J. Romberg, Proc. SPIE 5674, 76 (2005).
[CrossRef]

Sarvotham, S.

D. Takhar, J. Laska, M. Wakin, M. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. Baraniuk, Proc. SPIE 6065, 43 (2006).

Takhar, D.

D. Takhar, J. Laska, M. Wakin, M. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. Baraniuk, Proc. SPIE 6065, 43 (2006).

Tao, T.

E. Candès, J. Romberg, and T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
[CrossRef]

Temlyakov, V.

D. Donoho, M. Elad, and V. Temlyakov, IEEE Trans. Inf. Theory 52, 6 (2006).
[CrossRef]

Van Noorden, C. J. F.

R. A. Hoebe, C. H. Van Oven, T. W. J. Gadella, Jr., P. B. Dhonukshe, C. J. F. Van Noorden, and E. M. M. Manders, Nat. Biotechnol. 25, 249 (2007).
[CrossRef] [PubMed]

Van Oven, C. H.

R. A. Hoebe, C. H. Van Oven, T. W. J. Gadella, Jr., P. B. Dhonukshe, C. J. F. Van Noorden, and E. M. M. Manders, Nat. Biotechnol. 25, 249 (2007).
[CrossRef] [PubMed]

Wakin, M.

D. Takhar, J. Laska, M. Wakin, M. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. Baraniuk, Proc. SPIE 6065, 43 (2006).

Compte Rendus de l'Academie des Sciences, Paris, Serie I (1)

E. Candès, Compte Rendus de l'Academie des Sciences, Paris, Serie I 346, 589 (2008).

IEEE Trans. Image Process. (1)

C. Jackson, R. F. Murphy, and J. Kovačević, IEEE Trans. Image Process. 18, 2071 (2009).
[CrossRef]

IEEE Trans. Inf. Theory (3)

E. Candès, J. Romberg, and T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
[CrossRef]

D. L. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
[CrossRef]

D. Donoho, M. Elad, and V. Temlyakov, IEEE Trans. Inf. Theory 52, 6 (2006).
[CrossRef]

Inverse Probl. (1)

E. Candès and J. Romberg, Inverse Probl. 23, 969 (2007).
[CrossRef]

J. Opt. Soc. Am. A (1)

Magn. Reson. Med. (1)

M. Lustig, D. Donoho, and J. M. Pauly, Magn. Reson. Med. 58, 1182 (2007).
[CrossRef] [PubMed]

Nat. Biotechnol. (1)

R. A. Hoebe, C. H. Van Oven, T. W. J. Gadella, Jr., P. B. Dhonukshe, C. J. F. Van Noorden, and E. M. M. Manders, Nat. Biotechnol. 25, 249 (2007).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (1)

Proc. SPIE (3)

D. Takhar, J. Laska, M. Wakin, M. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. Baraniuk, Proc. SPIE 6065, 43 (2006).

E. Candès and J. Romberg, Proc. SPIE 5674, 76 (2005).
[CrossRef]

M. Marim, E. Angelini, and J.-C. Olivo-Marin, Proc. SPIE 7446, 744605 (2009).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Sketch of the experimental image acquisition setup.

Fig. 2
Fig. 2

CS reconstruction scheme.

Fig. 3
Fig. 3

(a) Standard holography, as described in Eq. (1). (b) CS reconstruction, using 7% of the Fresnel coefficients. (c) Gradient of g. (d) Residual from (a) and (b). (e), (f) Magnified views from (a) and (b).

Equations (6)

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

F = F ( f ) : C N C N ,
F p = 1 N n = 1 N f n e i ( α n 2 2 π n p / N ) ,
f p = 1 N n = 1 N F n e i ( α n 2 2 π n p / N ) .
g ̂ = arg   min g R N Ψ g 1     subject   to   F ̂ Γ = F Γ .
g ̂ = arg   min g R N g 1     subject   to   F ̂ Γ = F Γ .
g ̂ = arg   min g R N g 1     subject   to   F ̂ Γ F Γ 2 δ ,

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