## Video-rate compressive holographic microscopic tomography |

Optics Express, Vol. 19, Issue 8, pp. 7289-7298 (2011)

http://dx.doi.org/10.1364/OE.19.007289

Acrobat PDF (1734 KB)

### Abstract

Compressive holography enables 3D reconstruction from a single 2D holographic snapshot for objects that can be sparsely represented in some basis. The snapshot mode enables tomographic imaging of microscopic moving objects. We demonstrate video-rate tomographic image acquisition of two live water cyclopses with 5.2 μm spatial resolution and 60 μm axial resolution.

© 2011 OSA

## 1. Introduction

1. D. Gabor, “A new microscopic principle,” Nature **161**(4098), 777–778 (1948). [CrossRef] [PubMed]

2. E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. **52**(10), 1123–1130 (1962). [CrossRef]

3. D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express **17**(15), 13040–13049 (2009). [CrossRef] [PubMed]

4. E. J. Candès, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. **59**(8), 1207–1223 (2006). [CrossRef]

6. D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory **52**(4), 1289–1306 (2006). [CrossRef]

7. L. Denis, D. Lorenz, E. Thiébaut, C. Fournier, and D. Trede, “Inline hologram reconstruction with sparsity constraints,” Opt. Lett. **34**(22), 3475–3477 (2009). [CrossRef] [PubMed]

8. C. Fournier, L. Denis, and T. Fournel, “On the single point resolution of on-axis digital holography,” J. Opt. Soc. Am. A **27**(8), 1856–1862 (2010). [CrossRef]

9. A. F. Coskun, I. Sencan, T.-W. Su, and A. Ozcan, “Lensless wide-field fluorescent imaging on a chip using compressive decoding of sparse objects,” Opt. Express **18**(10), 10510–10523 (2010). [CrossRef] [PubMed]

10. M. M. Marim, M. Atlan, E. Angelini, and J.-C. Olivo-Marin, “Compressed sensing with off-axis frequency-shifting holography,” Opt. Lett. **35**(6), 871–873 (2010). [CrossRef] [PubMed]

11. Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel holography,” J. Disp. Technol. **6**(10), 506–509 (2010). [CrossRef]

12. H. Kim, S.-W. Min, B. Lee, and T.-C. Poon, “Optical sectioning for optical scanning holography using phase-space filtering with Wigner distribution functions,” Appl. Opt. **47**(19), D164–D175 (2008). [CrossRef] [PubMed]

13. E. Y. Lam, X. Zhang, H. Vo, T.-C. Poon, and G. Indebetouw, “Three-dimensional microscopy and sectional image reconstruction using optical scanning holography,” Appl. Opt. **48**(34), H113–H119 (2009). [CrossRef] [PubMed]

*NA*), the practical resolving power and reconstruction fidelity of compressive holography largely depend upon the

*effective NA*that is determined by object feature sizes. It is desirable that the diffraction signals fully occupy the detector area such that the

*NA*of an optical system is fully utilized. In practice, the combined effects of these factors may be reasonably estimated from the amounts of diffraction signals produced by the object features. Increasing diffraction signals, and equivalently the effective

*NA*, may be achieved by several ways. For example, a microscopic objective can be used to increase the numerical aperture of a system by magnifying the field entering the objective. Of course, the objective tends to increase system volume and decrease field of view.

14. J. Garcia-Sucerquia, W. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. **45**(5), 836–850 (2006). [CrossRef] [PubMed]

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

14. J. Garcia-Sucerquia, W. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. **45**(5), 836–850 (2006). [CrossRef] [PubMed]

## 2. System model

*z*can be written aswhere

*h*is the Huygens-Fresnel point-spread function [17]. Since the

*NA*of the system is relatively large, the Fresnel approximation may be inaccurate. Thus, the use of the angular spectrum transfer function for propagation would be desirable. However, the effective NA is smaller than the numerical aperture of the system, which allows us to use the Fresnel approximation without suffering from the loss of numerical accuracy. The concept of the effective NA is discussed in details in Sec. 3.

*Δ*denotes the sampling spacing in the FPA plane and

3. D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express **17**(15), 13040–13049 (2009). [CrossRef] [PubMed]

*f*and

*g*denote vectorized versions of

*H*denotes a measurement matrix whose element is given by

*C*is a matrix representing

*e*represents the measurement error resulting from the autocorrelation

*n*denotes additive noise.

*f*that minimizes the total variation (TV) [18

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

*f*in the variational domain. The minimum TV estimate

*f*can be obtained by solvingIn this equation,

*f*and is defined aswhere

*z*-th transverse slice. We solve the optimization problem in Eq. (9) by adapting the two-step iterative shrinkage/thresholding (TwIST) algorithm [19

19. J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new twIst: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. **16**(12), 2992–3004 (2007). [CrossRef] [PubMed]

*e*in Eq. (7)) and from the conjugate image is traditionally challenging for in-line holography. As previously reported in [3

3. D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express **17**(15), 13040–13049 (2009). [CrossRef] [PubMed]

## 3. Experimental results and discussion

*NA*manufactured by LOMO is used to generate a spherical wave for illuminating the sample. While the illumination source is not compact, we note that alternative sources using fiber or holographic components could achieve similar NA. The microscope objective is chosen to have a larger

*NA*(0.65) than the microscope system

*NA*(0.27) to ensure uniform illumination intensity on the FPA. A Lumenera CMOS sensor records the hologram. The sensor has

*g*in the algorithm in Eq. (8) to reconstruct the object scattering density

*f*(or equivalently,

*β*).

_{,}respectively. Figures 4(c) and 4(d) show transverse slices at the same axial positions as those in Figs. 4(a) and 4(b). It is clear that the compressive holography reconstructions show significantly better localization (or sectioning) capability. Also, the compressive holography reconstructions suffer less from the undesired background “noise” resulting in better image contrast. This reflects as better reconstruction fidelity. For example, the tails and antennae of both water cyclopses are remarkably sharper and more discernible in Fig. 4(c) and 4(d) compared to those in Fig. 4(a) and 4(b).

*NA*is defined as the half width of the FPA over the distance of objects to the FPA plane. The

*NA*is 0.27 in our experiments. From the equations, the theoretical resolution limits are estimated as

21. D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. **48**(9), 095801 (2009). [CrossRef]

23. D. L. Marks, “A family of approximations spanning the Born and Rytov scattering series,” Opt. Express **14**(19), 8837–8848 (2006). [CrossRef] [PubMed]

24. J. Chae and S. Nishida, “Integumental ultrastructure and colour patterns in the iridescent copepods of the family Sapphirinidae (Copepoda: Poecilostomatoida),” Mar. Biol. **119**(2), 205–210 (1994). [CrossRef]

*h*) represents Fig. 7(b), and the value (

*v*) represents Fig. 7(a). The saturation (

*s*) is set to 1.

_{.}The movement is illustrated by its color changes according to its depths.

## 4. Conclusion

## References and links

1. | D. Gabor, “A new microscopic principle,” Nature |

2. | E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. |

3. | D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express |

4. | E. J. Candès, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. |

5. | E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory |

6. | D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory |

7. | L. Denis, D. Lorenz, E. Thiébaut, C. Fournier, and D. Trede, “Inline hologram reconstruction with sparsity constraints,” Opt. Lett. |

8. | C. Fournier, L. Denis, and T. Fournel, “On the single point resolution of on-axis digital holography,” J. Opt. Soc. Am. A |

9. | A. F. Coskun, I. Sencan, T.-W. Su, and A. Ozcan, “Lensless wide-field fluorescent imaging on a chip using compressive decoding of sparse objects,” Opt. Express |

10. | M. M. Marim, M. Atlan, E. Angelini, and J.-C. Olivo-Marin, “Compressed sensing with off-axis frequency-shifting holography,” Opt. Lett. |

11. | Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel holography,” J. Disp. Technol. |

12. | H. Kim, S.-W. Min, B. Lee, and T.-C. Poon, “Optical sectioning for optical scanning holography using phase-space filtering with Wigner distribution functions,” Appl. Opt. |

13. | E. Y. Lam, X. Zhang, H. Vo, T.-C. Poon, and G. Indebetouw, “Three-dimensional microscopy and sectional image reconstruction using optical scanning holography,” Appl. Opt. |

14. | J. Garcia-Sucerquia, W. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. |

15. | S. Seo, T. W. Su, D. K. Tseng, A. Erlinger, and A. Ozcan, “Lensfree holographic imaging for on-chip cytometry and diagnostics,” Lab Chip |

16. | J. Hahn, S. Lim, K. Choi, R. Horisaki, D. L. Marks, and D. J. Brady, “Compressive Holographic Microscopy,” in |

17. | R. E. Blahurt, |

18. | L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D |

19. | J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new twIst: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. |

20. | D. J. Brady, |

21. | D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. |

22. | A. C. Kak, and M. Slaney, |

23. | D. L. Marks, “A family of approximations spanning the Born and Rytov scattering series,” Opt. Express |

24. | J. Chae and S. Nishida, “Integumental ultrastructure and colour patterns in the iridescent copepods of the family Sapphirinidae (Copepoda: Poecilostomatoida),” Mar. Biol. |

**OCIS Codes**

(070.0070) Fourier optics and signal processing : Fourier optics and signal processing

(090.1995) Holography : Digital holography

**ToC Category:**

Holography

**History**

Original Manuscript: November 1, 2010

Revised Manuscript: January 16, 2011

Manuscript Accepted: February 3, 2011

Published: March 31, 2011

**Virtual Issues**

Vol. 6, Iss. 5 *Virtual Journal for Biomedical Optics*

**Citation**

Joonku Hahn, Sehoon Lim, Kerkil Choi, Ryoichi Horisaki, and David J. Brady, "Video-rate compressive holographic microscopic tomography," Opt. Express **19**, 7289-7298 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-8-7289

Sort: Year | Journal | Reset

### References

- D. Gabor, “A new microscopic principle,” Nature 161(4098), 777–778 (1948). [CrossRef] [PubMed]
- E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. 52(10), 1123–1130 (1962). [CrossRef]
- D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17(15), 13040–13049 (2009). [CrossRef] [PubMed]
- E. J. Candès, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59(8), 1207–1223 (2006). [CrossRef]
- E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory 52(12), 5406–5425 (2006). [CrossRef]
- D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006). [CrossRef]
- L. Denis, D. Lorenz, E. Thiébaut, C. Fournier, and D. Trede, “Inline hologram reconstruction with sparsity constraints,” Opt. Lett. 34(22), 3475–3477 (2009). [CrossRef] [PubMed]
- C. Fournier, L. Denis, and T. Fournel, “On the single point resolution of on-axis digital holography,” J. Opt. Soc. Am. A 27(8), 1856–1862 (2010). [CrossRef]
- A. F. Coskun, I. Sencan, T.-W. Su, and A. Ozcan, “Lensless wide-field fluorescent imaging on a chip using compressive decoding of sparse objects,” Opt. Express 18(10), 10510–10523 (2010). [CrossRef] [PubMed]
- M. M. Marim, M. Atlan, E. Angelini, and J.-C. Olivo-Marin, “Compressed sensing with off-axis frequency-shifting holography,” Opt. Lett. 35(6), 871–873 (2010). [CrossRef] [PubMed]
- Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel holography,” J. Disp. Technol. 6(10), 506–509 (2010). [CrossRef]
- H. Kim, S.-W. Min, B. Lee, and T.-C. Poon, “Optical sectioning for optical scanning holography using phase-space filtering with Wigner distribution functions,” Appl. Opt. 47(19), D164–D175 (2008). [CrossRef] [PubMed]
- E. Y. Lam, X. Zhang, H. Vo, T.-C. Poon, and G. Indebetouw, “Three-dimensional microscopy and sectional image reconstruction using optical scanning holography,” Appl. Opt. 48(34), H113–H119 (2009). [CrossRef] [PubMed]
- J. Garcia-Sucerquia, W. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. 45(5), 836–850 (2006). [CrossRef] [PubMed]
- S. Seo, T. W. Su, D. K. Tseng, A. Erlinger, and A. Ozcan, “Lensfree holographic imaging for on-chip cytometry and diagnostics,” Lab Chip 9(6), 777–787 (2009). [CrossRef] [PubMed]
- J. Hahn, S. Lim, K. Choi, R. Horisaki, D. L. Marks, and D. J. Brady, “Compressive Holographic Microscopy,” in Biomedical Optics, OSA Technical Digest (CD) (Optical Society of America, 2010), paper JMA1.
- R. E. Blahurt, Theory of Remote Image Formation (Cambridge University Press, 2005).
- L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60(1–4), 259–268 (1992). [CrossRef]
- J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new twIst: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16(12), 2992–3004 (2007). [CrossRef] [PubMed]
- D. J. Brady, Optical Imaging and Spectroscopy (Wiley, 2009).
- D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48(9), 095801 (2009). [CrossRef]
- A. C. Kak, and M. Slaney, Principles of Computerized Tomographic Imaging (SIAM, 2001).
- D. L. Marks, “A family of approximations spanning the Born and Rytov scattering series,” Opt. Express 14(19), 8837–8848 (2006). [CrossRef] [PubMed]
- J. Chae and S. Nishida, “Integumental ultrastructure and colour patterns in the iridescent copepods of the family Sapphirinidae (Copepoda: Poecilostomatoida),” Mar. Biol. 119(2), 205–210 (1994). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

### Supplementary Material

» Media 1: MPEG (2668 KB)

» Media 2: MPEG (1792 KB)

» Media 3: MPEG (1774 KB)

» Media 4: MPEG (3884 KB)

» Media 5: MPEG (3846 KB)

« Previous Article | Next Article »

OSA is a member of CrossRef.