Abstract

We analyze the dependency and the accuracy of the refocusing criterion based on the integrated modulus amplitude in the case of amplitude object. Analytical dependencies on the defocus distance and the numerical aperture are found. This theoretical prediction for the refocusing criterion is well supported by simulation. We study also the robustness of the refocusing criterion by adding salt and pepper and speckle-type noises. We demonstrate that the refocusing criterion is robust up to an significant level of noise that can perturb the holograms.

© 2011 Optical Society of America

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  1. F. Dubois, M.-L. Novella Requena, C. Minetti, O. Monnom, and E. Istasse, "Partial spatial coherence effects in digital holographic microscopy with a laser source," Appl. Opt. 43, 1131-1139 (2004).
    [CrossRef] [PubMed]
  2. F. Dubois, L. Johannes, and J.-C. Legros, "Improved three-dimensional imaging with a digital holography microscope with a source of partial spatial coherence," Appl. Opt. 38, 7085-7094 (1999).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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  6. G. Popescu, L. P. Deflores, J. C. Vaughan, K. Badizadegan, H. Iwai, R. R. Dasari, and M. S. Feld, "Fourier phase microscopy for investigation of biological structures and dynamics," Opt. Lett. 29, 2503-2505 (2004).
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  7. 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, 836-850 (2006).
    [CrossRef] [PubMed]
  8. L. Yu, S. Mohanty, J. Zhang, S. Genc, M. K. Kim, M. W. Berns, and Z. Chen, "Digital holographic microscopy for quantitative cell dynamic evaluation during laser microsurgery," Appl. Opt. 17, 12031-12038 (2009).
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  10. M. Gustafsson, and M. Sebesta, "Refractometry of Microscopic Objects with Digital Holography," Appl. Opt. 43, 4796-4801 (2005).
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  11. D. Carl, B. Kemper, G. Wernicke, and G. von Bally, "Parameter-optimized digital holographic microscope for high-resolution living-cell analysis," Appl. Opt. 43, 6536-6544 (2004).
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  12. N. Warnasooriya, F. Joud, P. Bun, G. Tessier, M. Coppey-Moisan, P. Desbiolles, M. Atlan, M. Abboud, and M. Gross, "Imaging gold nanoparticles in living cell environments using heterodyne digital holographic microscopy," Opt. Express 18, 3264-3273 (2010).
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  13. D. Lebrun, A. Benkouider, S. Cotmellec, and M. Malek, "Particle field digital holographic reconstruction in arbitrary tilted planes," Opt. Express 11, 224-229 (2003).
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  14. N. Pandey, and B. Hennelly, "Fixed-point numerical-reconstruction for digital holographic microscopy," Opt. Lett. 35, 1076-1078 (2010).
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  15. C. Minetti, N. Callens, G. Coupier, T. Podgorski, and F. Dubois, "Fast measurements of concentration profiles inside deformable objects in microflows with spatial coherence digital holography," Appl. Opt. 45, 5305-5314 (2008).
    [CrossRef]
  16. F. Dubois, O. Monnom, C. Yourassowski, and J.-C. Legros, "Pattern recognition with digital holographic microscope working in partially coherent illumination," Appl. Opt. 41, 4108-4119 (2002).
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  17. B. Javidi, I. Moon, S. Yeom, and E. Carapezza, "Three-dimensional imaging and recognition of microorganism using single-exposure on-line (SEOL) digital holography," Opt. Express 13, 4492-4506 (2005).
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  21. F. Dubois, O. Monnom, C. Yourassowski, and J.-C. Legros, "Border processing in digital holography by extension of the digital hologram and reduction of the higher spatial frequencies," Appl. Opt. 41, 2621-2626 (2002).
    [CrossRef] [PubMed]
  22. T. Colomb, E. Cuche, F. Charrire, J. Khn, N. Aspert, F. Monfort, P. Marquet, and C. Despeursinge, "Automatic procedure for aberration compensation in digital holographic microscopy and applications to specimen shape compensation," Appl. Opt. 45, 851-863 (2006).
    [CrossRef] [PubMed]
  23. L. Yu, and L. Cai, "Iterative algorithm with a constraint condition for numerical reconstruction of a three-dimensional object from its hologram," J. Opt. Soc. Am. A 18, 1033-1045 (2001).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  26. P. Ferraro, G. Coppola, S. De Nicola, A. Finizio, and G. Pierattini, "Digital holographic microscope with automatic focus tracking by detecting sample displacement in real time," Opt. Lett. 28, 1257-1259 (2003).
    [CrossRef] [PubMed]
  27. Y. J. Choo, and B. S. Kang, "The characteristics of the particle position along an optical axis in particle holography," Meas. Sci. Technol. 17, 761-770 (2006).
    [CrossRef]
  28. Y. Yang, B. S. Kang, and Y. J. Choo, "Application of the correlation coefficient method for determination of the focus plane to digital particle holography," Appl. Opt. 47, 817-824 (2008).
    [CrossRef] [PubMed]
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    [CrossRef]
  31. F. Dubois, C. Schockaert, N. Callens, and C. Yourassowski, "Focus plane detection criteria in digital holography microscopy by amplitude analysis," Opt. Express 14, 5895-5908 (2006).
    [CrossRef] [PubMed]
  32. M. Antkowiak, N. Callens, C. Yourassowski, and F. Dubois, "Extended focused imaging of a microparticle field with digital holographic microscopy," Opt. Lett. 33, 1626-1628 (2008).
    [CrossRef] [PubMed]
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2010 (2)

2009 (1)

L. Yu, S. Mohanty, J. Zhang, S. Genc, M. K. Kim, M. W. Berns, and Z. Chen, "Digital holographic microscopy for quantitative cell dynamic evaluation during laser microsurgery," Appl. Opt. 17, 12031-12038 (2009).

2008 (4)

C. Minetti, N. Callens, G. Coupier, T. Podgorski, and F. Dubois, "Fast measurements of concentration profiles inside deformable objects in microflows with spatial coherence digital holography," Appl. Opt. 45, 5305-5314 (2008).
[CrossRef]

P. Langehanenberg, B. Kemper, D. Dirksen, and G. von Bally, "Autofocusing in digital holographic phase contrast microscopy on pure phase objects for live cell imaging," Appl. Opt. 47, 176-182 (2008).
[CrossRef]

Y. Yang, B. S. Kang, and Y. J. Choo, "Application of the correlation coefficient method for determination of the focus plane to digital particle holography," Appl. Opt. 47, 817-824 (2008).
[CrossRef] [PubMed]

M. Antkowiak, N. Callens, C. Yourassowski, and F. Dubois, "Extended focused imaging of a microparticle field with digital holographic microscopy," Opt. Lett. 33, 1626-1628 (2008).
[CrossRef] [PubMed]

2007 (1)

2006 (5)

2005 (3)

2004 (6)

2003 (2)

2002 (3)

2001 (1)

1999 (2)

1998 (1)

1994 (1)

1989 (1)

J. Gillespie, and R. A. King, "The use of self-entropy as a focus measure in digital holography," Pattern Recognit. Lett. 9, 19-25 (1989).
[CrossRef]

1980 (1)

Abboud, M.

Alfieri, D.

Antkowiak, M.

Aspert, N.

Atlan, M.

Badizadegan, K.

Benkouider, A.

Berns, M. W.

L. Yu, S. Mohanty, J. Zhang, S. Genc, M. K. Kim, M. W. Berns, and Z. Chen, "Digital holographic microscopy for quantitative cell dynamic evaluation during laser microsurgery," Appl. Opt. 17, 12031-12038 (2009).

Bevilacqua, F.

Bun, P.

Cai, L.

Callens, N.

Carapezza, E.

Carl, D.

Charrire, F.

Chen, Z.

L. Yu, S. Mohanty, J. Zhang, S. Genc, M. K. Kim, M. W. Berns, and Z. Chen, "Digital holographic microscopy for quantitative cell dynamic evaluation during laser microsurgery," Appl. Opt. 17, 12031-12038 (2009).

Choo, Y. J.

Y. Yang, B. S. Kang, and Y. J. Choo, "Application of the correlation coefficient method for determination of the focus plane to digital particle holography," Appl. Opt. 47, 817-824 (2008).
[CrossRef] [PubMed]

Y. J. Choo, and B. S. Kang, "The characteristics of the particle position along an optical axis in particle holography," Meas. Sci. Technol. 17, 761-770 (2006).
[CrossRef]

Colomb, T.

Coppey-Moisan, M.

Coppola, G.

Cotmellec, S.

Coupier, G.

C. Minetti, N. Callens, G. Coupier, T. Podgorski, and F. Dubois, "Fast measurements of concentration profiles inside deformable objects in microflows with spatial coherence digital holography," Appl. Opt. 45, 5305-5314 (2008).
[CrossRef]

Cuche, E.

Dasari, R. R.

Davis, C. S.

De Nicola, S.

Deflores, L. P.

Depeursinge, C.

Desbiolles, P.

Despeursinge, C.

Dirksen, D.

P. Langehanenberg, B. Kemper, D. Dirksen, and G. von Bally, "Autofocusing in digital holographic phase contrast microscopy on pure phase objects for live cell imaging," Appl. Opt. 47, 176-182 (2008).
[CrossRef]

Dubois, F.

M. Antkowiak, N. Callens, C. Yourassowski, and F. Dubois, "Extended focused imaging of a microparticle field with digital holographic microscopy," Opt. Lett. 33, 1626-1628 (2008).
[CrossRef] [PubMed]

C. Minetti, N. Callens, G. Coupier, T. Podgorski, and F. Dubois, "Fast measurements of concentration profiles inside deformable objects in microflows with spatial coherence digital holography," Appl. Opt. 45, 5305-5314 (2008).
[CrossRef]

F. Dubois, C. Schockaert, N. Callens, and C. Yourassowski, "Focus plane detection criteria in digital holography microscopy by amplitude analysis," Opt. Express 14, 5895-5908 (2006).
[CrossRef] [PubMed]

F. Dubois, M.-L. Novella Requena, C. Minetti, O. Monnom, and E. Istasse, "Partial spatial coherence effects in digital holographic microscopy with a laser source," Appl. Opt. 43, 1131-1139 (2004).
[CrossRef] [PubMed]

F. Dubois, O. Monnom, C. Yourassowski, and J.-C. Legros, "Pattern recognition with digital holographic microscope working in partially coherent illumination," Appl. Opt. 41, 4108-4119 (2002).
[CrossRef] [PubMed]

F. Dubois, O. Monnom, C. Yourassowski, and J.-C. Legros, "Border processing in digital holography by extension of the digital hologram and reduction of the higher spatial frequencies," Appl. Opt. 41, 2621-2626 (2002).
[CrossRef] [PubMed]

F. Dubois, L. Johannes, and J.-C. Legros, "Improved three-dimensional imaging with a digital holography microscope with a source of partial spatial coherence," Appl. Opt. 38, 7085-7094 (1999).
[CrossRef]

Feld, M. S.

Ferraro, P.

Finizio, A.

Garcia-Sucerquia, J.

Genc, S.

L. Yu, S. Mohanty, J. Zhang, S. Genc, M. K. Kim, M. W. Berns, and Z. Chen, "Digital holographic microscopy for quantitative cell dynamic evaluation during laser microsurgery," Appl. Opt. 17, 12031-12038 (2009).

Gillespie, J.

J. Gillespie, and R. A. King, "The use of self-entropy as a focus measure in digital holography," Pattern Recognit. Lett. 9, 19-25 (1989).
[CrossRef]

Gross, M.

Gustafsson, M.

Hennelly, B.

Hu, Q.

Istasse, E.

Iwai, H.

Javidi, B.

Jericho, M. H.

Jericho, S. K.

Jin, H.

L. Ma, H. Wang, Y. Li, and H. Jin, "Numerical reconstruction of digital holograms for three-dimensional shape measurement," J. Opt. A 6, 396-400 (2004).
[CrossRef]

Johannes, L.

Joud, F.

Jptner, W.

U. Schnars, and W. Jptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, 85-101 (2002).
[CrossRef]

U. Schnars, and W. Jptner, "Direct recording of holograms by a CCD target and numerical reconstruction," Appl. Opt. 33, 179-181 (1994).
[CrossRef] [PubMed]

Kang, B. S.

Y. Yang, B. S. Kang, and Y. J. Choo, "Application of the correlation coefficient method for determination of the focus plane to digital particle holography," Appl. Opt. 47, 817-824 (2008).
[CrossRef] [PubMed]

Y. J. Choo, and B. S. Kang, "The characteristics of the particle position along an optical axis in particle holography," Meas. Sci. Technol. 17, 761-770 (2006).
[CrossRef]

Kemper, B.

P. Langehanenberg, B. Kemper, D. Dirksen, and G. von Bally, "Autofocusing in digital holographic phase contrast microscopy on pure phase objects for live cell imaging," Appl. Opt. 47, 176-182 (2008).
[CrossRef]

D. Carl, B. Kemper, G. Wernicke, and G. von Bally, "Parameter-optimized digital holographic microscope for high-resolution living-cell analysis," Appl. Opt. 43, 6536-6544 (2004).
[CrossRef]

Khn, J.

Kim, M. K.

L. Yu, S. Mohanty, J. Zhang, S. Genc, M. K. Kim, M. W. Berns, and Z. Chen, "Digital holographic microscopy for quantitative cell dynamic evaluation during laser microsurgery," Appl. Opt. 17, 12031-12038 (2009).

King, R. A.

J. Gillespie, and R. A. King, "The use of self-entropy as a focus measure in digital holography," Pattern Recognit. Lett. 9, 19-25 (1989).
[CrossRef]

Klages, P.

Kreuzer, H. J.

Kuehn, J.

Langehanenberg, P.

P. Langehanenberg, B. Kemper, D. Dirksen, and G. von Bally, "Autofocusing in digital holographic phase contrast microscopy on pure phase objects for live cell imaging," Appl. Opt. 47, 176-182 (2008).
[CrossRef]

Lebrun, D.

Legros, J.-C.

Li, W.

Li, Y.

L. Ma, H. Wang, Y. Li, and H. Jin, "Numerical reconstruction of digital holograms for three-dimensional shape measurement," J. Opt. A 6, 396-400 (2004).
[CrossRef]

Liebling, M.

Loomis, N. C.

Ma, L.

L. Ma, H. Wang, Y. Li, and H. Jin, "Numerical reconstruction of digital holograms for three-dimensional shape measurement," J. Opt. A 6, 396-400 (2004).
[CrossRef]

Malek, M.

Marian, A.

Marquet, P.

Minetti, C.

C. Minetti, N. Callens, G. Coupier, T. Podgorski, and F. Dubois, "Fast measurements of concentration profiles inside deformable objects in microflows with spatial coherence digital holography," Appl. Opt. 45, 5305-5314 (2008).
[CrossRef]

F. Dubois, M.-L. Novella Requena, C. Minetti, O. Monnom, and E. Istasse, "Partial spatial coherence effects in digital holographic microscopy with a laser source," Appl. Opt. 43, 1131-1139 (2004).
[CrossRef] [PubMed]

Mohanty, S.

L. Yu, S. Mohanty, J. Zhang, S. Genc, M. K. Kim, M. W. Berns, and Z. Chen, "Digital holographic microscopy for quantitative cell dynamic evaluation during laser microsurgery," Appl. Opt. 17, 12031-12038 (2009).

Monfort, F.

Monnom, O.

Montford, F.

Moon, I.

Nazarathy, M.

Novella Requena, M.-L.

Pandey, N.

Pierattini, G.

Podgorski, T.

C. Minetti, N. Callens, G. Coupier, T. Podgorski, and F. Dubois, "Fast measurements of concentration profiles inside deformable objects in microflows with spatial coherence digital holography," Appl. Opt. 45, 5305-5314 (2008).
[CrossRef]

Popescu, G.

Schnars, U.

U. Schnars, and W. Jptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, 85-101 (2002).
[CrossRef]

U. Schnars, and W. Jptner, "Direct recording of holograms by a CCD target and numerical reconstruction," Appl. Opt. 33, 179-181 (1994).
[CrossRef] [PubMed]

Schockaert, C.

Sebesta, M.

Seesta, M.

Shamir, J.

Tessier, G.

Unser, M.

Vaughan, J. C.

von Bally, G.

P. Langehanenberg, B. Kemper, D. Dirksen, and G. von Bally, "Autofocusing in digital holographic phase contrast microscopy on pure phase objects for live cell imaging," Appl. Opt. 47, 176-182 (2008).
[CrossRef]

D. Carl, B. Kemper, G. Wernicke, and G. von Bally, "Parameter-optimized digital holographic microscope for high-resolution living-cell analysis," Appl. Opt. 43, 6536-6544 (2004).
[CrossRef]

Wang, H.

L. Ma, H. Wang, Y. Li, and H. Jin, "Numerical reconstruction of digital holograms for three-dimensional shape measurement," J. Opt. A 6, 396-400 (2004).
[CrossRef]

Warnasooriya, N.

Wernicke, G.

Xu, W.

Yamaguchi, I.

Yang, Y.

Yeom, S.

Yourassowski, C.

Yu, L.

L. Yu, S. Mohanty, J. Zhang, S. Genc, M. K. Kim, M. W. Berns, and Z. Chen, "Digital holographic microscopy for quantitative cell dynamic evaluation during laser microsurgery," Appl. Opt. 17, 12031-12038 (2009).

L. Yu, and L. Cai, "Iterative algorithm with a constraint condition for numerical reconstruction of a three-dimensional object from its hologram," J. Opt. Soc. Am. A 18, 1033-1045 (2001).
[CrossRef]

Zhang, J.

L. Yu, S. Mohanty, J. Zhang, S. Genc, M. K. Kim, M. W. Berns, and Z. Chen, "Digital holographic microscopy for quantitative cell dynamic evaluation during laser microsurgery," Appl. Opt. 17, 12031-12038 (2009).

Zhang, T.

Appl. Opt. (13)

L. Yu, S. Mohanty, J. Zhang, S. Genc, M. K. Kim, M. W. Berns, and Z. Chen, "Digital holographic microscopy for quantitative cell dynamic evaluation during laser microsurgery," Appl. Opt. 17, 12031-12038 (2009).

C. Minetti, N. Callens, G. Coupier, T. Podgorski, and F. Dubois, "Fast measurements of concentration profiles inside deformable objects in microflows with spatial coherence digital holography," Appl. Opt. 45, 5305-5314 (2008).
[CrossRef]

U. Schnars, and W. Jptner, "Direct recording of holograms by a CCD target and numerical reconstruction," Appl. Opt. 33, 179-181 (1994).
[CrossRef] [PubMed]

F. Dubois, L. Johannes, and J.-C. Legros, "Improved three-dimensional imaging with a digital holography microscope with a source of partial spatial coherence," Appl. Opt. 38, 7085-7094 (1999).
[CrossRef]

F. Dubois, O. Monnom, C. Yourassowski, and J.-C. Legros, "Border processing in digital holography by extension of the digital hologram and reduction of the higher spatial frequencies," Appl. Opt. 41, 2621-2626 (2002).
[CrossRef] [PubMed]

F. Dubois, O. Monnom, C. Yourassowski, and J.-C. Legros, "Pattern recognition with digital holographic microscope working in partially coherent illumination," Appl. Opt. 41, 4108-4119 (2002).
[CrossRef] [PubMed]

P. Langehanenberg, B. Kemper, D. Dirksen, and G. von Bally, "Autofocusing in digital holographic phase contrast microscopy on pure phase objects for live cell imaging," Appl. Opt. 47, 176-182 (2008).
[CrossRef]

F. Dubois, M.-L. Novella Requena, C. Minetti, O. Monnom, and E. Istasse, "Partial spatial coherence effects in digital holographic microscopy with a laser source," Appl. Opt. 43, 1131-1139 (2004).
[CrossRef] [PubMed]

M. Gustafsson, and M. Sebesta, "Refractometry of Microscopic Objects with Digital Holography," Appl. Opt. 43, 4796-4801 (2005).
[CrossRef]

D. Carl, B. Kemper, G. Wernicke, and G. von Bally, "Parameter-optimized digital holographic microscope for high-resolution living-cell analysis," Appl. Opt. 43, 6536-6544 (2004).
[CrossRef]

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, 836-850 (2006).
[CrossRef] [PubMed]

T. Colomb, E. Cuche, F. Charrire, J. Khn, N. Aspert, F. Monfort, P. Marquet, and C. Despeursinge, "Automatic procedure for aberration compensation in digital holographic microscopy and applications to specimen shape compensation," Appl. Opt. 45, 851-863 (2006).
[CrossRef] [PubMed]

Y. Yang, B. S. Kang, and Y. J. Choo, "Application of the correlation coefficient method for determination of the focus plane to digital particle holography," Appl. Opt. 47, 817-824 (2008).
[CrossRef] [PubMed]

J. Opt. A (1)

L. Ma, H. Wang, Y. Li, and H. Jin, "Numerical reconstruction of digital holograms for three-dimensional shape measurement," J. Opt. A 6, 396-400 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Meas. Sci. Technol. (2)

U. Schnars, and W. Jptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, 85-101 (2002).
[CrossRef]

Y. J. Choo, and B. S. Kang, "The characteristics of the particle position along an optical axis in particle holography," Meas. Sci. Technol. 17, 761-770 (2006).
[CrossRef]

Opt. Express (4)

Opt. Lett. (9)

M. Antkowiak, N. Callens, C. Yourassowski, and F. Dubois, "Extended focused imaging of a microparticle field with digital holographic microscopy," Opt. Lett. 33, 1626-1628 (2008).
[CrossRef] [PubMed]

N. Pandey, and B. Hennelly, "Fixed-point numerical-reconstruction for digital holographic microscopy," Opt. Lett. 35, 1076-1078 (2010).
[CrossRef] [PubMed]

F. Charrire, A. Marian, F. Montford, J. Kuehn, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, "Cell refractive index tomography by digital holographic microscopy," Opt. Lett. 31, 178-180 (2006).
[CrossRef]

M. Seesta, and M. Gustafsson, "Object characterization with refractometric digital Fourier holography," Opt. Lett. 30, 471-473 (2005).
[CrossRef]

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[CrossRef]

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Figures (15)

Fig. 1
Fig. 1

(a) Simulated particle of 5μm, (b) filtered by a numerical aperture of 0.3 and (c) defocused to −50μm. Sampling step : 1μm = 10 pixels. Image size: 256 × 256 pixels.

Fig. 2
Fig. 2

Evolution of the refocusing criterion as a function of ɛ (log-log) for NA=0.05, 0.1 and 0.15, showing the ɛ2-dependency.

Fig. 3
Fig. 3

Evolution of the refocusing criterion as a function of ɛ (log-log) for NA=0.2, 0.3, 0.4 and 0.5.

Fig. 4
Fig. 4

Evolution of the refocusing criterion as a function of ɛ for different values of NA.

Fig. 5
Fig. 5

(a) Intensity image of a particle of 220μm in the recorded plane (defocused by a distance of approximately −80 μm). (b) Reconstructed intensity image at a distance of 79 μm determined by the minimum of the refocusing criterion.

Fig. 6
Fig. 6

Experimental evolution of the refocusing criterion as a function of ɛ for NA = 0.10, showing the parabolic shape of the criterion and the minimum centered on the focus plane.

Fig. 7
Fig. 7

Experimental evolution of the refocusing criterion as a function of ɛ (log-log) for NA=0.10, showing the ɛ2-dependency.

Fig. 8
Fig. 8

Evolution of the refocusing criterion as a function of NA for ɛ = 0.1, 0.3 and 0.5μm.

Fig. 9
Fig. 9

Evolution of the refocusing criterion as a function of NA for ɛ = 1, 2 and 5μm.

Fig. 10
Fig. 10

Evolution of the refocusing criterion as a function of ɛ for NA = 0.3 and for different levels of salt and pepper noise.

Fig. 11
Fig. 11

Evolution of the deviation from the best focus plane as a function of level noise added for NA = 0.3 (and associated standard deviation - error bars), showing the robustness of the refocusing criterion up to 35% of noise added - A zoom is performed to show more clearly the zone corresponding to the 5–20% level of noise added.

Fig. 12
Fig. 12

(a) Illustration of 10%, (b) 20% and (c) 30% of noise added on a 5μm particle filtered by a numerical aperture of 0.3 and defocused to −50μm.

Fig. 13
Fig. 13

(a) Filtered noise by a numerical aperture of NA = 0.60 (level of speckle noise = 24%). (b) Particle of 5μm filtered by a numerical aperture of 0.60. (c) Corresponding noisy particle. (d) Defocused by a distance of −50μm.

Fig. 14
Fig. 14

Evolution of the refocusing criterion as a function of ɛ for NA = 0.60 and for different levels of speckle noise.

Fig. 15
Fig. 15

Evolution of the deviation from the best focus plane as a function of speckle noise level for NA = 0.60 (and associated standard deviation - error bars), showing the robustness of the refocusing criterion up to 35% of noise added - A zoom is performed to show more clearly the zone corresponding to the 12 – 36% level of speckle noise.

Tables (2)

Tables Icon

Table 1 Average Deviations from the Focus Plane (μ) for Different Values of Numerical Aperture and Different Levels of Salt and Pepper Noisea

Tables Icon

Table 2 Average Deviations from the Focus Plane (μ) for Different Values of Numerical Aperture and Different Levels of Speckle Noisea

Equations (10)

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t ɛ ( x , y ) = R [ ɛ ] t ( x , y ) = exp ( j k ɛ ) F 1 Q [ λ 2 ɛ ] F t ( x , y ) = exp ( j k ɛ ) F 1 exp ( j k λ 2 ɛ 2 ( ν x 2 + ν y 2 ) ) F t ( x , y )
t ɛ ( x , y ) exp ( j k ɛ ) ( t j k λ 2 ɛ 2 F 1 ( ν x 2 + ν y 2 ) F t ( x , y ) t )
| t ɛ ( x , y ) | d x d y t 1 + π 2 λ 2 ɛ 2 t 2 t 2 d x d y t ( 1 + π 2 λ 2 ɛ 2 2 t 2 t 2 d x d y )
| t ɛ ( x , y ) | d x d y = | t ( x , y ) | d x d y + π 2 λ 2 ɛ 2 2 | t | 2 t d x d y
| t ɛ ( x , y ) | d x d y | t ( x , y ) | d x d y π 2 λ 2 ɛ 3 2 t M | t | 2 d x d y
I = | T | 2 d ν x d ν y = ( ν x 2 + ν y 2 ) 2 | F t ( x , y ) | 2 d ν x d ν y
t f ( x , y ) = A F p ( λ f ν x , λ f ν y ) T ( ν x , ν y )
t f ( x , y ) c F p ( λ f ν x , λ f ν y )
I c ( ν x 2 + ν y 2 ) 2 { p ( λ f ν x , λ f ν y ) } 2 d ν x d ν y
I 2 c π 0 NA / 2 λ r 5 d r NA 6

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