Abstract

We propose and test a focus plane determination method that computes the digital refocus distance of an object investigated by digital holographic microscopy working in transmission. For this purpose we analyze the integrated amplitude modulus as a function of the digital holographic reconstruction distance. It is shown that when the focus distance is reached, the integrated amplitude is minimum for pure amplitude object and maximum for pure phase object. After a theoretical analysis, the method is demonstrated on actual digital holograms for the refocusing of pure amplitude and of pure phase microscopic samples.

© 2006 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  4. F. Dubois, L. Joannes, and J.-C. Legros, "Improved three-dimensional imaging with digital holography microscope using a partial spatial coherent source," Appl. Opt. 38, 7085-7094 (1999).
    [CrossRef]
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  6. M. Sebesta and M. Gustafsson, "Object characterization with refractometric digital Fourier holography," Opt. Lett. 30, 471-473 (2005).
    [CrossRef] [PubMed]
  7. E. Malkiel, O. Alquaddoomi and J. Katz, "Measurements of plankton distribution in the ocean using submersible holography," Meas. Sci. Technol. 10, 1142-1152 (1999).
    [CrossRef]
  8. 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]
  9. 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).
    [CrossRef] [PubMed]
  10. T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, "Hilbert phase microscopy for investigating fast dynamics in transparent systems," Opt. Lett. 30, 1165-1167 (2005).
    [CrossRef] [PubMed]
  11. L. Martínez-León, G. Pedrini, and W. Osten, "Applications of short-coherence digital holography in microscopy," Appl. Opt. 44, 3977-3984 (2005).
    [CrossRef] [PubMed]
  12. D. Dirksena, H. Drostea, B. Kempera, H. Delerlea, M. Deiwickb, H. H. Scheldb, and G. von Bally, "Lensless fourier holography for digital holographic interferometry on biological samples," Opt. Lasers Eng. 36, 241-249 (2001).
    [CrossRef]
  13. P. Marquet, B. Rappaz, and P. J. Magistretti, E. Cuche and Y. Emery, T. Colomb and C. Depeursinge, "Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy," Opt. Lett. 30, 468-470 (2005).
    [CrossRef] [PubMed]
  14. F. Charrière, A. Marian, F. Montfort, J. Kuehn, and T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, "Cell refractive index tomography by digital holographic microscopy," Opt. Lett. 31, 178-180 (2006).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  22. F. Dubois, C. Minetti, O. Monnom, C. Yourassowsky, and J.-C. Legros, "Pattern recognition with digital holographic microscope working in partially coherent illumination," Appl. Opt. 41, 4108-4119 (2002).
    [CrossRef] [PubMed]
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    [CrossRef]
  24. F. Dubois, O. Monnom, C. Yourassowsky 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]
  25. S.-G. Kim, B. Lee, and E.-S. Kim, "Removal of bias and the conjugate image in incoherent on-axis triangular holography and real-time reconstruction of the complex hologram," Appl. Opt. 36, 4784-4791 (1997).
    [CrossRef]
  26. T.-C. Poon, T. Kim, G. Indebetouw, M. H. Wu, K. Shinoda, and Y. Suzuki, "Twin-image elimination experiments for three-dimensional images in optical scanning Holography," Opt. Lett. 25, 215-217 (2000).
    [CrossRef]
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    [CrossRef] [PubMed]
  28. T. Colomb, E. Cuche, F. Charrière, J. Kühn, N. Aspert, F. Monfort, P. Marquet and C. Depeursinge, "Automatic procedure for aberration compensation in digital holographic microscopy and applications to specimen shape compensation," Appl. Opt. 45, 851-863 (2006).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  31. L. Ma, H. Wang, Y. Li and H. Jin, "Numerical reconstruction of digital holograms for three-dimensional shape measurement," J. Opt. A: Pure Appl. Opt. 6, 396-400 (2004).
    [CrossRef]
  32. P. Ferraro and 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]
  33. M. Liebling and M. Unser, "Autofocus for digital Fresnel Holograms by use of a Fresnelet-Sparsity Criterion," J. Opt. Soc. Am. 21, 2424-2430 (2004).
    [CrossRef]
  34. 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]
  35. F. Dubois, N. Callens, C. Yourassowsky, M. Hoyos, P. Kurowski, and O. Monnom, "Digital holographic microscopy with reduced spatial coherence for three-dimensional particle flow analysis," Appl. Opt. 45, 864-871 (2006).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  38. C. J. Mann, L. Yu, and M. K. Kim, "Movies of cellular and sub-cellular motion by digital holographic microscopy," Biomed. Eng. Online 5:21 (2006)
    [CrossRef] [PubMed]

2006

2005

2004

2003

2002

2001

2000

1999

1998

1997

1989

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

1986

Alfieri, D.

Alquaddoomi, O.

E. Malkiel, O. Alquaddoomi and J. Katz, "Measurements of plankton distribution in the ocean using submersible holography," Meas. Sci. Technol. 10, 1142-1152 (1999).
[CrossRef]

Aspert, N.

Badizadegan, K.

Benkouider, A.

D. Lebrun, A. Benkouider, S. Coëtmellec, and M. Malek, "Particle field digital holographic reconstruction in arbitrary tilted planes," Opt. Express. 11, 224-229 (2003).
[CrossRef] [PubMed]

Bevilacqua, F.

Cai, L.

Callens, N.

Carl, D.

Charrière, F.

Coëtmellec, S.

D. Lebrun, A. Benkouider, S. Coëtmellec, and M. Malek, "Particle field digital holographic reconstruction in arbitrary tilted planes," Opt. Express. 11, 224-229 (2003).
[CrossRef] [PubMed]

Colomb, T.

Coppola, G.

Cuche, E.

Dasari, R. R.

De Nicola, S.

Deflores, L. P.

Deiwickb, M.

D. Dirksena, H. Drostea, B. Kempera, H. Delerlea, M. Deiwickb, H. H. Scheldb, and G. von Bally, "Lensless fourier holography for digital holographic interferometry on biological samples," Opt. Lasers Eng. 36, 241-249 (2001).
[CrossRef]

Delerlea, H.

D. Dirksena, H. Drostea, B. Kempera, H. Delerlea, M. Deiwickb, H. H. Scheldb, and G. von Bally, "Lensless fourier holography for digital holographic interferometry on biological samples," Opt. Lasers Eng. 36, 241-249 (2001).
[CrossRef]

Depeursinge, C.

Despeuringe, C.

E. Cuche, P. Marquet and C. Despeuringe, "Aperture Apodization using cubic spline interpolation: application in digital holography microscopy," Opt. Commun. 182, 59-69 (2000).
[CrossRef]

Dirksena, D.

D. Dirksena, H. Drostea, B. Kempera, H. Delerlea, M. Deiwickb, H. H. Scheldb, and G. von Bally, "Lensless fourier holography for digital holographic interferometry on biological samples," Opt. Lasers Eng. 36, 241-249 (2001).
[CrossRef]

Drostea, H.

D. Dirksena, H. Drostea, B. Kempera, H. Delerlea, M. Deiwickb, H. H. Scheldb, and G. von Bally, "Lensless fourier holography for digital holographic interferometry on biological samples," Opt. Lasers Eng. 36, 241-249 (2001).
[CrossRef]

Dubois, F.

Emery, Y.

Feld, M. S.

Ferraro, P.

Finizio, A.

Garcia-Sucerquia, J.

Gillespie, J.

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

Grilli, S.

Gustafsson, M.

Hoyos, M.

Ikeda, T.

Indebetouw, G.

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: Pure Appl. Opt. 6, 396-400 (2004).
[CrossRef]

Joannes, L.

Kato, J.-I.

Katz, J.

E. Malkiel, O. Alquaddoomi and J. Katz, "Measurements of plankton distribution in the ocean using submersible holography," Meas. Sci. Technol. 10, 1142-1152 (1999).
[CrossRef]

Kemper, B.

Kempera, B.

D. Dirksena, H. Drostea, B. Kempera, H. Delerlea, M. Deiwickb, H. H. Scheldb, and G. von Bally, "Lensless fourier holography for digital holographic interferometry on biological samples," Opt. Lasers Eng. 36, 241-249 (2001).
[CrossRef]

Kim, E.-S.

Kim, M. K.

C. J. Mann, L. Yu, and M. K. Kim, "Movies of cellular and sub-cellular motion by digital holographic microscopy," Biomed. Eng. Online 5:21 (2006)
[CrossRef] [PubMed]

C. J. Mann, L. Yu, C.-M. Lo, and M. K. Kim, "High-resolution quantitative phase-contrast microscopy by digital holography," Opt. Express 13, 8693-8697 (2005).
[CrossRef] [PubMed]

Kim, S.-G.

Kim, T.

King, R. A.

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

Klages, P.

Klysubun, P.

Kreis, T.

Kreuzer, H. J.

Kuehn, J.

Kühn, J.

Kurowski, P.

Lebrun, D.

D. Lebrun, A. Benkouider, S. Coëtmellec, and M. Malek, "Particle field digital holographic reconstruction in arbitrary tilted planes," Opt. Express. 11, 224-229 (2003).
[CrossRef] [PubMed]

Lee, B.

Legros, J.-C.

Li, Y.

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

Liebling, M.

M. Liebling and M. Unser, "Autofocus for digital Fresnel Holograms by use of a Fresnelet-Sparsity Criterion," J. Opt. Soc. Am. 21, 2424-2430 (2004).
[CrossRef]

Lo, C.-M.

Ma, L.

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

Magistretti, P. J.

Magro, C.

Malek, M.

D. Lebrun, A. Benkouider, S. Coëtmellec, and M. Malek, "Particle field digital holographic reconstruction in arbitrary tilted planes," Opt. Express. 11, 224-229 (2003).
[CrossRef] [PubMed]

Malkiel, E.

E. Malkiel, O. Alquaddoomi and J. Katz, "Measurements of plankton distribution in the ocean using submersible holography," Meas. Sci. Technol. 10, 1142-1152 (1999).
[CrossRef]

Mann, C. J.

C. J. Mann, L. Yu, and M. K. Kim, "Movies of cellular and sub-cellular motion by digital holographic microscopy," Biomed. Eng. Online 5:21 (2006)
[CrossRef] [PubMed]

C. J. Mann, L. Yu, C.-M. Lo, and M. K. Kim, "High-resolution quantitative phase-contrast microscopy by digital holography," Opt. Express 13, 8693-8697 (2005).
[CrossRef] [PubMed]

Marian, A.

Marquet, P.

Martínez-León, L.

Minetti, C.

Mizuno, J.

Monfort, F.

Monnom, O.

Montfort, F.

Novella Requena, M.-L.

Osten, W.

Otha, S.

Pedrini, G.

Pierattini, G.

Poon, T.-C.

Popescu, G.

Rappaz, B.

Scheldb, H. H.

D. Dirksena, H. Drostea, B. Kempera, H. Delerlea, M. Deiwickb, H. H. Scheldb, and G. von Bally, "Lensless fourier holography for digital holographic interferometry on biological samples," Opt. Lasers Eng. 36, 241-249 (2001).
[CrossRef]

Sebesta, M.

Shinoda, K.

Suzuki, Y.

Tajahuerce, E.

Unser, M.

M. Liebling and M. Unser, "Autofocus for digital Fresnel Holograms by use of a Fresnelet-Sparsity Criterion," J. Opt. Soc. Am. 21, 2424-2430 (2004).
[CrossRef]

Vaughan, J. C.

von Bally, G.

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]

D. Dirksena, H. Drostea, B. Kempera, H. Delerlea, M. Deiwickb, H. H. Scheldb, and G. von Bally, "Lensless fourier holography for digital holographic interferometry on biological samples," Opt. Lasers Eng. 36, 241-249 (2001).
[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: Pure Appl. Opt. 6, 396-400 (2004).
[CrossRef]

Wernicke, G.

Wu, M. H.

Xu, W.

Yamaguchi, I.

Yourassowsky, C.

Yu, L.

Zhang, T.

Appl. Opt.

I. Yamaguchi, J.-I. Kato, S. Otha and J. Mizuno, "Image formation in phase-shifting digital holography and applications to microscopy," Appl. Opt. 40, 6177-6186 (2001).
[CrossRef]

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

L. Martínez-León, G. Pedrini, and W. Osten, "Applications of short-coherence digital holography in microscopy," Appl. Opt. 44, 3977-3984 (2005).
[CrossRef] [PubMed]

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]

T.-C. Poon and T. Kim, "Optical image recognition of three-dimensional objects," Appl. Opt. 38, 370-381 (1999).
[CrossRef]

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

P. Ferraro, S. De Nicola, A. Finizio, G. Coppola, S. Grilli, C. Magro, and G. Pierattini, "Compensation of the inherent wave front curvature in digital holographic coherent microscopy for quantitative phase-contrast imaging," Appl. Opt. 42, 1938-1946 (2003).
[CrossRef] [PubMed]

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

F. Dubois, O. Monnom, C. Yourassowsky 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]

S.-G. Kim, B. Lee, and E.-S. Kim, "Removal of bias and the conjugate image in incoherent on-axis triangular holography and real-time reconstruction of the complex hologram," Appl. Opt. 36, 4784-4791 (1997).
[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]

F. Dubois, N. Callens, C. Yourassowsky, M. Hoyos, P. Kurowski, and O. Monnom, "Digital holographic microscopy with reduced spatial coherence for three-dimensional particle flow analysis," Appl. Opt. 45, 864-871 (2006).
[CrossRef] [PubMed]

Biomed. Eng. Online

C. J. Mann, L. Yu, and M. K. Kim, "Movies of cellular and sub-cellular motion by digital holographic microscopy," Biomed. Eng. Online 5:21 (2006)
[CrossRef] [PubMed]

J. Opt. A: Pure Appl. Opt.

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

J. Opt. Soc. Am.

M. Liebling and M. Unser, "Autofocus for digital Fresnel Holograms by use of a Fresnelet-Sparsity Criterion," J. Opt. Soc. Am. 21, 2424-2430 (2004).
[CrossRef]

J. Opt. Soc. Am. A

Meas. Sci. Technol.

E. Malkiel, O. Alquaddoomi and J. Katz, "Measurements of plankton distribution in the ocean using submersible holography," Meas. Sci. Technol. 10, 1142-1152 (1999).
[CrossRef]

Opt. Commun.

E. Cuche, P. Marquet and C. Despeuringe, "Aperture Apodization using cubic spline interpolation: application in digital holography microscopy," Opt. Commun. 182, 59-69 (2000).
[CrossRef]

Opt. Express

Opt. Express.

D. Lebrun, A. Benkouider, S. Coëtmellec, and M. Malek, "Particle field digital holographic reconstruction in arbitrary tilted planes," Opt. Express. 11, 224-229 (2003).
[CrossRef] [PubMed]

Opt. Lasers Eng.

D. Dirksena, H. Drostea, B. Kempera, H. Delerlea, M. Deiwickb, H. H. Scheldb, and G. von Bally, "Lensless fourier holography for digital holographic interferometry on biological samples," Opt. Lasers Eng. 36, 241-249 (2001).
[CrossRef]

Opt. Lett.

P. Marquet, B. Rappaz, and P. J. Magistretti, E. Cuche and Y. Emery, T. Colomb and C. Depeursinge, "Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy," Opt. Lett. 30, 468-470 (2005).
[CrossRef] [PubMed]

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

E. Cuche, F. Bevilacqua and C. Depeursinge, "Digital holography for quantitative phase contrast imaging," Opt. Lett. 24, 291-293 (1999).
[CrossRef]

P. Ferraro, S. De Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, "Controlling image size as a function of distance and wavelength in Fresnel-transform reconstruction of digital holograms," Opt. Lett. 29, 854-856 (2004).
[CrossRef] [PubMed]

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

T. Zhang and I. Yamaguchi, "Three-dimensional microscopy with phase-shifting digital holography," Opt. Lett. 23, 1221-1223 (1998).
[CrossRef]

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).
[CrossRef] [PubMed]

T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, "Hilbert phase microscopy for investigating fast dynamics in transparent systems," Opt. Lett. 30, 1165-1167 (2005).
[CrossRef] [PubMed]

P. Ferraro and 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]

T.-C. Poon, T. Kim, G. Indebetouw, M. H. Wu, K. Shinoda, and Y. Suzuki, "Twin-image elimination experiments for three-dimensional images in optical scanning Holography," Opt. Lett. 25, 215-217 (2000).
[CrossRef]

B. Javidi and E. Tajahuerce, "Three-dimensional object recognition by use of digital holography," Opt. Lett. 25, 610-612 (2000).
[CrossRef]

Pattern Recogn. Lett.

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

Other

F. Dubois, C. Yourassowsky, and O. Monnom, "Microscopie en holographie digitale avec une source partiellement cohérente," in Imagerie et Photonique pour les sciences du vivant et la médecine, M. Faupel, P. Smigielski and R. Grzymala, eds. (Fontis Media, 2004).

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Figures (9)

Fig. 1.
Fig. 1.

(a) A pure real valued amplitude object. It represents a back-illuminated aperture by constant amplitude field; (b) The same object that is defocused by the digital holographic propagation; (c) Horizontal amplitude profile across the middle of (a); (d) Horizontal amplitude profile across the middle of (b).

Fig. 3.
Fig. 3.

Intensity image of the ruler in the recorded plane.

Fig. 4.
Fig. 4.

Reconstructed intensity image at a distance of 66 µm according to the minimization of the amplitude focus criterion. A border processing to avoid the diffraction effects is applied.

Fig. 5.
Fig. 5.

Evolution of the focus criterion that is minimal for the plane at the position z=-66µm.

Fig. 6.
Fig. 6.

Defocus intensity image of 5 µm particle in the recorded plane. ROI of 209×201 pixels.

Fig. 7.
Fig. 7.

Reconstructed intensity image of the ROI of Fig. 6 with d=55 µm computed by the focus criterion.

Fig. 8.
Fig. 8.

Focus criterion as a function of the reconstruction distance. It is minimal at the distance d=55 µm.

Fig. 9.
Fig. 9.

(a) Intensity image of a cell at a defocusing distance of -30 µm. The ROI size is 111×153 pixels; (b) Phase image corresponding to the image of (a); (c) Intensity image corresponding to (a) reconstructed at a distance d=30 µm. We observe that, in intensity, the object is almost invisible due to the fact that it is a phase object; (d) Phase image corresponding to the image of (c).

Fig. 10.
Fig. 10.

Md as a function of the refocusing distance.

Equations (21)

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v d ( x ' , y ' ) = exp ( j k d ) F ( C ) x ' , y ' 1 exp ( j k d λ 2 2 ( ν x 2 + ν y 2 ) ) F ( C ) ν x , ν y + 1 u 0 ( x , y )
F ( C ) η , ξ ± 1 g ( α , β ) = exp { 2 j π ( α η + β ξ ) } g ( α , β ) d α d β
u d ( x ' , y ' ) = F ( C ) x ' , y ' 1 exp ( j k d λ 2 2 ( ν x 2 + ν y 2 ) ) F ( C ) ν x , ν y + 1 u 0 ( x , y )
U d ( ν x , ν y ) 2 = U 0 ( ν x , ν y ) 2
u d ( x ' , y ' ) 2 d x ' d y ' = u 0 ( x , y ) 2 d x d y = E
( u 0 2 + A 2 + A * u 0 + A u 0 * ) d x ' d y ' = ( u d 2 + A 2 + A * u d + A u d * ) d x d y
( A * u 0 + A u 0 * ) d x ' d y ' = ( A * u d + A u d * ) d x d y
u d ( x ' , y ' ) d x ' d y ' = u 0 ( x , y ) d x d y = B
B u d ( x , y ) d x d y = M d
u d ( x , y ) d x d y is minimum
u d ( x , y ) = t ( x , y ) C
u d ( x , y ) d x d y = u d ( x , y ) d x d y = u d ( x , y ) d x d y
u d ( x , y ) d x d y = u d ' ( x ' , y ' ) d x ' d y '
u d ' ( x ' , y ' ) d x ' d y ' u d ' ( x ' , y ' ) d x ' d y '
u d ( x , y ) d x d y u d ' ( x ' , y ' ) d x ' d y '
u d ( x , y ) = a exp { j φ ( x , y ) }
u d ( x , y ) d x d y is maximum
u s = a s exp { j φ s }
E = s = 0 N 1 a s 2 = N a 2
M d = s = 0 N 1 a s
f ( a s ) = s = 0 N 1 a s + Γ s = 0 N 1 a s 2

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