Abstract

We address the problem of exact complex-wave reconstruction in digital holography. We show that, by confining the object-wave modulation to one quadrant of the frequency domain, and by maintaining a reference-wave intensity higher than that of the object, one can achieve exact complex-wave reconstruction in the absence of noise. A feature of the proposed technique is that the zero-order artifact, which is commonly encountered in hologram reconstruction, can be completely suppressed in the absence of noise. The technique is noniterative and nonlinear. We also establish a connection between the reconstruction technique and homomorphic signal processing, which enables an interpretation of the technique from the perspective of deconvolution. Another key contribution of this paper is a direct link between the reconstruction technique and the two-dimensional Hilbert transform formalism proposed by Hahn. We show that this connection leads to explicit Hilbert transform relations between the magnitude and phase of the complex wave encoded in the hologram. We also provide results on simulated as well as experimental data to validate the accuracy of the reconstruction technique.

© 2011 Optical Society of America

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References

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    [CrossRef] [PubMed]
  5. P. Marquet, B. Rappaz, P. Magistretti, E. Cuche, 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]
  6. 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]
  8. J. Kühn, F. Charriére, T. Colomb, E. Cuche, F. Montfort, Y. Emery, P. Marquet, and C. Depeursinge, “Axial sub-nanometer accuracy in digital holographic microscopy,” Meas. Sci. Technol. 19, 074007 (2008).
    [CrossRef]
  9. F. Montfort, Y. Emery, E. Solanas, E. Cuche, N. Aspert, P. Marquet, C. Joris, J. Kühn, and C. Depeursinge, “Surface roughness parameters measurements by Digital Holographic Microscopy (DHM),” in Third International Symposium on Precision Mechanical Measurements (SPIE, 2006), Vol.  6280 I, pp. 62800V-1–62800V-6.
  10. F. Montfort, Y. Emery, F. Marquet, E. Cuche, N. Aspert, E. Solanas, A. Mehdaoui, A. Ionescu, and C. Depeursinge, “Process engineering and failure analysis of MEMS and MOEMS by digital holography microscopy (DHM),” in International Symposium on MOEMS–MEMS 2007 Micro and Nanofabrication: Reliability, Packaging, Testing, and Characterization of MEMS/MOEMS VI, A.L.Hartzell and R.Rameshuni (SPIE, 2007), Vol.  6463, pp. 64630G-1–64630G-7.
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  16. T. M. Kreis and W. P. O. Jueptner, “Suppression of the dc term in digital holography,” Opt. Eng. 36, 2357–2360 (1997).
    [CrossRef]
  17. C. S. Seelamantula, N. Pavillon, C. Depeursinge, and M. Unser, “Zero-order-free image reconstruction in digital holographic microscopy,” in IEEE International Symposium on Biomedical Imaging: From Nano to Micro (IEEE, 2009), pp. 201–204.
    [CrossRef]
  18. N. Pavillon, C. S. Seelamantula, J. Kühn, M. Unser, and C. Depeursinge, “Suppression of the zero-order in off-axis digital holography through nonlinear filtering,” Appl. Opt. 48, H186–H195 (2009).
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  21. F. Montfort, F. Charrière, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, “Purely numerical compensation for microscope objective phase curvature in digital holographic microscopy: influence of digital phase mask position,” J. Opt. Soc. Am. A 23, 2944–2953 (2006).
    [CrossRef]
  22. B. P. Bogert, M. J. R. Healy, and J. W. Tukey, “The frequency analysis of time series for echoes: cepstrum, pseudo-autocovariance, cross-cepstrum, and saphe cracking,” in Time Series Analysis, M.Rosenblatt, ed. (Wiley, 1963), ch. 15, pp. 209–243.
  23. J. Lim, Two-Dimensional Signal and Image Processing(Prentice-Hall, 1990).
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2009 (1)

2008 (2)

J. Kühn, F. Charriére, T. Colomb, E. Cuche, F. Montfort, Y. Emery, P. Marquet, and C. Depeursinge, “Axial sub-nanometer accuracy in digital holographic microscopy,” Meas. Sci. Technol. 19, 074007 (2008).
[CrossRef]

B. Kemper and G. Von Bally, “Digital holographic microscopy for live cell applications and technical inspection,” Appl. Opt. 47, A52–A61 (2008).
[CrossRef] [PubMed]

2006 (2)

2005 (1)

2002 (2)

W. Osten, T. Baumbach, and W. Jüptner, “Comparative digital holography,” Opt. Lett. 27, 1764–1766 (2002).
[CrossRef]

U. Schnars and W. P. O. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

2000 (1)

1999 (1)

1997 (2)

T. M. Kreis and W. P. O. Jueptner, “Suppression of the dc term in digital holography,” Opt. Eng. 36, 2357–2360 (1997).
[CrossRef]

I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22, 1268–1270 (1997).
[CrossRef] [PubMed]

1995 (1)

1994 (1)

1992 (1)

S. L. Hahn, “Multidimensional complex signals with single-orthant spectra,” Proc. IEEE 80, 1287–1300 (1992).
[CrossRef]

1982 (2)

1967 (1)

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
[CrossRef]

1962 (1)

Aspert, N.

F. Montfort, Y. Emery, F. Marquet, E. Cuche, N. Aspert, E. Solanas, A. Mehdaoui, A. Ionescu, and C. Depeursinge, “Process engineering and failure analysis of MEMS and MOEMS by digital holography microscopy (DHM),” in International Symposium on MOEMS–MEMS 2007 Micro and Nanofabrication: Reliability, Packaging, Testing, and Characterization of MEMS/MOEMS VI, A.L.Hartzell and R.Rameshuni (SPIE, 2007), Vol.  6463, pp. 64630G-1–64630G-7.

F. Montfort, Y. Emery, E. Solanas, E. Cuche, N. Aspert, P. Marquet, C. Joris, J. Kühn, and C. Depeursinge, “Surface roughness parameters measurements by Digital Holographic Microscopy (DHM),” in Third International Symposium on Precision Mechanical Measurements (SPIE, 2006), Vol.  6280 I, pp. 62800V-1–62800V-6.

Baumbach, T.

Bogert, B. P.

B. P. Bogert, M. J. R. Healy, and J. W. Tukey, “The frequency analysis of time series for echoes: cepstrum, pseudo-autocovariance, cross-cepstrum, and saphe cracking,” in Time Series Analysis, M.Rosenblatt, ed. (Wiley, 1963), ch. 15, pp. 209–243.

Bracewell, R.

R. Bracewell, The Fourier Transform and Its Applications, 3rd ed. (McGraw-Hill, 1999).

Callens, N.

Charriére, F.

J. Kühn, F. Charriére, T. Colomb, E. Cuche, F. Montfort, Y. Emery, P. Marquet, and C. Depeursinge, “Axial sub-nanometer accuracy in digital holographic microscopy,” Meas. Sci. Technol. 19, 074007 (2008).
[CrossRef]

Charrière, F.

Colomb, T.

Conde, R.

Coquoz, O.

Cuche, E.

J. Kühn, F. Charriére, T. Colomb, E. Cuche, F. Montfort, Y. Emery, P. Marquet, and C. Depeursinge, “Axial sub-nanometer accuracy in digital holographic microscopy,” Meas. Sci. Technol. 19, 074007 (2008).
[CrossRef]

F. Montfort, F. Charrière, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, “Purely numerical compensation for microscope objective phase curvature in digital holographic microscopy: influence of digital phase mask position,” J. Opt. Soc. Am. A 23, 2944–2953 (2006).
[CrossRef]

P. Marquet, B. Rappaz, P. Magistretti, E. Cuche, 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]

E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39, 4070–4075 (2000).
[CrossRef]

E. Cuche, P. Marquet, and C. Depeursinge, “Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms,” Appl. Opt. 38, 6994–7001 (1999).
[CrossRef]

F. Montfort, Y. Emery, E. Solanas, E. Cuche, N. Aspert, P. Marquet, C. Joris, J. Kühn, and C. Depeursinge, “Surface roughness parameters measurements by Digital Holographic Microscopy (DHM),” in Third International Symposium on Precision Mechanical Measurements (SPIE, 2006), Vol.  6280 I, pp. 62800V-1–62800V-6.

F. Montfort, Y. Emery, F. Marquet, E. Cuche, N. Aspert, E. Solanas, A. Mehdaoui, A. Ionescu, and C. Depeursinge, “Process engineering and failure analysis of MEMS and MOEMS by digital holography microscopy (DHM),” in International Symposium on MOEMS–MEMS 2007 Micro and Nanofabrication: Reliability, Packaging, Testing, and Characterization of MEMS/MOEMS VI, A.L.Hartzell and R.Rameshuni (SPIE, 2007), Vol.  6463, pp. 64630G-1–64630G-7.

Depeursinge, C.

N. Pavillon, C. S. Seelamantula, J. Kühn, M. Unser, and C. Depeursinge, “Suppression of the zero-order in off-axis digital holography through nonlinear filtering,” Appl. Opt. 48, H186–H195 (2009).
[CrossRef] [PubMed]

J. Kühn, F. Charriére, T. Colomb, E. Cuche, F. Montfort, Y. Emery, P. Marquet, and C. Depeursinge, “Axial sub-nanometer accuracy in digital holographic microscopy,” Meas. Sci. Technol. 19, 074007 (2008).
[CrossRef]

F. Montfort, F. Charrière, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, “Purely numerical compensation for microscope objective phase curvature in digital holographic microscopy: influence of digital phase mask position,” J. Opt. Soc. Am. A 23, 2944–2953 (2006).
[CrossRef]

P. Marquet, B. Rappaz, P. Magistretti, E. Cuche, 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]

E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39, 4070–4075 (2000).
[CrossRef]

E. Cuche, P. Marquet, and C. Depeursinge, “Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms,” Appl. Opt. 38, 6994–7001 (1999).
[CrossRef]

O. Coquoz, R. Conde, F. Taleblou, and C. Depeursinge, “Performance of endoscopic holography with a multicore optical fiber,” Appl. Opt. 34, 7186–7193 (1995).
[CrossRef] [PubMed]

F. Montfort, Y. Emery, E. Solanas, E. Cuche, N. Aspert, P. Marquet, C. Joris, J. Kühn, and C. Depeursinge, “Surface roughness parameters measurements by Digital Holographic Microscopy (DHM),” in Third International Symposium on Precision Mechanical Measurements (SPIE, 2006), Vol.  6280 I, pp. 62800V-1–62800V-6.

F. Montfort, Y. Emery, F. Marquet, E. Cuche, N. Aspert, E. Solanas, A. Mehdaoui, A. Ionescu, and C. Depeursinge, “Process engineering and failure analysis of MEMS and MOEMS by digital holography microscopy (DHM),” in International Symposium on MOEMS–MEMS 2007 Micro and Nanofabrication: Reliability, Packaging, Testing, and Characterization of MEMS/MOEMS VI, A.L.Hartzell and R.Rameshuni (SPIE, 2007), Vol.  6463, pp. 64630G-1–64630G-7.

C. S. Seelamantula, N. Pavillon, C. Depeursinge, and M. Unser, “Zero-order-free image reconstruction in digital holographic microscopy,” in IEEE International Symposium on Biomedical Imaging: From Nano to Micro (IEEE, 2009), pp. 201–204.
[CrossRef]

Dubois, F.

Emery, Y.

J. Kühn, F. Charriére, T. Colomb, E. Cuche, F. Montfort, Y. Emery, P. Marquet, and C. Depeursinge, “Axial sub-nanometer accuracy in digital holographic microscopy,” Meas. Sci. Technol. 19, 074007 (2008).
[CrossRef]

P. Marquet, B. Rappaz, P. Magistretti, E. Cuche, 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. Montfort, Y. Emery, F. Marquet, E. Cuche, N. Aspert, E. Solanas, A. Mehdaoui, A. Ionescu, and C. Depeursinge, “Process engineering and failure analysis of MEMS and MOEMS by digital holography microscopy (DHM),” in International Symposium on MOEMS–MEMS 2007 Micro and Nanofabrication: Reliability, Packaging, Testing, and Characterization of MEMS/MOEMS VI, A.L.Hartzell and R.Rameshuni (SPIE, 2007), Vol.  6463, pp. 64630G-1–64630G-7.

F. Montfort, Y. Emery, E. Solanas, E. Cuche, N. Aspert, P. Marquet, C. Joris, J. Kühn, and C. Depeursinge, “Surface roughness parameters measurements by Digital Holographic Microscopy (DHM),” in Third International Symposium on Precision Mechanical Measurements (SPIE, 2006), Vol.  6280 I, pp. 62800V-1–62800V-6.

Fienup, J. R.

Goodman, J. W.

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
[CrossRef]

Hahn, S. L.

S. L. Hahn, “Multidimensional complex signals with single-orthant spectra,” Proc. IEEE 80, 1287–1300 (1992).
[CrossRef]

S. L. Hahn, Hilbert Transforms in Signal Processing (Artech House, 1996).

Healy, M. J. R.

B. P. Bogert, M. J. R. Healy, and J. W. Tukey, “The frequency analysis of time series for echoes: cepstrum, pseudo-autocovariance, cross-cepstrum, and saphe cracking,” in Time Series Analysis, M.Rosenblatt, ed. (Wiley, 1963), ch. 15, pp. 209–243.

Hoyos, M.

Ina, H.

Ionescu, A.

F. Montfort, Y. Emery, F. Marquet, E. Cuche, N. Aspert, E. Solanas, A. Mehdaoui, A. Ionescu, and C. Depeursinge, “Process engineering and failure analysis of MEMS and MOEMS by digital holography microscopy (DHM),” in International Symposium on MOEMS–MEMS 2007 Micro and Nanofabrication: Reliability, Packaging, Testing, and Characterization of MEMS/MOEMS VI, A.L.Hartzell and R.Rameshuni (SPIE, 2007), Vol.  6463, pp. 64630G-1–64630G-7.

Joris, C.

F. Montfort, Y. Emery, E. Solanas, E. Cuche, N. Aspert, P. Marquet, C. Joris, J. Kühn, and C. Depeursinge, “Surface roughness parameters measurements by Digital Holographic Microscopy (DHM),” in Third International Symposium on Precision Mechanical Measurements (SPIE, 2006), Vol.  6280 I, pp. 62800V-1–62800V-6.

Jueptner, W. P. O.

T. M. Kreis and W. P. O. Jueptner, “Suppression of the dc term in digital holography,” Opt. Eng. 36, 2357–2360 (1997).
[CrossRef]

Jüptner, W.

Jüptner, W. P. O.

U. Schnars and W. P. O. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

Kemper, B.

Kobayashi, S.

Kreis, T. M.

T. M. Kreis and W. P. O. Jueptner, “Suppression of the dc term in digital holography,” Opt. Eng. 36, 2357–2360 (1997).
[CrossRef]

Kühn, J.

N. Pavillon, C. S. Seelamantula, J. Kühn, M. Unser, and C. Depeursinge, “Suppression of the zero-order in off-axis digital holography through nonlinear filtering,” Appl. Opt. 48, H186–H195 (2009).
[CrossRef] [PubMed]

J. Kühn, F. Charriére, T. Colomb, E. Cuche, F. Montfort, Y. Emery, P. Marquet, and C. Depeursinge, “Axial sub-nanometer accuracy in digital holographic microscopy,” Meas. Sci. Technol. 19, 074007 (2008).
[CrossRef]

F. Montfort, Y. Emery, E. Solanas, E. Cuche, N. Aspert, P. Marquet, C. Joris, J. Kühn, and C. Depeursinge, “Surface roughness parameters measurements by Digital Holographic Microscopy (DHM),” in Third International Symposium on Precision Mechanical Measurements (SPIE, 2006), Vol.  6280 I, pp. 62800V-1–62800V-6.

Kurowski, P.

Lawrence, R. W.

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
[CrossRef]

Leith, E. N.

Lim, J.

J. Lim, Two-Dimensional Signal and Image Processing(Prentice-Hall, 1990).

Magistretti, P.

Marquet, F.

F. Montfort, Y. Emery, F. Marquet, E. Cuche, N. Aspert, E. Solanas, A. Mehdaoui, A. Ionescu, and C. Depeursinge, “Process engineering and failure analysis of MEMS and MOEMS by digital holography microscopy (DHM),” in International Symposium on MOEMS–MEMS 2007 Micro and Nanofabrication: Reliability, Packaging, Testing, and Characterization of MEMS/MOEMS VI, A.L.Hartzell and R.Rameshuni (SPIE, 2007), Vol.  6463, pp. 64630G-1–64630G-7.

Marquet, P.

J. Kühn, F. Charriére, T. Colomb, E. Cuche, F. Montfort, Y. Emery, P. Marquet, and C. Depeursinge, “Axial sub-nanometer accuracy in digital holographic microscopy,” Meas. Sci. Technol. 19, 074007 (2008).
[CrossRef]

F. Montfort, F. Charrière, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, “Purely numerical compensation for microscope objective phase curvature in digital holographic microscopy: influence of digital phase mask position,” J. Opt. Soc. Am. A 23, 2944–2953 (2006).
[CrossRef]

P. Marquet, B. Rappaz, P. Magistretti, E. Cuche, 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]

E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39, 4070–4075 (2000).
[CrossRef]

E. Cuche, P. Marquet, and C. Depeursinge, “Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms,” Appl. Opt. 38, 6994–7001 (1999).
[CrossRef]

F. Montfort, Y. Emery, E. Solanas, E. Cuche, N. Aspert, P. Marquet, C. Joris, J. Kühn, and C. Depeursinge, “Surface roughness parameters measurements by Digital Holographic Microscopy (DHM),” in Third International Symposium on Precision Mechanical Measurements (SPIE, 2006), Vol.  6280 I, pp. 62800V-1–62800V-6.

Mehdaoui, A.

F. Montfort, Y. Emery, F. Marquet, E. Cuche, N. Aspert, E. Solanas, A. Mehdaoui, A. Ionescu, and C. Depeursinge, “Process engineering and failure analysis of MEMS and MOEMS by digital holography microscopy (DHM),” in International Symposium on MOEMS–MEMS 2007 Micro and Nanofabrication: Reliability, Packaging, Testing, and Characterization of MEMS/MOEMS VI, A.L.Hartzell and R.Rameshuni (SPIE, 2007), Vol.  6463, pp. 64630G-1–64630G-7.

Monnom, O.

Montfort, F.

J. Kühn, F. Charriére, T. Colomb, E. Cuche, F. Montfort, Y. Emery, P. Marquet, and C. Depeursinge, “Axial sub-nanometer accuracy in digital holographic microscopy,” Meas. Sci. Technol. 19, 074007 (2008).
[CrossRef]

F. Montfort, F. Charrière, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, “Purely numerical compensation for microscope objective phase curvature in digital holographic microscopy: influence of digital phase mask position,” J. Opt. Soc. Am. A 23, 2944–2953 (2006).
[CrossRef]

F. Montfort, Y. Emery, F. Marquet, E. Cuche, N. Aspert, E. Solanas, A. Mehdaoui, A. Ionescu, and C. Depeursinge, “Process engineering and failure analysis of MEMS and MOEMS by digital holography microscopy (DHM),” in International Symposium on MOEMS–MEMS 2007 Micro and Nanofabrication: Reliability, Packaging, Testing, and Characterization of MEMS/MOEMS VI, A.L.Hartzell and R.Rameshuni (SPIE, 2007), Vol.  6463, pp. 64630G-1–64630G-7.

F. Montfort, Y. Emery, E. Solanas, E. Cuche, N. Aspert, P. Marquet, C. Joris, J. Kühn, and C. Depeursinge, “Surface roughness parameters measurements by Digital Holographic Microscopy (DHM),” in Third International Symposium on Precision Mechanical Measurements (SPIE, 2006), Vol.  6280 I, pp. 62800V-1–62800V-6.

Oppenheim, A. V.

A. V. Oppenheim and R. W. Schafer, Discrete Time Signal Processing, 2nd ed. (Prentice Hall, 1999).

Osten, W.

Papoulis, A.

A. Papoulis, Signal Analysis (McGraw-Hill, 1977).

Pavillon, N.

N. Pavillon, C. S. Seelamantula, J. Kühn, M. Unser, and C. Depeursinge, “Suppression of the zero-order in off-axis digital holography through nonlinear filtering,” Appl. Opt. 48, H186–H195 (2009).
[CrossRef] [PubMed]

C. S. Seelamantula, N. Pavillon, C. Depeursinge, and M. Unser, “Zero-order-free image reconstruction in digital holographic microscopy,” in IEEE International Symposium on Biomedical Imaging: From Nano to Micro (IEEE, 2009), pp. 201–204.
[CrossRef]

Rappaz, B.

Schafer, R. W.

A. V. Oppenheim and R. W. Schafer, Discrete Time Signal Processing, 2nd ed. (Prentice Hall, 1999).

Schnars, U.

U. Schnars and W. P. O. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

U. Schnars, “Direct phase determination in hologram interferometry with use of digitally recorded holograms,” J. Opt. Soc. Am. A 11, 2011–2015 (1994).
[CrossRef]

Seelamantula, C. S.

N. Pavillon, C. S. Seelamantula, J. Kühn, M. Unser, and C. Depeursinge, “Suppression of the zero-order in off-axis digital holography through nonlinear filtering,” Appl. Opt. 48, H186–H195 (2009).
[CrossRef] [PubMed]

C. S. Seelamantula, N. Pavillon, C. Depeursinge, and M. Unser, “Zero-order-free image reconstruction in digital holographic microscopy,” in IEEE International Symposium on Biomedical Imaging: From Nano to Micro (IEEE, 2009), pp. 201–204.
[CrossRef]

Solanas, E.

F. Montfort, Y. Emery, F. Marquet, E. Cuche, N. Aspert, E. Solanas, A. Mehdaoui, A. Ionescu, and C. Depeursinge, “Process engineering and failure analysis of MEMS and MOEMS by digital holography microscopy (DHM),” in International Symposium on MOEMS–MEMS 2007 Micro and Nanofabrication: Reliability, Packaging, Testing, and Characterization of MEMS/MOEMS VI, A.L.Hartzell and R.Rameshuni (SPIE, 2007), Vol.  6463, pp. 64630G-1–64630G-7.

F. Montfort, Y. Emery, E. Solanas, E. Cuche, N. Aspert, P. Marquet, C. Joris, J. Kühn, and C. Depeursinge, “Surface roughness parameters measurements by Digital Holographic Microscopy (DHM),” in Third International Symposium on Precision Mechanical Measurements (SPIE, 2006), Vol.  6280 I, pp. 62800V-1–62800V-6.

Takeda, M.

Taleblou, F.

Tukey, J. W.

B. P. Bogert, M. J. R. Healy, and J. W. Tukey, “The frequency analysis of time series for echoes: cepstrum, pseudo-autocovariance, cross-cepstrum, and saphe cracking,” in Time Series Analysis, M.Rosenblatt, ed. (Wiley, 1963), ch. 15, pp. 209–243.

Unser, M.

N. Pavillon, C. S. Seelamantula, J. Kühn, M. Unser, and C. Depeursinge, “Suppression of the zero-order in off-axis digital holography through nonlinear filtering,” Appl. Opt. 48, H186–H195 (2009).
[CrossRef] [PubMed]

C. S. Seelamantula, N. Pavillon, C. Depeursinge, and M. Unser, “Zero-order-free image reconstruction in digital holographic microscopy,” in IEEE International Symposium on Biomedical Imaging: From Nano to Micro (IEEE, 2009), pp. 201–204.
[CrossRef]

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Yourassowsky, C.

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[CrossRef]

F. Montfort, Y. Emery, E. Solanas, E. Cuche, N. Aspert, P. Marquet, C. Joris, J. Kühn, and C. Depeursinge, “Surface roughness parameters measurements by Digital Holographic Microscopy (DHM),” in Third International Symposium on Precision Mechanical Measurements (SPIE, 2006), Vol.  6280 I, pp. 62800V-1–62800V-6.

F. Montfort, Y. Emery, F. Marquet, E. Cuche, N. Aspert, E. Solanas, A. Mehdaoui, A. Ionescu, and C. Depeursinge, “Process engineering and failure analysis of MEMS and MOEMS by digital holography microscopy (DHM),” in International Symposium on MOEMS–MEMS 2007 Micro and Nanofabrication: Reliability, Packaging, Testing, and Characterization of MEMS/MOEMS VI, A.L.Hartzell and R.Rameshuni (SPIE, 2007), Vol.  6463, pp. 64630G-1–64630G-7.

B. P. Bogert, M. J. R. Healy, and J. W. Tukey, “The frequency analysis of time series for echoes: cepstrum, pseudo-autocovariance, cross-cepstrum, and saphe cracking,” in Time Series Analysis, M.Rosenblatt, ed. (Wiley, 1963), ch. 15, pp. 209–243.

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Figures (9)

Fig. 1
Fig. 1

Off-axis digital holography—recording.

Fig. 2
Fig. 2

Off-axis digital holography—reconstruction.

Fig. 3
Fig. 3

Spectral occupancy of (a) the object wave, (b) the modulated object wave, and (c) the hologram i ( x ) .

Fig. 4
Fig. 4

(a) Homomorphic system with convolution operation at the input as well as at the output, (b) its canonical representation.

Fig. 5
Fig. 5

(a) Characteristic system D { * , + } for convolution, (b) its inverse D 1 { + , * } .

Fig. 6
Fig. 6

Phase of the phantom used in simulations. (a) Ground truth phase used in hologram simulation, (b) phase obtained by the standard Fourier method, and (c) phase of the nonlinear reconstruction.

Fig. 7
Fig. 7

Phase difference between the ground truth and that obtained by (a) the standard Fourier reconstruction, and (b) the nonlinear reconstruction technique.

Fig. 8
Fig. 8

Fourier transform of a hologram of yew pollens. The dashed zone highlights the filtered region used in the reconstructions of Fig. 9.

Fig. 9
Fig. 9

Specimen: solution of yew pollens. The hologram is reconstructed with (a), (c) standard Fourier filtering and (b), (d) with the proposed nonlinear technique, where amplitude (a), (b) and phase (c), (d) images are shown.

Equations (30)

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i ( x ) = | r ( x ) + o ( x ) | 2 = | r ( x ) | 2 + | o ( x ) | 2 + r * ( x ) o ( x ) + r ( x ) o * ( x ) .
ψ o ( x ) = u ( x ) i ( x ) = i r u ( x ) + u ( x ) i o ( x ) zero-order terms + u ( x ) r * ( x ) o ( x ) virtual image + u ( x ) r ( x ) o * ( x ) real image .
ψ i ( ξ ) = A exp ( i π λ d ( ξ 2 + η 2 ) ) ψ o ( x ) exp ( i π λ d ( x 2 + y 2 ) ) exp ( i 2 π λ d ( x ξ + y η ) ) d x = A exp ( i π λ d ( ξ 2 + η 2 ) ) F 1 { ψ o ( x ) exp ( i π λ d ( x 2 + y 2 ) ) } ,
ψ i [ m ] = A exp ( i π λ d ( m T Δ 1 m ) ) · F p { ψ o [ p ] exp ( i π λ d ( p T Δ 2 p ) ) } [ m ] ,
ψ i [ m ] = ψ o ( x ) · 1 [ 0 , x L x ] × [ 0 , y L y ] | x = ( m Δ x , n Δ y ) ,
i ( x ) = | A exp ( i k , x ) + o ( x ) | 2 = | A | 2 | 1 + 1 A exp ( i k , x ) o ( x ) o ˜ ( x ) = o ( x ) r ( x ) | 2 .
F { i } ( ω ) = | A | 2 { δ ( ω ) + F { o } ( ω + k ) + F { o } ( ω k ) + q ( ω ) } ,
log ( 1 + o ˜ ( x ) ) = n = 1 ( 1 ) n 1 n o ˜ n ( x ) .
F { log ( 1 + o ˜ ) } ( ω ) = n = 1 ( 1 ) n 1 n F { o ˜ n } ( ω ) .
F { o ˜ n } ( ω ) = ( F { o ˜ } * F { o ˜ } * F { o ˜ } * * F { o ˜ } ) n times ( ω ) .
log | 1 + o ˜ | 2 = log ( 1 + o ˜ ) + log ( 1 + o ˜ * ) .
c ( ω ) = F { log | 1 + o ˜ | 2 } ( ω ) = F { log ( 1 + o ˜ ) } ( ω ) + F { log ( 1 + o ˜ * ) } ( ω ) .
F { log ( 1 + o ˜ ) } ( ω ) = F { log | 1 + o ˜ | 2 } ( ω ) · 1 [ 0 , + ) × [ 0 , + ) ,
log ( 1 + o ˜ ( x ) ) = F 1 { F { log | 1 + o ˜ | 2 } · 1 [ 0 , + ) × [ 0 , + ) } ( x ) , o ˜ ( x ) = exp ( F 1 { F { log | 1 + o ˜ | 2 } · 1 [ 0 , + ) × [ 0 , + ) } ) ( x ) 1 .
c f ( x ) = F 1 { log F { f } } ( x ) .
c f r ( x ) = F 1 { log | F { f } | } ( x ) .
T { f 1 * f 2 } ( x ) = T { f 1 } * T { f 2 } ( x ) ( linearity ) ,
T { c · f 1 } ( x ) = c · T { f 1 } ( x ) ( scaling property ) ,
H { h } ( x ) F i sign ( ω ) F { h } ( ω ) ,
F { a h } ( ω ) = ( 1 + sign ( ω ) ) F { h } ( ω ) .
F { a h } ( ω ) = ( 1 + sign ( ω x ) ) ( 1 + sign ( ω y ) ) F { h } ( ω ) = ( 1 + sign ( ω x ) + sign ( ω y ) + sign ( ω x ) sign ( ω y ) ) F { h } ( ω ) = 4 · 1 [ 0 , + ) × [ 0 , + ) ( ω ) F { h } ( ω ) ,
F { a h } ( ω ) = F { h } ( ω ) + i { i sign ( ω x ) i sign ( ω y ) i sign ( ω x ) sign ( ω y ) } F { h } ( ω ) ,
a h ( x ) = Re { a h } ( x ) + i Im { a h } ( x ) ,
Re { a h } ( x ) = h ( x ) ( H x H y ) h ( x ) ,
Im { a h } ( x ) = H x h ( x ) + H y h ( x ) .
Re { a h } , Im { a h } L 2 ( R × R ) = 0 .
H x { Re { a h } } ( x ) = H y { Re { a h } } ( x ) = Im { a h } ( x ) .
H x { Im { a h } } ( x ) = H y { Im { a h } } ( x ) = Re { a h } ( x ) .
log ( 1 + o ˜ ( x ) ) = F 1 { F { log | 1 + o ˜ | 2 } · 1 [ 0 , + ) × [ 0 , + ) } ( x ) ,
log ( 1 + o ˜ ( x ) ) = log | 1 + o ˜ ( x ) | + i ( 1 + o ˜ ( x ) ) ,

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