Abstract

We present a new digital two-step reconstruction method for off-axis holograms recorded on a CCD camera. First, we retrieve the complex object wave in the acquisition plane from the hologram’s samples. In a second step, if required, we propagate the wave front by using a digital Fresnel transform to achieve proper focus. This algorithm is sufficiently general to be applied to sophisticated optical setups that include a microscope objective. We characterize and evaluate the algorithm by using simulated data sets and demonstrate its applicability to real-world experimental conditions by reconstructing optically acquired holograms.

© 2004 Optical Society of America

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2003

M. Liebling, T. Blu, M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
[CrossRef]

2002

T. Kreis, “Frequency analysis of digital holography,” Opt. Eng. 41, 771–778 (2002).
[CrossRef]

C. Liu, Y. Li, X. Cheng, Z. Liu, F. Bo, J. Zhu, “Elimination of zero-order diffraction in digital holography,” Opt. Eng. 41, 2434–2437 (2002).
[CrossRef]

T. Colomb, P. Dahlgren, D. Beghuin, E. Cuche, P. Marquet, Ch. Depeursinge, “Polarization imaging by use of digital holography,” Appl. Opt. 41, 27–37 (2002).
[CrossRef] [PubMed]

2001

2000

1999

1998

1997

1996

1995

C. Rathjen, “Statistical properties of phase-shift algorithms,” J. Opt. Soc. Am. A 12, 1997–2008 (1995).
[CrossRef]

A. J. Moore, F. Mendoza-Santoyo, “Phase demodulation in the space domain without a fringe carrier,” Opt. Lasers Eng. 23, 319–330 (1995).
[CrossRef]

M. Servin, F. J. Cuevas, “A novel technique for spatial phase-shifting interferometry,” J. Mod. Opt. 42, 1853–1862 (1995).
[CrossRef]

P. H. Chan, P. J. Bryanston-Cross, S. C. Parker, “Fringe-pattern analysis using a spatial phase-stepping method with automatic phase unwrapping,” Meas. Sci. Technol. 6, 1250–1259 (1995).
[CrossRef]

R. Windecker, H. J. Tiziani, “Semispatial, robust, and accurate phase evaluation algorithm,” Appl. Opt. 34, 7321–7326 (1995).
[CrossRef] [PubMed]

M. Pirga, M. Kujawińska, “Two directional, spatial-carrier, phase-shifting method for analysis of crossed closed fringe patterns,” Opt. Eng. 34, 2459–2466 (1995).
[CrossRef]

1994

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

M. Unser, A. Aldroubi, S. Schiff, “Fast implementation of the continuous wavelet transform with integer scales,” IEEE Trans. Signal Process. 42, 3519–3523 (1994).
[CrossRef]

1991

1987

1986

1985

1984

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
[CrossRef]

1983

1982

1972

Y. Ichioka, M. Inuiya, “Direct phase detecting system,” Appl. Opt. 11, 1507–1514 (1972).
[CrossRef] [PubMed]

M. A. Kronrod, N. S. Merzlyakov, L. Yaroslavskii, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys. 17, 333–334 (1972).

1967

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

1948

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

Adams, M.

T. M. Kreis, M. Adams, W. P. O. Jüptner, “Methods of digital holography: A comparison,” in Optical Inspection and Micromeasurements II, C. Gorecki, ed., Proc. SPIE3098, 224–233 (1997).
[CrossRef]

Aldroubi, A.

M. Unser, A. Aldroubi, S. Schiff, “Fast implementation of the continuous wavelet transform with integer scales,” IEEE Trans. Signal Process. 42, 3519–3523 (1994).
[CrossRef]

Bachor, H.-A.

Beghuin, D.

Bevilacqua, F.

Blu, T.

M. Liebling, T. Blu, M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
[CrossRef]

Bo, F.

C. Liu, Y. Li, X. Cheng, Z. Liu, F. Bo, J. Zhu, “Elimination of zero-order diffraction in digital holography,” Opt. Eng. 41, 2434–2437 (2002).
[CrossRef]

Bone, D. J.

Bothe, T.

Bruning, J. H.

J. E. Greivenkamp, J. H. Bruning, “Phase-shifting interferometry,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 501–598.

Bryanston-Cross, P. J.

P. H. Chan, P. J. Bryanston-Cross, S. C. Parker, “Fringe-pattern analysis using a spatial phase-stepping method with automatic phase unwrapping,” Meas. Sci. Technol. 6, 1250–1259 (1995).
[CrossRef]

Burke, J.

Cha, S. S.

Chan, P. H.

P. H. Chan, P. J. Bryanston-Cross, S. C. Parker, “Fringe-pattern analysis using a spatial phase-stepping method with automatic phase unwrapping,” Meas. Sci. Technol. 6, 1250–1259 (1995).
[CrossRef]

Cheng, X.

C. Liu, Y. Li, X. Cheng, Z. Liu, F. Bo, J. Zhu, “Elimination of zero-order diffraction in digital holography,” Opt. Eng. 41, 2434–2437 (2002).
[CrossRef]

Colomb, T.

Creath, K.

J. Schmit, K. Creath, “Window function influence on phase error in phase-shifting algorithms,” Appl. Opt. 35, 5642–5649 (1996).
[CrossRef] [PubMed]

K. Creath, J. Schmit, “N-point spatial phase-measurement techniques for non-destructive testing,” Opt. Lasers Eng. 24, 365–379 (1996).
[CrossRef]

Cuche, E.

Cuevas, F. J.

Dahlgren, P.

Depeursinge, Ch.

Farrant, D. I.

Fessler, J. A.

S. Sotthivirat, J. A. Fessler, “Relaxed ordered subsets algorithm for image restoration of confocal microscopy,” in Proceedings of the 2002 IEEE International Symposium on Biomedical Imaging: Macro to Nano (ISBI’02) (Institute of Electrical and Electronics Engineers, New York, 2002), Vol. 3, pp. 1051–1054.

S. Sotthivirat, J. A. Fessler, “Penalized-likelihood image reconstruction for digital holography,” J. Opt. Soc. Am. A (to be published).

Gabor, D.

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

Goodman, J. W.

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

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Greivenkamp, J. E.

J. E. Greivenkamp, J. H. Bruning, “Phase-shifting interferometry,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 501–598.

Helmers, H.

Hibino, K.

Ichioka, Y.

Ina, H.

Inuiya, M.

Jüptner, W.

Jüptner, W. P. O.

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

T. M. Kreis, M. Adams, W. P. O. Jüptner, “Methods of digital holography: A comparison,” in Optical Inspection and Micromeasurements II, C. Gorecki, ed., Proc. SPIE3098, 224–233 (1997).
[CrossRef]

Kobayashi, S.

Kokal, J.

Kreis, T.

T. Kreis, “Frequency analysis of digital holography,” Opt. Eng. 41, 771–778 (2002).
[CrossRef]

T. Kreis, “Digital holographic interference-phase measurement using the Fourier-transform method,” J. Opt. Soc. Am. A 3, 847–855 (1986).
[CrossRef]

T. Kreis, Holographic Interferometry, Vol. 1 of Optical Metrology Series (Akademie Verlag, Berlin, 1996).

T. Kreis, “Computer-aided evaluation of holographic interferograms,” in Holographic Interferometry: Principles and Methods, P. K. Rastogi, ed. (Springer, Heidelberg, Germany, 1994), pp. 151–212.

Kreis, T. M.

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

T. M. Kreis, M. Adams, W. P. O. Jüptner, “Methods of digital holography: A comparison,” in Optical Inspection and Micromeasurements II, C. Gorecki, ed., Proc. SPIE3098, 224–233 (1997).
[CrossRef]

Kronrod, M. A.

M. A. Kronrod, N. S. Merzlyakov, L. Yaroslavskii, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys. 17, 333–334 (1972).

Kujawinska, M.

M. Pirga, M. Kujawińska, “Two directional, spatial-carrier, phase-shifting method for analysis of crossed closed fringe patterns,” Opt. Eng. 34, 2459–2466 (1995).
[CrossRef]

M. Kujawińska, “Spatial phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds. (Bristol Institute of Physics, Bristol, UK, 1993), pp. 141–193.

M. Kujawińska, J. Wójciak, “Spatial phase-shifting techniques of fringe pattern analysis in photomechanics,” in Second International Conference on Photomechanics and Speckle Metrology: Moiré Techniques, Holographic Interferometry, Optical NDT, and Applications to Fluid Mechanics, Parts 1 and 2, F.-P. Chiang, ed., Proc. SPIE1554B, 503–513 (1991).

Lai, G.

Larkin, K. G.

Lawrence, R. W.

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

Li, Y.

C. Liu, Y. Li, X. Cheng, Z. Liu, F. Bo, J. Zhu, “Elimination of zero-order diffraction in digital holography,” Opt. Eng. 41, 2434–2437 (2002).
[CrossRef]

Liebling, M.

M. Liebling, T. Blu, M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
[CrossRef]

Liu, C.

C. Liu, Y. Li, X. Cheng, Z. Liu, F. Bo, J. Zhu, “Elimination of zero-order diffraction in digital holography,” Opt. Eng. 41, 2434–2437 (2002).
[CrossRef]

Liu, Z.

C. Liu, Y. Li, X. Cheng, Z. Liu, F. Bo, J. Zhu, “Elimination of zero-order diffraction in digital holography,” Opt. Eng. 41, 2434–2437 (2002).
[CrossRef]

Macy, W. W.

Malacara, D.

V. I. Vlad, D. Malacara, “Direct spatial reconstruction of optical phase from phase-modulated images,” in Progress in Optics, Vol. XXXIII, E. Wolf, ed. (Elsevier, Amsterdam, 1994), pp. 261–317.

Marian, A.

A. Marian, E. Cuche, Ch. Depeursinge, “Point spread function model for microscopic image deconvolution in digital holographic microscopy,” in Novel Optical Instrumentation for Biomedical Applications, A.-C. Boccara, ed., Proc. SPIE5143, 202–209 (2003).
[CrossRef]

Marquet, P.

Marroquin, J. L.

Mendoza-Santoyo, F.

A. J. Moore, F. Mendoza-Santoyo, “Phase demodulation in the space domain without a fringe carrier,” Opt. Lasers Eng. 23, 319–330 (1995).
[CrossRef]

Mertz, L.

Merzlyakov, N. S.

M. A. Kronrod, N. S. Merzlyakov, L. Yaroslavskii, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys. 17, 333–334 (1972).

L. P. Yaroslavskii, N. S. Merzlyakov, Methods of Digital Holography (Consultants Bureau, New York, 1980).

Moore, A. J.

A. J. Moore, F. Mendoza-Santoyo, “Phase demodulation in the space domain without a fringe carrier,” Opt. Lasers Eng. 23, 319–330 (1995).
[CrossRef]

Nugent, K. A.

Oldfield, M. A.

Onural, L.

Oreb, B. F.

Paez, G.

Parker, S. C.

P. H. Chan, P. J. Bryanston-Cross, S. C. Parker, “Fringe-pattern analysis using a spatial phase-stepping method with automatic phase unwrapping,” Meas. Sci. Technol. 6, 1250–1259 (1995).
[CrossRef]

Pirga, M.

M. Pirga, M. Kujawińska, “Two directional, spatial-carrier, phase-shifting method for analysis of crossed closed fringe patterns,” Opt. Eng. 34, 2459–2466 (1995).
[CrossRef]

Ransom, P. L.

Rathjen, C.

Roddier, C.

Roddier, F.

Rodriguez-Vera, R.

Sandeman, R. J.

Schiff, S.

M. Unser, A. Aldroubi, S. Schiff, “Fast implementation of the continuous wavelet transform with integer scales,” IEEE Trans. Signal Process. 42, 3519–3523 (1994).
[CrossRef]

Schmit, J.

J. Schmit, K. Creath, “Window function influence on phase error in phase-shifting algorithms,” Appl. Opt. 35, 5642–5649 (1996).
[CrossRef] [PubMed]

K. Creath, J. Schmit, “N-point spatial phase-measurement techniques for non-destructive testing,” Opt. Lasers Eng. 24, 365–379 (1996).
[CrossRef]

Schnars, U.

Servin, M.

Sotthivirat, S.

S. Sotthivirat, J. A. Fessler, “Penalized-likelihood image reconstruction for digital holography,” J. Opt. Soc. Am. A (to be published).

S. Sotthivirat, J. A. Fessler, “Relaxed ordered subsets algorithm for image restoration of confocal microscopy,” in Proceedings of the 2002 IEEE International Symposium on Biomedical Imaging: Macro to Nano (ISBI’02) (Institute of Electrical and Electronics Engineers, New York, 2002), Vol. 3, pp. 1051–1054.

Strojnik, M.

Surrel, Y.

Takeda, M.

Tiziani, H. J.

Unser, M.

M. Liebling, T. Blu, M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
[CrossRef]

M. Unser, “Splines: a perfect fit for signal and image processing,” IEEE Signal Process. Mag. 16, 22–38 (1999).
[CrossRef]

M. Unser, A. Aldroubi, S. Schiff, “Fast implementation of the continuous wavelet transform with integer scales,” IEEE Trans. Signal Process. 42, 3519–3523 (1994).
[CrossRef]

Vlad, V. I.

V. I. Vlad, D. Malacara, “Direct spatial reconstruction of optical phase from phase-modulated images,” in Progress in Optics, Vol. XXXIII, E. Wolf, ed. (Elsevier, Amsterdam, 1994), pp. 261–317.

Windecker, R.

Wójciak, J.

M. Kujawińska, J. Wójciak, “Spatial phase-shifting techniques of fringe pattern analysis in photomechanics,” in Second International Conference on Photomechanics and Speckle Metrology: Moiré Techniques, Holographic Interferometry, Optical NDT, and Applications to Fluid Mechanics, Parts 1 and 2, F.-P. Chiang, ed., Proc. SPIE1554B, 503–513 (1991).

Womack, K. H.

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
[CrossRef]

Yaroslavskii, L.

M. A. Kronrod, N. S. Merzlyakov, L. Yaroslavskii, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys. 17, 333–334 (1972).

Yaroslavskii, L. P.

L. P. Yaroslavskii, N. S. Merzlyakov, Methods of Digital Holography (Consultants Bureau, New York, 1980).

Yatagai, T.

Yu, E.

Zhu, J.

C. Liu, Y. Li, X. Cheng, Z. Liu, F. Bo, J. Zhu, “Elimination of zero-order diffraction in digital holography,” Opt. Eng. 41, 2434–2437 (2002).
[CrossRef]

Appl. Opt.

Y. Ichioka, M. Inuiya, “Direct phase detecting system,” Appl. Opt. 11, 1507–1514 (1972).
[CrossRef] [PubMed]

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Figures (8)

Fig. 1
Fig. 1

Schematic view of the off-axis geometry.

Fig. 2
Fig. 2

Information separation of the interferogram using the Fourier, respectively Fresnel, transform.

Fig. 3
Fig. 3

Schematic hologram with a grid denoting the pixels’ centers.

Fig. 4
Fig. 4

Weights computed from the tensor product of two cubic B-splines and a window size L=9, M=81.

Fig. 5
Fig. 5

(a) Simulated hologram; (b) original phase; (c) reconstructed phase; (d) reconstructed phase with weighted algorithm.

Fig. 6
Fig. 6

(a) Simulated hologram; (b) Window width table visualization; maximum width, 80 pixels (white regions); minimum width, 7 pixels (black regions). (c) Weighted hologram. (d) Test wave’s phase; in black regions it equals π/30, in bright regions 2π/30. (e) Reconstructed phase. (f) Difference image: The gray scale covers the range [0, π/30].

Fig. 7
Fig. 7

Phase aberration compensation and phase retrieval in the case of a microscopy setup. The CCD is not in the image plane.

Fig. 8
Fig. 8

(a) Measured hologram; data courtesy of E. Cuche, École Polytechnique Fédérale de Lausanne, and P. Marquet, Université de Lausanne. (b) Reconstructed amplitude in the CCD plane. Reconstructed amplitude (c) and phase (d) with adjusted focus. Reconstructed amplitude (e) and phase (f) with an alternative technique.8 All images are 360×360 pixels. The phase images’ gray scale covers the range (-π, π].

Equations (31)

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fˆ(ν)=-f(x)exp(-2iπxν)dx,
f(x)=-fˆ(ν)exp(2iπxν)dν,
R(d){U}(x)=exp(ikd)iλd-U(ξ, η)expiπλd[(ξ-x)2+(η-y)2]dξdη,
I(x)=|R(x)+Ψ(x)|2=|R(x)|2+|Ψ(x)|2zero-order+R*(x)Ψ(x)+Ψ*(x)R(x),
I(x+xm)=|Ψ(x)+A(x)exp[iθ(x+xm)]|2,
Im=|Ψ+A exp(iθm)|2=|Ψ|2+A2+2R(Rm*Ψ).
arg minA+*,Ψmwm|Im-[|Ψ|2+A2+2R(Rm*Ψ)]|2.
w(k, l)=βn(k/s)βn(l/s),with s=(L-1)/(n+1),
Ψ=Z/A,
U=|Z|2/A2+A2.
arg minU,Zmwm|Im-U-2R(VmZ)|2,
Ψ=Z/A,
A±=+U±U2-4|Z|221/2.
mwm[Im-U-2R(VmZ)]=0,mwmVm[Im-U-2R(VmZ)]=0,mwmVm*[Im-U-2R(VmZ)]=0,
mwmIm=U+2RZmwmVm,mwmVmIm=UmwmVm+ZmwmVm2+Z*,mwmVm*Im=UmwmVm*+Z+Z*mwmVm*2.
1αα*αβ1α*1β*UZZ*=mwmImmwmVmImmwmVm*Im.
R(x, z)=exp[i(kxx+kyy+kzz)],
kx<π/T,    ky<π/T.
sin(θmax)=(kx2+ky2)1/22π/λλ2T.
ΨI(x)=ΨxMexpiπx2Mλf,
I(x)=A(x)exp[i(kxx+kyy)]+ΨxMexpiπx2Mλf2,
I(x)=A(x)expi1Dx-xc2+C+ΨxM2,
L(x, y)=  L_if K(x, y)>2π/L_  L¯if K(x, y)<2π/L¯2π/K(x, y)otherwise,
ψI(x)=ψxMexpiπx2Mλf.
ψ˜(x)=R(-d)ψ·Mexpiπ·2Mλf(x)=Ψ(x)expix2D,
Ψ(x)=R(-d)ψ·M(x),
d=d-d2Mf,
D=λ(Mf-d)π,
M=Mf-df.
I(x)=A(x)exp[i(kxx+kyy)]+Ψ(x)expi1Dx22,
ψxM=R(d){Ψ}(x).

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