Abstract

Holographic rendering of off-axis intensity digital holograms is discussed. A review of some of the main numerical processing methods, based either on the Fourier transform interpretation of the propagation integral or on its linear system counterpart, is reported. Less common methods such as adjustable magnification reconstruction schemes and Fresnelet decomposition are presented and applied to the digital treatment of off-axis holograms. The influence of experimental parameters on the classical hologram reconstruction methods is assessed, offering guidelines for optimal image rendering regarding the hologram recording conditions.

© 2011 Optical Society of America

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2011 (3)

2010 (6)

2009 (6)

2008 (6)

2007 (4)

2006 (8)

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]

I. Yamaguchi, K. Yamamoto, G. A. Mills, and M. Yokota, “Image reconstruction only by phase data in phase-shifting holography,” Appl. Opt. 45, 975–983 (2006).
[CrossRef] [PubMed]

A. Asundi and V. R. Singh, “Time-averaged in-line digital holographic interferometry for vibration analysis,” Appl. Opt. 45, 2391–2395 (2006).
[CrossRef] [PubMed]

F. Charrière, N. Pavillon, T. Colomb, C. Depeursinge, T. J. Heger, E. A. D. Mitchell, P. Marquet, and B. Rappaz, “Living specimen tomography by digital holographic microscopy: morphometry of testate amoeba,” Opt. Express 14, 7005–7013(2006).
[CrossRef] [PubMed]

T. Colomb, F. Montfort, J. Kühn, N. Aspert, E. Cuche, A. Marian, F. Charrière, S. Bourquin, P. Marquet, and C. Depeursinge, “Numerical parametric lens for shifting, magnification, and complete aberration compensation in digital holographic microscopy,” J. Opt. Soc. Am. A 23, 3177–3190(2006).
[CrossRef]

U. Iemma, L. Morino, and M. Diez, “Digital holography and Karhunen-Loève decomposition for the modal analysis of two-dimensional vibrating structures,” J. Sound Vib. 291, 107–131 (2006).
[CrossRef]

M. Atlan and M. Gross, “Laser Doppler imaging, revisited,” Rev. Sci. Instrum. 77, 116103 (2006).
[CrossRef]

E. Darakis and J. J. Soraghan, “Use of Fresnelets for phase-shift digital hologram compression,” IEEE Trans. Image Process. 15, 3804–3811 (2006).
[CrossRef] [PubMed]

2005 (6)

2004 (6)

2003 (3)

M. Liebling, T. Blu, and M. Unser, “Non-linear Fresnelet approximation for interference term suppression in digital holography,” Proc. SPIE 5207, 553–559 (2003).
[CrossRef]

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

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]

2002 (1)

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

2001 (2)

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

W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. USA 98, 11301–11305 (2001).
[CrossRef] [PubMed]

2000 (4)

1999 (1)

A. Lozano, J. Kostas, and J. Soria, “Use of holography in particle image velocimetry measurements of a swirling flow,” Exp. Fluids 27, 251–261 (1999).
[CrossRef]

1997 (1)

1994 (1)

1993 (1)

M. Unser, A. Aldroubi, and M. Eden, “A family of polynomial spline wavelet transforms,” Signal Process. 30, 141–162 (1993).
[CrossRef]

1992 (1)

1987 (2)

G. Liu and P. D. Scott, “Phase retrieval and twin-image elimination for in-line Fresnel holograms,” J. Opt. Soc. Am. A 4, 159–165 (1987).
[CrossRef]

L. Onural and P. D. Scott, “Digital recording of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).

1982 (1)

1972 (1)

M. A. Kronrod, N. S. Merzlyakov, and L. P. Yaroslavsky, “Reconstruction of holograms with a computer,” Sov. Phys. Tech. Phys. 17, 419–420 (1972).

1971 (1)

1970 (1)

L. Bleustein, “Linear filtering approach to the computation of the discrete Fourier transform,” IEEE Trans. Audio Electroacoust. 18, 451–455 (1970).
[CrossRef]

1967 (1)

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

1965 (2)

J. W. Cooley and J. W. Tukey, “An algorithm for the machine computation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

R. L. Powell and K. A. Stetson, “Interferometric vibration analysis by wavefront reconstruction,” J. Opt. Soc. Am. 55, 1593–1597 (1965).
[CrossRef]

1963 (1)

1962 (1)

1949 (1)

D. Gabor, “Microscopy by reconstructed wave-fronts,” Proc. R. Soc. Lond. A 197, 454–487 (1949).
[CrossRef]

1948 (1)

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

Absil, E.

Ahrenberg, L.

L. Ahrenberg, A. J. Page, B. M. Hennelly, J. B. McDonald, and T. J. Naughton, “Using commodity graphics hardware for real-time digital hologram view-reconstruction,” IEEE J. Display Technol. 5, 111–119 (2009).
[CrossRef]

Aldroubi, A.

M. Unser, A. Aldroubi, and M. Eden, “A family of polynomial spline wavelet transforms,” Signal Process. 30, 141–162 (1993).
[CrossRef]

Aleksoff, C. C.

Alfieri, D.

Angelini, E.

Aspert, N.

Asundi, A.

Atlan, M.

Benkouider, A.

Bleustein, L.

L. Bleustein, “Linear filtering approach to the computation of the discrete Fourier transform,” IEEE Trans. Audio Electroacoust. 18, 451–455 (1970).
[CrossRef]

Blu, T.

M. Liebling, T. Blu, and M. Unser, “Complex-wave retrieval from a single off-axis hologram,” J. Opt. Soc. Am. A 21, 367–377 (2004).
[CrossRef]

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

M. Liebling, T. Blu, and M. Unser, “Non-linear Fresnelet approximation for interference term suppression in digital holography,” Proc. SPIE 5207, 553–559 (2003).
[CrossRef]

Boileau, J-P.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Borza, D.

D. Borza, “Mechanical vibration measurement by high-resolution time-averaged digital holography,” Meas. Sci. Technol. 16, 1853–1864 (2005).
[CrossRef]

Bourquin, S.

Brunel, M.

Callens, N.

Charrière, F.

Coëtmellec, S.

Colomb, T.

Cooley, J. W.

J. W. Cooley and J. W. Tukey, “An algorithm for the machine computation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Coppola, G.

Cuche, E.

Darakis, E.

E. Darakis, T. J. Naughton, and J. J. Soraghan, “Compression defect in different reconstructions from phase-shifting digital holographic data,” Appl. Opt. 46, 4579–4586(2007).
[CrossRef] [PubMed]

E. Darakis and J. J. Soraghan, “Use of Fresnelets for phase-shift digital hologram compression,” IEEE Trans. Image Process. 15, 3804–3811 (2006).
[CrossRef] [PubMed]

De Nicola, S.

Denis, L.

Depeursinge, C.

Desse, J-M.

Diez, M.

U. Iemma, L. Morino, and M. Diez, “Digital holography and Karhunen-Loève decomposition for the modal analysis of two-dimensional vibrating structures,” J. Sound Vib. 291, 107–131 (2006).
[CrossRef]

Dubois, F.

Eden, M.

M. Unser, A. Aldroubi, and M. Eden, “A family of polynomial spline wavelet transforms,” Signal Process. 30, 141–162 (1993).
[CrossRef]

Ferraro, P.

Fienup, J. R.

Finizio, A.

Fournier, C.

Gabor, D.

D. Gabor, “Microscopy by reconstructed wave-fronts,” Proc. R. Soc. Lond. A 197, 454–487 (1949).
[CrossRef]

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

Garcia-Sucerquia, J.

Goepfert, C.

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]

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005).

Gougeon, S.

Gross, M.

Hare, J.

Heger, T. J.

Hennelly, B.

B. Hennelly, D. Kelly, N. Pandey, and D. Monaghan, “Zooming algorithms for digital holography,” J. Phys. Conf. Ser. 206, 012027 (2010).
[CrossRef]

Hennelly, B. M.

L. Ahrenberg, A. J. Page, B. M. Hennelly, J. B. McDonald, and T. J. Naughton, “Using commodity graphics hardware for real-time digital hologram view-reconstruction,” IEEE J. Display Technol. 5, 111–119 (2009).
[CrossRef]

Hoyos, M.

Ichihashi, Y.

T. Shimobaba, N. Masuda, Y. Ichihashi, and T. Ito, “Real-time digital holographic microscopy observable in multi-view and multi-resolution,” J. Opt. 12, 065402 (2010).
[CrossRef]

Iemma, U.

U. Iemma, L. Morino, and M. Diez, “Digital holography and Karhunen-Loève decomposition for the modal analysis of two-dimensional vibrating structures,” J. Sound Vib. 291, 107–131 (2006).
[CrossRef]

Ito, T.

T. Shimobaba, N. Masuda, Y. Ichihashi, and T. Ito, “Real-time digital holographic microscopy observable in multi-view and multi-resolution,” J. Opt. 12, 065402 (2010).
[CrossRef]

T. Shimobaba, Y. Sato, J. Miura, M. Takenouchi, and T. Ito, “Real-time digital holographic microscopy using the graphic processing unit,” Opt. Express 16, 11776–11781(2008).
[CrossRef] [PubMed]

Janssen, A. J. E. M.

Jericho, M. H.

W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. USA 98, 11301–11305 (2001).
[CrossRef] [PubMed]

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[CrossRef]

Signal Process. (1)

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[CrossRef]

Sov. Phys. Tech. Phys. (1)

M. A. Kronrod, N. S. Merzlyakov, and L. P. Yaroslavsky, “Reconstruction of holograms with a computer,” Sov. Phys. Tech. Phys. 17, 419–420 (1972).

Other (3)

L. P. Yaroslvsky and N. S. Merzlyakov, Methods of Digital Holography (Springer, 1980).

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M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

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Figures (8)

Fig. 1
Fig. 1

(a) Hologram recording in off-axis configuration, (b) spatial frequency representation of off-axis holograms.

Fig. 2
Fig. 2

Angular acceptance of digital holographic reconstruction process. Solid lines are associated with the single-FFT reconstruction, and dashed lines correspond to the convolution approaches.

Fig. 3
Fig. 3

Reconstruction of a hologram for γ = 0.8 , 1 , 2.5 × γ 0 . (a), (c), (e) Quadratic lens method, (b), (d), (f) Fresnel–Bluestein method.

Fig. 4
Fig. 4

Illustrations of alias and replica phenomena. (a) Reconstruction with γ = 0.5 × γ 0 , (c) reconstruction with γ = 4 × γ 0 , (d) same as (a) with replica removal, (f) same as (c) with alias filtering, (b), (e) reconstruction with γ = γ 0 .

Fig. 5
Fig. 5

(a) Synoptics of the antialias procedure, (b) replica removal scheme.

Fig. 6
Fig. 6

Experimental procedure for the holographic reconstruction benchmarking.

Fig. 7
Fig. 7

Holographic reconstructions of the USAF target located at different distances.

Fig. 8
Fig. 8

Fresnelet decomposition of the hologram recorded for Δ ξ > Δ x . (a) Fresnelet coefficients computed within the single-FFT scheme, (b) Fresnelet coefficients computed within the three-FFT scheme, (c) hologram reconstruction from (a), (d) hologram reconstruction from (b).

Equations (34)

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E ( x , y ) = | R ( x , y ) | 2 + | O ( x , y ) | 2 + O * ( x , y ) R ( x , y ) + O ( x , y ) R * ( x , y ) ,
α max λ 2 Δ x ,
E rec ( ξ , η ) = i z λ R 2 E ( x , y ) exp ( i k r ) r d x d y .
r = z 2 + ( x ξ ) 2 + ( y η ) 2 ,
r = z [ 1 + 1 2 ( x ξ z ) 2 + 1 2 ( y η z ) 2 ] ,
E rec ( ξ , η ) = exp ( i 2 π λ z ) i λ z R 2 E ( x , y ) exp { i π λ z [ ( x ξ ) 2 + ( y η ) 2 ] } d x d y .
E rec ( p ) = exp ( i 2 π λ z ) i λ z exp ( i π λ z p 2 Δ ξ 2 ) n = 0 N 1 E ( n ) exp ( i π λ z n 2 Δ x 2 ) exp ( i 2 π λ z n p Δ x Δ ξ ) ,
Δ ξ = λ z N Δ x .
E rec ( p ) = exp ( i 2 π λ z ) i λ z exp ( i π λ z p 2 N 2 Δ x 2 ) n = 0 N 1 E ( n ) exp ( i π λ z n 2 Δ x 2 ) exp ( i 2 π n p N ) .
E rec ( ξ ) = exp ( i 2 π λ z ) i λ z exp ( i π λ z p 2 N 2 Δ x 2 ) F { E ( x ) exp ( i π λ z x 2 ) } .
h z ( x ) = exp ( i π λ z x 2 ) .
E rec ( ξ ) = exp ( i 2 π λ z ) i λ z F 1 [ F { E ( x ) } × F { h z ( x ) } ] ,
H ( u ) exp [ 2 i π z λ ( 1 1 2 λ 2 u 2 ) ] ,
E rec ( ξ ) = 1 i λ z F 1 [ F { E ( x ) } × H ( u ) ] .
L ( x ) = exp ( i π λ R c x 2 ) ,
R c = γ z γ 1 .
E rec ( ξ ) = exp ( i 2 π λ z ) i λ z F 1 [ F { E ( x ) L ( x ) } × F { h z ( x ) } ] ,
E rec ( ξ ) = exp ( i 2 π λ z ) i λ z F 1 [ F { E ( x ) L ( x ) } × H ( u ) ] ,
E rec ( p ) = exp ( i 2 π λ z ) i λ z exp [ i π λ z Δ ξ ( Δ x Δ ξ ) p 2 ] n = 0 N E ( n ) exp [ i π λ z Δ x ( Δ x Δ ξ ) n 2 ] × exp [ i π λ z Δ x Δ ξ ( p n ) 2 ] .
E rec ( p ) = exp ( i 2 π λ z ) i λ z exp [ i π λ z γ ( 1 γ ) Δ x 2 p 2 ] n = 0 N E ( n ) exp [ i π λ z ( 1 γ ) n 2 Δ x 2 ] × exp [ i π λ z γ ( p n ) 2 Δ x 2 ] .
f ( n ) = E ( n ) exp [ i π λ z ( 1 γ ) n 2 Δ x 2 ]
g ( n ) = exp ( i π λ z γ n 2 Δ x 2 ) .
E rec ( ξ ) = exp ( i 2 π λ z ) i λ z exp [ i π λ z γ ( 1 γ ) Δ x 2 p 2 ] F 1 [ F { f ( x ) } × F { g ( x ) } ] .
γ < γ 0
γ > γ 0
C ( x ) = exp ( i π λ z x 2 ) .
β n ( x ) = β 0 * * β 0 n + 1 ( x ) ,
β 0 ( x ) = { 1 , 0 < x < 1 1 2 , x = 0     or     x = 1 0 , otherwise ,
β n ( x 2 ) = k Z h ( k ) β n ( x k ) .
{ ψ j , k n = 2 j 2 ψ n ( 2 j x k ) } j , k Z ,
ψ n ( x 2 ) = k Z g ( k ) β n ( x k ) .
{ ψ ˜ j , k n = 2 j 2 ψ ˜ n ( 2 j x k ) } j , k Z ,
ψ ˜ n ( x 2 ) = k Z g ( k ) β ˜ n ( x k ) ,
β ˜ n ( x 2 ) = k Z h ( k ) β ˜ n ( x k ) .

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