Abstract

This paper discusses convolution algorithms to reconstruct off-axis digital holograms. The problem of convolution is addressed by considering the spatial spectral properties of digital holograms, especially the unusual localization property of the Fourier spectrum of the hologram, in regard to the physical object space. After deriving the sampling requirements for the transfer functions, three approaches are considered with the concept of spatial bandwidth extension: zero padding, spectrum scanning, and adjustable magnification. The theoretical discussion is completed by experimental illustrations that enable the algorithms to be objectively compared.

© 2012 Optical Society of America

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2012

2011

P. Tankam and P. Picart, “Use of digital color holography for crack investigation in electronic components,” Opt. Lasers Eng. 49, 1335–1342 (2011).
[CrossRef]

N. Verrier and M. Atlan, “Off-axis digital hologram reconstruction: some practical considerations,” Appl. Opt. 50, H136–H146 (2011).
[CrossRef]

2010

2009

2008

J. Zhao, H. Jiang, and J. Di, “Recording and reconstruction of a color holographic image by using digital lensless Fourier transform holography,” Opt. Express 16, 2514–2519 (2008).
[CrossRef]

J. M. Desse, P. Picart, and P. Tankam, “Digital three-color holographic interferometry for flow analysis,” Opt. Express 16, 5471–5480 (2008).
[CrossRef]

A. Wada, M. Kato, and Y. Ishii, “Multiple-wavelength digital holographic interferometry using tunable laser diodes,” Appl. Opt. 47, 2053–2060 (2008).
[CrossRef]

A. Wada, M. Kato, and Y. Ishii, “Large step-height measurements using multiple-wavelength holographic interferometry with tunable laser diodes,” J. Opt. Soc. Am. A 25, 3013–3020 (2008).
[CrossRef]

U. P. Kumar, B. Bhaduri, N. K. Mohan, M. P. Kothiyal, and A. K. Asundi, “Microscopic TV holography for MEMS deflection and 3-D surface profile characterization,” Opt. Lasers Eng. 46, 687–694 (2008).
[CrossRef]

C. J. Mann, P. R. Bingham, V. C. Paquit, and K. W. Tobin, “Quantitative phase imaging by three-wavelength digital holography,” Opt. Express 16, 9753–9764 (2008).
[CrossRef]

A. Khmaladze, M. Kim, and C.-M. Lo, “Phase imaging of cells by simultaneous dual wavelength reflection digital holography,” Opt. Express 16, 10900–10911 (2008).
[CrossRef]

Y. Awatsuji, T. Tahara, A. Kaneko, T. Koyama, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Parallel two-step phase-shifting digital holography,” Appl. Opt. 47, D183–D189 (2008).
[CrossRef]

P. Picart and J. Leval, “General theoretical formulation of image formation in digital Fresnel holography,” J. Opt. Soc. Am. A 25, 1744–1761 (2008).
[CrossRef]

2007

2006

2005

2004

2003

2002

I. Yamaguchi, T. Matsumura, and J. Kato, “Phase shifting color digital holography,” Opt. Lett. 27, 1108–1110 (2002).
[CrossRef]

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

Th. Kreis, “Frequency analysis of digital holography with reconstruction by convolution,” Opt. Eng. 41, 1829–1839 (2002).
[CrossRef]

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

2001

2000

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wave front reconstruction and multi-wavelength contouring,” Opt. Eng. 39, 79–85(2000).
[CrossRef]

1999

1997

Th. Kreis, M. Adams, and W. Jüptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224–233 (1997).
[CrossRef]

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

1994

1993

Adams, M.

Th. Kreis, M. Adams, and W. Jüptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224–233 (1997).
[CrossRef]

Alfieri, D.

Asundi, A.

Asundi, A. K.

U. P. Kumar, B. Bhaduri, N. K. Mohan, M. P. Kothiyal, and A. K. Asundi, “Microscopic TV holography for MEMS deflection and 3-D surface profile characterization,” Opt. Lasers Eng. 46, 687–694 (2008).
[CrossRef]

Atlan, M.

Awatsuji, Y.

Bernardo, L. M.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Bhaduri, B.

U. P. Kumar, B. Bhaduri, N. K. Mohan, M. P. Kothiyal, and A. K. Asundi, “Microscopic TV holography for MEMS deflection and 3-D surface profile characterization,” Opt. Lasers Eng. 46, 687–694 (2008).
[CrossRef]

Bingham, P. R.

Blun, T.

Charriere, F.

Colomb, T.

Coppola, G.

Cuche, E.

De Nicola, S.

Demoli, N.

Depeursinge, C.

Desse, J. M.

Di, J.

Emery, Y.

Ferraro, P.

Ferreira, C.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Finizio, A.

Fröning, P.

Fu, Y.

J. C. Li, Z. Peng, and Y. Fu, “Diffraction transfer function and its calculation of classic diffraction formula,” Opt. Commun. 280, 243–248 (2007).
[CrossRef]

Garcia, J.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Garcia-Sucerquia, J.

Goodman, J. W.

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

Guo, C. S.

C. S. Guo, L. Zhang, Z. Y. Rong, and H. T. Wang, “Effect of the fill factor of CCD pixels on digital holograms: comment on the paper,” Opt. Eng. 42, 2768–2772 (2003).
[CrossRef]

Guo, Z.

Gusev, M. E.

Ida, T.

Ishii, Y.

Ito, K.

T. Kakue, K. Ito, Y. Awatsuji, K. Nishio, T. Kubota, and O. Maoba, “Parallel phase-shifting digital holography capable of simultaneously capturing visible and invisible three-dimensional information,” J. Disp. Technol. 6, 472–478(2010).
[CrossRef]

T. Kakue, T. Tahara, K. Ito, Y. Shimozato, Y. Awatsuji, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Parallel phase-shifting color digital holography using two phase shifts,” Appl. Opt. 48, H244–H250 (2009).
[CrossRef]

Jericho, M.

Jericho, S.

Jiang, H.

Jüptner, W.

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

C. Wagner, S. Seebacher, W. Osten, and W. Jüptner, “Digital recording and numerical reconstruction of lens less Fourier holograms in optical metrology,” Appl. Opt. 38, 4812–4820 (1999).
[CrossRef]

Th. Kreis, M. Adams, and W. Jüptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224–233 (1997).
[CrossRef]

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

Kakue, T.

T. Kakue, K. Ito, Y. Awatsuji, K. Nishio, T. Kubota, and O. Maoba, “Parallel phase-shifting digital holography capable of simultaneously capturing visible and invisible three-dimensional information,” J. Disp. Technol. 6, 472–478(2010).
[CrossRef]

T. Kakue, T. Tahara, K. Ito, Y. Shimozato, Y. Awatsuji, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Parallel phase-shifting color digital holography using two phase shifts,” Appl. Opt. 48, H244–H250 (2009).
[CrossRef]

Kaneko, A.

Karray, M.

Kato, J.

Kato, M.

Khmaladze, A.

Kim, M.

Kim, M. K.

Klages, P.

Kothiyal, M. P.

U. P. Kumar, B. Bhaduri, N. K. Mohan, M. P. Kothiyal, and A. K. Asundi, “Microscopic TV holography for MEMS deflection and 3-D surface profile characterization,” Opt. Lasers Eng. 46, 687–694 (2008).
[CrossRef]

Koyama, T.

Kreis, Th.

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

Th. Kreis, “Frequency analysis of digital holography with reconstruction by convolution,” Opt. Eng. 41, 1829–1839 (2002).
[CrossRef]

Th. Kreis, M. Adams, and W. Jüptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224–233 (1997).
[CrossRef]

Kreuzer, H. J.

Kubota, T.

Kuhn, J.

Kumar, U. P.

U. P. Kumar, B. Bhaduri, N. K. Mohan, M. P. Kothiyal, and A. K. Asundi, “Microscopic TV holography for MEMS deflection and 3-D surface profile characterization,” Opt. Lasers Eng. 46, 687–694 (2008).
[CrossRef]

Leval, J.

Li, J. C.

Liebling, M.

Lo, C.-M.

Mann, C. J.

Maoba, O.

T. Kakue, K. Ito, Y. Awatsuji, K. Nishio, T. Kubota, and O. Maoba, “Parallel phase-shifting digital holography capable of simultaneously capturing visible and invisible three-dimensional information,” J. Disp. Technol. 6, 472–478(2010).
[CrossRef]

Marinho, F.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Marquet, P.

Mas, D.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Matoba, O.

Matsumura, T.

Mia, J.

Mizuno, J.

Mohan, N. K.

U. P. Kumar, B. Bhaduri, N. K. Mohan, M. P. Kothiyal, and A. K. Asundi, “Microscopic TV holography for MEMS deflection and 3-D surface profile characterization,” Opt. Lasers Eng. 46, 687–694 (2008).
[CrossRef]

Montfort, F.

Mounier, D.

Nishio, K.

Ohta, S.

Onural, L.

Osten, W.

Paquit, V. C.

Pedrini, G.

Peng, X.

Peng, Z.

J. C. Li, Z. Peng, and Y. Fu, “Diffraction transfer function and its calculation of classic diffraction formula,” Opt. Commun. 280, 243–248 (2007).
[CrossRef]

Picart, P.

Pierattini, G.

Restrepo, J. F.

Rong, Z. Y.

C. S. Guo, L. Zhang, Z. Y. Rong, and H. T. Wang, “Effect of the fill factor of CCD pixels on digital holograms: comment on the paper,” Opt. Eng. 42, 2768–2772 (2003).
[CrossRef]

Schnars, U.

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

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

Seebacher, S.

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wave front reconstruction and multi-wavelength contouring,” Opt. Eng. 39, 79–85(2000).
[CrossRef]

C. Wagner, S. Seebacher, W. Osten, and W. Jüptner, “Digital recording and numerical reconstruction of lens less Fourier holograms in optical metrology,” Appl. Opt. 38, 4812–4820 (1999).
[CrossRef]

Shimozato, Y.

Song, Q.

Tahara, T.

Tankam, P.

Tiziani, H. J.

Tobin, K. W.

Torzynski, M.

Unser, M.

Ura, S.

Verrier, N.

Vukicevic, D.

Wada, A.

Wagner, C.

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wave front reconstruction and multi-wavelength contouring,” Opt. Eng. 39, 79–85(2000).
[CrossRef]

C. Wagner, S. Seebacher, W. Osten, and W. Jüptner, “Digital recording and numerical reconstruction of lens less Fourier holograms in optical metrology,” Appl. Opt. 38, 4812–4820 (1999).
[CrossRef]

Wang, H. T.

C. S. Guo, L. Zhang, Z. Y. Rong, and H. T. Wang, “Effect of the fill factor of CCD pixels on digital holograms: comment on the paper,” Opt. Eng. 42, 2768–2772 (2003).
[CrossRef]

Xu, L.

Xu, W.

Yamaguchi, I.

Yamashita, K.

Yaroslavsky, L. P.

Yokota, M.

Yu, L.

Zhang, F.

Zhang, L.

C. S. Guo, L. Zhang, Z. Y. Rong, and H. T. Wang, “Effect of the fill factor of CCD pixels on digital holograms: comment on the paper,” Opt. Eng. 42, 2768–2772 (2003).
[CrossRef]

Zhang, T.

Zhang, Y.

Zhao, J.

Appl. Opt.

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

T. Kakue, T. Tahara, K. Ito, Y. Shimozato, Y. Awatsuji, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Parallel phase-shifting color digital holography using two phase shifts,” Appl. Opt. 48, H244–H250 (2009).
[CrossRef]

I. Yamaguchi, T. Ida, M. Yokota, and K. Yamashita, “Surface shape measurement by phase-shifting digital holography with a wavelength shift,” Appl. Opt. 45, 7610–7616 (2006).
[CrossRef]

A. Wada, M. Kato, and Y. Ishii, “Multiple-wavelength digital holographic interferometry using tunable laser diodes,” Appl. Opt. 47, 2053–2060 (2008).
[CrossRef]

G. Pedrini, P. Fröning, H. J. Tiziani, and M. E. Gusev, “Pulsed digital holography for high-speed contouring that uses a two-wavelength method,” Appl. Opt. 38, 3460–3467 (1999).
[CrossRef]

P. Tankam, P. Picart, D. Mounier, J. M. Desse, and J. C. Li, “Method of digital holographic recording and reconstruction using a stacked color image sensor,” Appl. Opt. 49, 320–328 (2010).
[CrossRef]

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

N. Verrier and M. Atlan, “Off-axis digital hologram reconstruction: some practical considerations,” Appl. Opt. 50, H136–H146 (2011).
[CrossRef]

Y. Awatsuji, T. Tahara, A. Kaneko, T. Koyama, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Parallel two-step phase-shifting digital holography,” Appl. Opt. 47, D183–D189 (2008).
[CrossRef]

Y. Zhang, G. Pedrini, W. Osten, and H. J. Tiziani, “Image reconstruction for in-line holography with the Yang-Gu algorithm,” Appl. Opt. 42, 6452–6457 (2003).
[CrossRef]

J. Garcia-Sucerquia, W. Xu, S. Jericho, P. Klages, M. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. 45, 836–850 (2006).
[CrossRef]

C. Wagner, S. Seebacher, W. Osten, and W. Jüptner, “Digital recording and numerical reconstruction of lens less Fourier holograms in optical metrology,” Appl. Opt. 38, 4812–4820 (1999).
[CrossRef]

J. F. Restrepo and J. Garcia-Sucerquia, “Magnified reconstruction of digitally recorded holograms by Fresnel-Bluestein transform,” Appl. Opt. 49, 6430–6435 (2010).
[CrossRef]

J. Disp. Technol.

T. Kakue, K. Ito, Y. Awatsuji, K. Nishio, T. Kubota, and O. Maoba, “Parallel phase-shifting digital holography capable of simultaneously capturing visible and invisible three-dimensional information,” J. Disp. Technol. 6, 472–478(2010).
[CrossRef]

J. Opt. Soc. Am. A

Meas. Sci. Technol.

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

Opt. Commun.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

J. C. Li, Z. Peng, and Y. Fu, “Diffraction transfer function and its calculation of classic diffraction formula,” Opt. Commun. 280, 243–248 (2007).
[CrossRef]

Opt. Eng.

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

Th. Kreis, “Frequency analysis of digital holography with reconstruction by convolution,” Opt. Eng. 41, 1829–1839 (2002).
[CrossRef]

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Figures (10)

Fig. 1.
Fig. 1.

Hologram spectrum and adaptation of spatial bandwidth of the convolution kernel: (a) hologram spectrum, (b) Fourier transform of the impulse response, (c) spatial frequency bandwidth adaptation, and (d) zero order due to the spherical wavefront.

Fig. 2.
Fig. 2.

(a) Discrete Fresnel transform and (b) experimental spatial frequency spectrum.

Fig. 3.
Fig. 3.

Reconstruction with convolution: (a) Fourier transform of impulse response, (b) angular spectrum transfer function, (c) object zone reconstructed with impulse response, and (d) object zone reconstructed with angular spectrum transfer function.

Fig. 4.
Fig. 4.

Anisotropic reconstruction with convolution: (a) Fourier transform of impulse response, (b) angular spectrum transfer function, (c) object zone reconstructed with impulse response, and (d) object zone reconstructed with angular spectrum transfer function.

Fig. 5.
Fig. 5.

Principle of the spectrum scanning algorithm: (a) hologram spectrum paving (white square line), (b) parcel obtained with the central filter, and (c) full-field object reconstructed with adjacent parcels.

Fig. 6.
Fig. 6.

Reconstructed object using spectrum scanning: (a) with the impulse response, (b) with the angular spectrum, and (c) highlight of the edge effects leading to aliasing.

Fig. 7.
Fig. 7.

Overlapping of transfer functions of the filter bank: (a) profiles of three adjacent filters and (b) overlapping of the spectral bandwidth of the filter bank.

Fig. 8.
Fig. 8.

Full object reconstructed with overlapping of the transfer functions: (a) object zone reconstructed with impulse response and (b) object zone reconstructed with angular spectrum transfer function.

Fig. 9.
Fig. 9.

Reconstructions with adjustable magnification: (a) hologram spectrum and spatial bandwidth limits, (b) angular spectrum transfer function, and (c) reconstructed object.

Fig. 10.
Fig. 10.

Comparison of the reconstructed algorithms.

Tables (2)

Tables Icon

Table 1. Values of the Spatial Frequencies for the Filter Bank

Tables Icon

Table 2. Values of the Reconstruction Parameters for K=L=2048 Data Points

Equations (39)

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H=|R|2+|O|2+R*O+RO*,
O(x,y,d0)=(A*h)(x,y),
h(x,y,d0)=id0λexp[2iπ/λd02+x2+y2]d02+x2+y2.
Ar(x,y,dr)=(H*h)(x,y).
Ar=FT1[FT[H]×FT[h]],
Ar=FT1[FT[H]×G].
G(u,v,dr)=exp[2iπdrλ1λ2u2λ2v2].
Ar(x,y,dr)=iλdrexp(2iπdrλ)exp[iπλdr(x2+y2)]×++H(x,y)exp[iπλdr(x2+y2)]exp[2iπλdr(xx+yy)]dxdy.
limdriλdrexp(2iπdr/λ)Ar=FT[H].
{ui=12πΘhx=xλdr2+x2+y2xλdrvi=12πΘhy=yλdr2+x2+y2yλdr.
Δuh×Δvh=Lpxλdr×Kpyλdr.
drsup{Lpx2λ,Kpy2λ}.
{Lλdrpx2Kλdrpy2.
{Tix=12πΘGu=12πuπλdr(u2+v2)=λdruTiy=12πΘGv=12πvπλdr(u2+v2)=λdrv.
{ΔTx=2Tixmax=2λdrumax=λdrpxΔTy=2Tiymax=2λdrvmax=λdrpy.
{δu=1Lpxpxλdrδv=1Kpypyλdr.
{Lλdrpx2Kλdrpy2.
{Tixmax=λd0umax=12ΔAxTiymax=λd0vmax=12ΔAy.
{LΔAxpxKΔAypy.
FT[H](u,v)=C0(u,v)+C1(uu0,vv0)+C1*(uu0,vv0).
Δuobject×Δvobject=(ΔAxλdr)×(ΔAyλdr).
Δukernel×ΔvkernelΔuobject×Δvobject=(ΔAxλdr)×(ΔAyλdr).
Δukernel=Lpxλdr=Δuobject=ΔAxλdr,
hkernel(x,y,dr)=h(x,y,dr)×exp[+2iπ(u0x+v0y)].
Gkernel(u,v,dr)={G(uu0,vv0,dr)if|uu0|Lpx/2λdrand|vv0|Kpy/2λdr0elsewhere.
{nx=ΔuobjectΔukernel=(ΔAxλdr)/(Lpxλdr)=ΔAxLpxny=ΔvobjectΔvkernel=(ΔAyλdr)/(Kpxλdr)=ΔAyKpx.
hpq(x,y,dr)=h(x,y,dr)×exp[+2iπ(upx+vqy)].
FT[hpq]=FT[h](uup,vvq).
Gpq(u,v,dr)={G(uup,vvq,dr)if|uup|Lpx/2λdrand|vvq|Kpy/2λdr0elsewhere.
1dr=1ds1d0+1Rc.
γ=drd0.
Δuobject=ΔAxλd0=γΔAxλdr=Lpxλdr=Δukernel.
Δuw×Δvw=Npxλ|Rc|×Mpyλ|Rc|.
{Npx2λ|Rc|+Lpx2λ|dr|<|u0|Mpy2λ|Rc|+Kpy2λ|dr|<|v0|.
{(1γ)Npx2λγd0+Lpx2λγd0<|u0|(1γ)Mpy2λγd0+Kpy2λγd0<|v0|.
max{(L+N)px2λd0|u0|+Npx,(K+M)py2λd0|v0|+Mpy}<γ<min{LpxΔAx,KpyΔAy}.
hkernel(x,y,dr)={h(x,y,dr)×exp[+2iπ(u0x+v0y)]ifx2+y2γ2ΔA2/40elsewhere.
Gkernel(u,v,dr)={G(uu0,vv0,dr)if(uu0)2+(vv0)2γ2ΔA2/40elsewhere.
{ρx=λdrNpx=γλd0Npx=γρxρy=λdrMpy=γλd0Mpy=γρy,

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