Abstract

Methods of three-dimensional deconvolution with a point-spread function as frequently employed in optical microscopy to reconstruct true three-dimensional distribution of objects are extended to holographic reconstructions. Two such schemes have been developed and are discussed: an instant deconvolution using the Wiener filter as well as an iterative deconvolution routine. The instant 3d-deconvolution can be applied to restore the positions of volume-spread objects such as small particles. The iterative deconvolution can be applied to restore the distribution of complex and extended objects. Simulated and experimental examples are presented and demonstrate artifact and noise free three-dimensional reconstructions from a single two-dimensional holographic record.

© 2010 OSA

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  1. J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11(3), 77–79 (1967).
    [CrossRef]
  2. L. P. Yaroslavsky, and N. S. Merzlykaov, Methods of digital holography, (Consultants Bureau, New York, 1980).
  3. U. Schnars, and W. Jueptner, Digital holography. Digital hologram recording, numerical reconstruction, and related techniques, (Springer Berlin Heidelberg, 2010).
  4. P. Ferraro, S. Grilli, D. Alfieri, S. De Nicola, A. Finizio, G. Pierattini, B. Javidi, G. Coppola, and V. Striano, “Extended focused image in microscopy by digital Holography,” Opt. Express 13(18), 6738–6749 (2005).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  7. J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999).
    [CrossRef] [PubMed]
  8. P. J. Shaw, “Comparison of wide-field/deconvolution and confocal microscopy for 3D imaging,” in Handbook of biological confocal microscopy, J. B. Pawley, ed., (Plenum Press, New York and London, 1995).
  9. J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999).
    [CrossRef] [PubMed]
  10. P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Process. Mag. 23(3), 32–45 (2006).
    [CrossRef]
  11. W. Wallace, L. H. Schaefer, and J. R. Swedlow, “A workingperson’s guide to deconvolution in light microscopy,” Biotechniques 31(5), 1076–1097 (2001).
    [PubMed]
  12. I. T. Young, “Image fidelity: Characterizing the imaging transfer function,” Methods Cell. Biol. 30, 2–47 (1989).
  13. T. M. Kreis, “Frequency analysis of digital holography with reconstruction by convolution,” Opt. Eng. 41(8), 1829–1839 (2002).
    [CrossRef]
  14. H. Meng and F. Hussain, “Holographic particle velocimetry: a 3D measurement technique for vortex interactions, coherent structures and turbulence,” Fluid Dyn. Res. 8(1-4), 33–52 (1991).
    [CrossRef]
  15. S. Murata and N. Yasuda, “Potential of digital holography in particle measurement,” Opt. Laser Technol. 32(7-8), 567–574 (2000).
    [CrossRef]
  16. R. Gold, “An iterative unfolding method for matrices,” Tech. Rep. ANL-6984 Argonne National Laboratory, Argonne, Illinois.
  17. J. W. Goodman, “Introduction to Fourier optics,” (Roberts & Company Publishers, Engelwood, Colorado) 2005, Chap. 3.
  18. S. A. Saq'an, A. S. Ayesh, A. M. Zihlif, E. Martuscelli, and G. Ragosta, “Physical properties of polystyrene/alum composites,” Polym. Test. 23(7), 739–745 (2004).
    [CrossRef]

2010

2006

P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Process. Mag. 23(3), 32–45 (2006).
[CrossRef]

J. Garcia-Sucerquia, W. B. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. 45(5), 836–850 (2006).
[CrossRef] [PubMed]

2005

2004

S. A. Saq'an, A. S. Ayesh, A. M. Zihlif, E. Martuscelli, and G. Ragosta, “Physical properties of polystyrene/alum composites,” Polym. Test. 23(7), 739–745 (2004).
[CrossRef]

2002

T. M. Kreis, “Frequency analysis of digital holography with reconstruction by convolution,” Opt. Eng. 41(8), 1829–1839 (2002).
[CrossRef]

2001

W. Wallace, L. H. Schaefer, and J. R. Swedlow, “A workingperson’s guide to deconvolution in light microscopy,” Biotechniques 31(5), 1076–1097 (2001).
[PubMed]

2000

S. Murata and N. Yasuda, “Potential of digital holography in particle measurement,” Opt. Laser Technol. 32(7-8), 567–574 (2000).
[CrossRef]

1999

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999).
[CrossRef] [PubMed]

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999).
[CrossRef] [PubMed]

1991

H. Meng and F. Hussain, “Holographic particle velocimetry: a 3D measurement technique for vortex interactions, coherent structures and turbulence,” Fluid Dyn. Res. 8(1-4), 33–52 (1991).
[CrossRef]

1989

I. T. Young, “Image fidelity: Characterizing the imaging transfer function,” Methods Cell. Biol. 30, 2–47 (1989).

1967

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11(3), 77–79 (1967).
[CrossRef]

Alfieri, D.

Ayesh, A. S.

S. A. Saq'an, A. S. Ayesh, A. M. Zihlif, E. Martuscelli, and G. Ragosta, “Physical properties of polystyrene/alum composites,” Polym. Test. 23(7), 739–745 (2004).
[CrossRef]

Conchello, J. A.

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999).
[CrossRef] [PubMed]

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999).
[CrossRef] [PubMed]

Cooper, J.

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999).
[CrossRef] [PubMed]

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999).
[CrossRef] [PubMed]

Coppola, G.

De Nicola, S.

Ferraro, P.

Finizio, A.

Garcia-Sucerquia, J.

Goodman, J. W.

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11(3), 77–79 (1967).
[CrossRef]

Grilli, S.

Hussain, F.

H. Meng and F. Hussain, “Holographic particle velocimetry: a 3D measurement technique for vortex interactions, coherent structures and turbulence,” Fluid Dyn. Res. 8(1-4), 33–52 (1991).
[CrossRef]

Javidi, B.

Jericho, M. H.

Jericho, S. K.

Karpova, T.

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999).
[CrossRef] [PubMed]

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999).
[CrossRef] [PubMed]

Klages, P.

Kreis, T. M.

T. M. Kreis, “Frequency analysis of digital holography with reconstruction by convolution,” Opt. Eng. 41(8), 1829–1839 (2002).
[CrossRef]

Kreuzer, H. J.

Lawrence, R. W.

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11(3), 77–79 (1967).
[CrossRef]

Martuscelli, E.

S. A. Saq'an, A. S. Ayesh, A. M. Zihlif, E. Martuscelli, and G. Ragosta, “Physical properties of polystyrene/alum composites,” Polym. Test. 23(7), 739–745 (2004).
[CrossRef]

McNally, J. G.

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999).
[CrossRef] [PubMed]

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999).
[CrossRef] [PubMed]

Meng, H.

H. Meng and F. Hussain, “Holographic particle velocimetry: a 3D measurement technique for vortex interactions, coherent structures and turbulence,” Fluid Dyn. Res. 8(1-4), 33–52 (1991).
[CrossRef]

Murata, S.

S. Murata and N. Yasuda, “Potential of digital holography in particle measurement,” Opt. Laser Technol. 32(7-8), 567–574 (2000).
[CrossRef]

Nehorai, A.

P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Process. Mag. 23(3), 32–45 (2006).
[CrossRef]

Panigrahi, P. K.

Pierattini, G.

Ragosta, G.

S. A. Saq'an, A. S. Ayesh, A. M. Zihlif, E. Martuscelli, and G. Ragosta, “Physical properties of polystyrene/alum composites,” Polym. Test. 23(7), 739–745 (2004).
[CrossRef]

Saq'an, S. A.

S. A. Saq'an, A. S. Ayesh, A. M. Zihlif, E. Martuscelli, and G. Ragosta, “Physical properties of polystyrene/alum composites,” Polym. Test. 23(7), 739–745 (2004).
[CrossRef]

Sarder, P.

P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Process. Mag. 23(3), 32–45 (2006).
[CrossRef]

Schaefer, L. H.

W. Wallace, L. H. Schaefer, and J. R. Swedlow, “A workingperson’s guide to deconvolution in light microscopy,” Biotechniques 31(5), 1076–1097 (2001).
[PubMed]

Singh, D. K.

Striano, V.

Swedlow, J. R.

W. Wallace, L. H. Schaefer, and J. R. Swedlow, “A workingperson’s guide to deconvolution in light microscopy,” Biotechniques 31(5), 1076–1097 (2001).
[PubMed]

Wallace, W.

W. Wallace, L. H. Schaefer, and J. R. Swedlow, “A workingperson’s guide to deconvolution in light microscopy,” Biotechniques 31(5), 1076–1097 (2001).
[PubMed]

Xu, W. B.

Yasuda, N.

S. Murata and N. Yasuda, “Potential of digital holography in particle measurement,” Opt. Laser Technol. 32(7-8), 567–574 (2000).
[CrossRef]

Young, I. T.

I. T. Young, “Image fidelity: Characterizing the imaging transfer function,” Methods Cell. Biol. 30, 2–47 (1989).

Zihlif, A. M.

S. A. Saq'an, A. S. Ayesh, A. M. Zihlif, E. Martuscelli, and G. Ragosta, “Physical properties of polystyrene/alum composites,” Polym. Test. 23(7), 739–745 (2004).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11(3), 77–79 (1967).
[CrossRef]

Biotechniques

W. Wallace, L. H. Schaefer, and J. R. Swedlow, “A workingperson’s guide to deconvolution in light microscopy,” Biotechniques 31(5), 1076–1097 (2001).
[PubMed]

Fluid Dyn. Res.

H. Meng and F. Hussain, “Holographic particle velocimetry: a 3D measurement technique for vortex interactions, coherent structures and turbulence,” Fluid Dyn. Res. 8(1-4), 33–52 (1991).
[CrossRef]

IEEE Signal Process. Mag.

P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Process. Mag. 23(3), 32–45 (2006).
[CrossRef]

Methods

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999).
[CrossRef] [PubMed]

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999).
[CrossRef] [PubMed]

Methods Cell. Biol.

I. T. Young, “Image fidelity: Characterizing the imaging transfer function,” Methods Cell. Biol. 30, 2–47 (1989).

Opt. Eng.

T. M. Kreis, “Frequency analysis of digital holography with reconstruction by convolution,” Opt. Eng. 41(8), 1829–1839 (2002).
[CrossRef]

Opt. Express

Opt. Laser Technol.

S. Murata and N. Yasuda, “Potential of digital holography in particle measurement,” Opt. Laser Technol. 32(7-8), 567–574 (2000).
[CrossRef]

Polym. Test.

S. A. Saq'an, A. S. Ayesh, A. M. Zihlif, E. Martuscelli, and G. Ragosta, “Physical properties of polystyrene/alum composites,” Polym. Test. 23(7), 739–745 (2004).
[CrossRef]

Other

R. Gold, “An iterative unfolding method for matrices,” Tech. Rep. ANL-6984 Argonne National Laboratory, Argonne, Illinois.

J. W. Goodman, “Introduction to Fourier optics,” (Roberts & Company Publishers, Engelwood, Colorado) 2005, Chap. 3.

L. P. Yaroslavsky, and N. S. Merzlykaov, Methods of digital holography, (Consultants Bureau, New York, 1980).

U. Schnars, and W. Jueptner, Digital holography. Digital hologram recording, numerical reconstruction, and related techniques, (Springer Berlin Heidelberg, 2010).

P. J. Shaw, “Comparison of wide-field/deconvolution and confocal microscopy for 3D imaging,” in Handbook of biological confocal microscopy, J. B. Pawley, ed., (Plenum Press, New York and London, 1995).

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Figures (12)

Fig. 1
Fig. 1

Scheme of hologram recording and illustration to the used symbols.

Fig. 2
Fig. 2

Arrangement of numerical experiments for the simulation of holograms of a particle distribution and of an individual point scatterer. (a) Simulation of the hologram of a particle distribution showing the transmission functions in the three planes at which the particles are arranged. (b) Simulation of the hologram of a point scatterer.

Fig. 3
Fig. 3

(a) Three-dimensional representation of the amplitude of the reconstructed complex wave-front. (b) Amplitude of the complex PSF. (c) Result of the instant 3d-deconvolution.

Fig. 4
Fig. 4

Results of the instant 3d-deconvolution of the intensities reconstructed from the simulated hologram of a particle distribution. (a) Original objects used in the simulation at the three z-planes. (b) Reconstructed amplitude of the object wave-front at the same z-planes. (c) Result of instant 3d-deconvolution. (d) Intensity profiles along z-direction for the particle indicated by the red arrow in (c) before (black) and after (red) instant 3d-deconvolution.

Fig. 5
Fig. 5

Results of the iterative 3d-deconvolution of the complex fields reconstructed from the simulated hologram of a particle distribution. (a) Amplitude distributions of the deconvoluted fields at the three planes. (b) Intensity profiles for a 16mm span around the in-focus position along z-direction for the particle indicated by an arrow in (a) before and after deconvolution. (c) Error as a function of the iteration number.

Fig. 6
Fig. 6

Results of the iterative 3d-deconvolution of the complex fields reconstructed from the simulated hologram of continuous objects. (a) Original objects used in the simulation at the three different planes. (b) Reconstructed objects at the same planes. (c) Result of the 3d-deconvolution. (d) Three-dimensional representation of the reconstructed amplitude of the object wave-front and amplitude of the deconvoluted object function. (e) Intensity profiles for a 16mm span around the in-focus position along z-direction for the point of the γ-object indicated by the red arrow in (c) before (black) and after (red) iterative 3d-deconvolution.

Fig. 7
Fig. 7

Experimental scheme for in inline holographic recording with plane waves.

Fig. 8
Fig. 8

Experimentally recorded hologram H0 of polystyrene spheres (left), the background image B (center) and the normalized hologram H (right).

Fig. 9
Fig. 9

3d-deconvolution of experimental holographic reconstruction. (a) Normalized hologram. (b) Hologram for the PSF as a cut out of the experimental hologram related to just a single sphere. (c) Amplitude reconstruction of the hologram for the PSF, slice in xz-plane. (d) Amplitude reconstruction at both sides of the microscopic slide, at the S1 and S2 plane. In-focus and out-of of focus spheres can be distinguished. Right: 3d-representation of the reconstructed amplitude distribution. (e) Results after instant 3d-deconvolution. Amplitude distributions at the S1 and S2 plane together with the 3d-distribution. (f) Results after iterative 3d-deconvolution. Amplitude distributions at the S1 and S2 plane and 3d-distribution.

Fig. 10
Fig. 10

Magnified images of the reconstructed amplitude before and after iterative 3d-deconvolution at plane S1.

Fig. 11
Fig. 11

3d-deconvolution of the experimental holographic reconstruction with a simulated PSF. (a) Simulated hologram of a point scatterer. (b) Results of instant 3d-deconvolution at S1 and S2 planes and as 3d-representation. (b) Results of iterative 3d-deconvolution at S1 and S2 planes and as 3d-representation.

Fig. 12
Fig. 12

Results of iterative 3d-deconvolution of the optical field reconstructed from an experimental hologram with a simulated PSF. (a) Selected fragment of an experimental hologram of spheres. (b) Reconstructed amplitude distribution at planes S1 and S2 and its three-dimensional representation. (c) Results of the iterative 3d-deconvolution at the plane S1 and S2 together with its three-dimensional representations. (d) Amplitude profiles along z-direction at the spheres’ positions before (black) and after (red) 3d-deconvolution.

Equations (36)

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O ( r ) M ( r ) .
δ ( r ) PSF ( r ) .
O ( r ) = O ( s ) δ ( r s ) d s
M ( r ) = O ( s ) PSF ( r s ) d s .
M ( r ) = O ( r ) PSF ( r ) .
U Object ( r ) = O ( r O ) exp ( i k | r O r | ) | r O r | d r O = O ( r ) exp ( i k r ) r
H ˜ ( r S ) ~ U ˜ Object ( r S ) U ˜ R * ( r S ) + U ˜ Objext * ( r S ) U ˜ R ( r S ) .
U O ( r ) = U ˜ Object ( r S ) δ ( z S z D ) exp ( i k | r S r | ) | r S r | d r S
U P ( r ) = U ˜ Point ( r S ) δ ( z S z D ) exp ( i k | r S r | ) | r S r | d r S .
U O ( r ) = O ( r ) U P ( r ) ,
FT( U O ( r )) = FT ( O ( r ) ) FT( U P ( r )) .
O ( r ) = FT 1 ( FT( U O ( r )) FT( U P ( r )) ) .
FT( U O ( r ))=FT( U ˜ Object ( r ) δ ( z - z D ))FT ( exp ( i k r ) r )
FT( U P ( r ))=FT( U ˜ Point ( r ) δ ( z - z D ))FT ( exp ( i k r ) r ) ,
FT ( U ( r ) ) = U ( r ) exp ( 2 π i r q ) d r
FT ( U ( x , y , z ) ) = U ( x , y , z ) exp ( 2 π i ( x μ + y ν + z η ) ) d x d y d z
O ( r ) = FT 1 ( FT( U ˜ Object ( r ) δ ( z - z D )) FT( U ˜ Point ( r ) δ ( z - z D )) ) ,
FT( U ˜ Object ( r ) δ ( z - z D )) = exp(-2 π i z D η ) U ˜ Object ( x , y )exp(-2 π i ( x μ + y ν ))d x d y
FT( U ˜ Point ( r ) δ ( z - z D )) = exp(-2 π i z D η ) U ˜ Point ( x , y )exp(-2 π i ( x μ + y ν ))d x d y .
O ( r ) = FT 1 ( U ˜ Object ( x , y )exp(-2 π i ( x μ + y ν ))d x d y U ˜ Point ( x , y )exp(-2 π i ( x μ + y ν ))d x d y ) .
O ( r ) = FT 1 ( FT  ( | U O ( r ) | 2 ) FT  ( | U P ( r ) | 2 + β ) ) ,
U A ( r S ) exp ( i k x S 2 + y S 2 + z S 2 ) z S
U B ( r S ) exp ( i k ( x S x B ) 2 + ( y S y B ) 2 + ( z S z B ) 2 ) ( z S z B ) .
FT( U ˜ A ( r ) δ ( z - z D )) = exp(-2 π i z D η ) exp ( π i λ z D ( μ 2 + ν 2 ) )
FT( U ˜ B ( r ) δ ( z - z D )) =
= exp(-2 π i z D η )exp(-2 π i ( x B μ + y B ν ) )exp(- π i λ ( z D z B ) ( μ 2 + ν 2 ) ) .
FT ( exp ( i k r ) r ) δ ( η + λ 2 ( μ 2 + ν 2 ) ) ,
exp(-2 π i ( x B μ + y B ν ) )exp( π i λ z B ( μ 2 + ν 2 ) ) δ ( η + λ 2 ( μ 2 + ν 2 ) ) =
η = λ 2 ( μ 2 + ν 2 )
O ( 1 ) ( r ) = U O ( r )
U O ( k ) ( r ) = O ( k ) ( r ) U P ( r )
O ( k + 1 ) ( r ) = O ( k ) ( r ) U O ( r ) U O ( k ) ( r )
k = k + 1
O ( k + 1 ) ( r ) = O ( k ) ( r ) U O ( r ) ( U O ( k ) ( r ) ) * | U O ( k ) ( r ) | 2 + β
Error ( k ) = i,j,l | | U O ( k ) ( i,j,l ) | | U O ( i,j,l ) | | i,j,l | U O ( i,j,l ) |
GLPF ( μ , ν , η ) = exp ( μ 2 + ν 2 + η 2 2 D 2 )

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