Abstract

This paper discusses the reconstruction of sectional images from a hologram generated by optical scanning holography. We present a mathematical model for the holographic image capture, which facilitates the use of inverse imaging techniques to recover individual sections. This framework is much more flexible than existing work, in the sense that it can handle objects with multiple sections, and possibly corrupted with white Gaussian noise. Simulation results show that the algorithm is capable of recovering a prescribed section while suppressing the other ones as defocus noise. The proposed algorithm is applicable to on-axis holograms acquired by conventional holography as well as phase-shifting holography.

© 2008 Optical Society of America

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References

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  1. B. D. Duncan and T.-C. Poon, "Gaussian Beam Analysis of Optical Scanning Holography," J. Opt. Soc. Am. A 9, 229-236 (1992).
    [CrossRef]
  2. B. W. Schilling and G. C. Templeton, "Three-dimensional Remote Sensing by Optical Scanning Holography," Appl. Opt. 40, 5474-5481 (2001).
    [CrossRef]
  3. T. Kim, T.-C. Poon, and G. Indebetouw, "Depth Detection and Image Recovery in Remote Sensing by Optical Scanning Holography," Opt. Eng. 41, 1331-1338 (2002).
    [CrossRef]
  4. P. P. Banerjee and R. M. Misra, "Dependence of Photorefractive Beam Fanning on Beam Parameters," Opt. Commun. 100, 166-172 (1993).
    [CrossRef]
  5. T.-C. Poon, "Recent Progress in Optical Scanning Holography," J. Holography Speckle 1, 6-25 (2004).
    [CrossRef]
  6. G. Indebetouw and W. Zhong, "Scanning Holographic Microscopy of Three-dimensional Fluorescent Specimens," J. Opt. Soc. Am. A 23, 1699-1707 (2006).
    [CrossRef]
  7. T.-C. Poon, "Scanning Holography and Two-dimensional Image Processing by Acousto-optic Two-pupil Synthesis," J. Opt. Soc. Am. A 2, 521-527 (1985).
    [CrossRef]
  8. C. J. Kuo, "Electronic Holography," Opt. Eng. 35, 1528 (1996).
    [CrossRef]
  9. K. M. Johnson, M. Armstrong, L. Hesselink, and J. W. Goodman, "Multiple Multiple-exposure Hologram," Applied Optics 24, 4467-4472 (1985).
    [CrossRef] [PubMed]
  10. G. Indebetouw, "Properties of a Scanning Holographic Microscopy: Improved Resolution, Extended Depth-offocus, and/or Optical Sectioning," J. Mod. Opt. 49, 1479-1500 (2002).
    [CrossRef]
  11. T. Kim, "Optical Sectioning by Optical Scanning Holography and a Wiener Filter," Appl. Opt. 45, 872-879 (2006).
    [CrossRef] [PubMed]
  12. H. Kim, S.-W. Min, B. Lee, and T.-C. Poon, "Optical Sectioning for Optical Scanning Holography Using Phasespace Filtering with Wigner Distribution Functions," Appl. Opt. 47, 164-175 (2008).
    [CrossRef]
  13. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform: with Applications in Optics and Signal Processing, 1st ed. (Wiley, Chichester, 2001).
  14. T.-C. Poon, Optical Scanning Holography with MATLAB, 1st ed. (Springer-Verlag, New York, 2007).
    [CrossRef]
  15. J. Swoger, M. Martínez-Corral, J. Huisken, and E. Stelzer, "Optical Scanning Holography as a Technique for High-resolution Three-dimensional Biological Microscopy," J. Opt. Soc. Am. A 19, 1910-1918 (2002).
    [CrossRef]
  16. G. Indebetouw, W. Zhong, and D. Chamberlin-Long, "Point-spread Function Synthesis in Scanning Holographic Microscopy," J. Opt. Soc. Am. A 23, 1708-1717 (2006).
    [CrossRef]
  17. M. R. Banham and A. K. Katsaggelos, "Digital Image Restoration," IEEE Signal Processing Magazine 14, 24-41 (1997).
    [CrossRef]
  18. J.M. Blackledge, Digital Image Processing: Mathematical and Computational Methods, 1st ed. (Horwood,West Sussex, 2005).
  19. A. Tikhonov and V. Arsenin, Solutions of Ill-posed Problems, 1st ed. (V.H. Winston and Sons, Washington, 1977).
  20. F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction, 1st ed. (SIAM, Philadelphia, 2001).
    [CrossRef]
  21. L. Vese, "A Study in the BV Space of a Denoising-deblurring Variational Problem," Appl. Math. Optimization 44, 131-161 (2001).
    [CrossRef]
  22. G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and Calculus of Variations, 2nd ed. (Springer-Verlag, New York, 2006).
    [PubMed]
  23. C. R. Vogel, Computational Methods for Inverse Problems, 1st ed. (SIAM, Philadelphia, 2002).
    [CrossRef]

2008 (1)

2006 (3)

2004 (1)

T.-C. Poon, "Recent Progress in Optical Scanning Holography," J. Holography Speckle 1, 6-25 (2004).
[CrossRef]

2002 (3)

T. Kim, T.-C. Poon, and G. Indebetouw, "Depth Detection and Image Recovery in Remote Sensing by Optical Scanning Holography," Opt. Eng. 41, 1331-1338 (2002).
[CrossRef]

G. Indebetouw, "Properties of a Scanning Holographic Microscopy: Improved Resolution, Extended Depth-offocus, and/or Optical Sectioning," J. Mod. Opt. 49, 1479-1500 (2002).
[CrossRef]

J. Swoger, M. Martínez-Corral, J. Huisken, and E. Stelzer, "Optical Scanning Holography as a Technique for High-resolution Three-dimensional Biological Microscopy," J. Opt. Soc. Am. A 19, 1910-1918 (2002).
[CrossRef]

2001 (2)

B. W. Schilling and G. C. Templeton, "Three-dimensional Remote Sensing by Optical Scanning Holography," Appl. Opt. 40, 5474-5481 (2001).
[CrossRef]

L. Vese, "A Study in the BV Space of a Denoising-deblurring Variational Problem," Appl. Math. Optimization 44, 131-161 (2001).
[CrossRef]

1997 (1)

M. R. Banham and A. K. Katsaggelos, "Digital Image Restoration," IEEE Signal Processing Magazine 14, 24-41 (1997).
[CrossRef]

1996 (1)

C. J. Kuo, "Electronic Holography," Opt. Eng. 35, 1528 (1996).
[CrossRef]

1993 (1)

P. P. Banerjee and R. M. Misra, "Dependence of Photorefractive Beam Fanning on Beam Parameters," Opt. Commun. 100, 166-172 (1993).
[CrossRef]

1992 (1)

1985 (2)

K. M. Johnson, M. Armstrong, L. Hesselink, and J. W. Goodman, "Multiple Multiple-exposure Hologram," Applied Optics 24, 4467-4472 (1985).
[CrossRef] [PubMed]

T.-C. Poon, "Scanning Holography and Two-dimensional Image Processing by Acousto-optic Two-pupil Synthesis," J. Opt. Soc. Am. A 2, 521-527 (1985).
[CrossRef]

Armstrong, M.

K. M. Johnson, M. Armstrong, L. Hesselink, and J. W. Goodman, "Multiple Multiple-exposure Hologram," Applied Optics 24, 4467-4472 (1985).
[CrossRef] [PubMed]

Banerjee, P. P.

P. P. Banerjee and R. M. Misra, "Dependence of Photorefractive Beam Fanning on Beam Parameters," Opt. Commun. 100, 166-172 (1993).
[CrossRef]

Banham, M. R.

M. R. Banham and A. K. Katsaggelos, "Digital Image Restoration," IEEE Signal Processing Magazine 14, 24-41 (1997).
[CrossRef]

Chamberlin-Long, D.

Duncan, B. D.

Goodman, J. W.

K. M. Johnson, M. Armstrong, L. Hesselink, and J. W. Goodman, "Multiple Multiple-exposure Hologram," Applied Optics 24, 4467-4472 (1985).
[CrossRef] [PubMed]

Hesselink, L.

K. M. Johnson, M. Armstrong, L. Hesselink, and J. W. Goodman, "Multiple Multiple-exposure Hologram," Applied Optics 24, 4467-4472 (1985).
[CrossRef] [PubMed]

Huisken, J.

Indebetouw, G.

G. Indebetouw and W. Zhong, "Scanning Holographic Microscopy of Three-dimensional Fluorescent Specimens," J. Opt. Soc. Am. A 23, 1699-1707 (2006).
[CrossRef]

G. Indebetouw, W. Zhong, and D. Chamberlin-Long, "Point-spread Function Synthesis in Scanning Holographic Microscopy," J. Opt. Soc. Am. A 23, 1708-1717 (2006).
[CrossRef]

T. Kim, T.-C. Poon, and G. Indebetouw, "Depth Detection and Image Recovery in Remote Sensing by Optical Scanning Holography," Opt. Eng. 41, 1331-1338 (2002).
[CrossRef]

G. Indebetouw, "Properties of a Scanning Holographic Microscopy: Improved Resolution, Extended Depth-offocus, and/or Optical Sectioning," J. Mod. Opt. 49, 1479-1500 (2002).
[CrossRef]

Johnson, K. M.

K. M. Johnson, M. Armstrong, L. Hesselink, and J. W. Goodman, "Multiple Multiple-exposure Hologram," Applied Optics 24, 4467-4472 (1985).
[CrossRef] [PubMed]

Katsaggelos, A. K.

M. R. Banham and A. K. Katsaggelos, "Digital Image Restoration," IEEE Signal Processing Magazine 14, 24-41 (1997).
[CrossRef]

Kim, H.

Kim, T.

T. Kim, "Optical Sectioning by Optical Scanning Holography and a Wiener Filter," Appl. Opt. 45, 872-879 (2006).
[CrossRef] [PubMed]

T. Kim, T.-C. Poon, and G. Indebetouw, "Depth Detection and Image Recovery in Remote Sensing by Optical Scanning Holography," Opt. Eng. 41, 1331-1338 (2002).
[CrossRef]

Kuo, C. J.

C. J. Kuo, "Electronic Holography," Opt. Eng. 35, 1528 (1996).
[CrossRef]

Lee, B.

Martínez-Corral, M.

Min, S.-W.

Misra, R. M.

P. P. Banerjee and R. M. Misra, "Dependence of Photorefractive Beam Fanning on Beam Parameters," Opt. Commun. 100, 166-172 (1993).
[CrossRef]

Poon, T.-C.

Schilling, B. W.

Stelzer, E.

Swoger, J.

Templeton, G. C.

Vese, L.

L. Vese, "A Study in the BV Space of a Denoising-deblurring Variational Problem," Appl. Math. Optimization 44, 131-161 (2001).
[CrossRef]

Zhong, W.

Appl. Math. Optimization (1)

L. Vese, "A Study in the BV Space of a Denoising-deblurring Variational Problem," Appl. Math. Optimization 44, 131-161 (2001).
[CrossRef]

Appl. Opt. (3)

Applied Optics (1)

K. M. Johnson, M. Armstrong, L. Hesselink, and J. W. Goodman, "Multiple Multiple-exposure Hologram," Applied Optics 24, 4467-4472 (1985).
[CrossRef] [PubMed]

IEEE Signal Processing Magazine (1)

M. R. Banham and A. K. Katsaggelos, "Digital Image Restoration," IEEE Signal Processing Magazine 14, 24-41 (1997).
[CrossRef]

J. Holography Speckle (1)

T.-C. Poon, "Recent Progress in Optical Scanning Holography," J. Holography Speckle 1, 6-25 (2004).
[CrossRef]

J. Mod. Opt. (1)

G. Indebetouw, "Properties of a Scanning Holographic Microscopy: Improved Resolution, Extended Depth-offocus, and/or Optical Sectioning," J. Mod. Opt. 49, 1479-1500 (2002).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Commun. (1)

P. P. Banerjee and R. M. Misra, "Dependence of Photorefractive Beam Fanning on Beam Parameters," Opt. Commun. 100, 166-172 (1993).
[CrossRef]

Opt. Eng. (2)

T. Kim, T.-C. Poon, and G. Indebetouw, "Depth Detection and Image Recovery in Remote Sensing by Optical Scanning Holography," Opt. Eng. 41, 1331-1338 (2002).
[CrossRef]

C. J. Kuo, "Electronic Holography," Opt. Eng. 35, 1528 (1996).
[CrossRef]

Other (7)

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform: with Applications in Optics and Signal Processing, 1st ed. (Wiley, Chichester, 2001).

T.-C. Poon, Optical Scanning Holography with MATLAB, 1st ed. (Springer-Verlag, New York, 2007).
[CrossRef]

G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and Calculus of Variations, 2nd ed. (Springer-Verlag, New York, 2006).
[PubMed]

C. R. Vogel, Computational Methods for Inverse Problems, 1st ed. (SIAM, Philadelphia, 2002).
[CrossRef]

J.M. Blackledge, Digital Image Processing: Mathematical and Computational Methods, 1st ed. (Horwood,West Sussex, 2005).

A. Tikhonov and V. Arsenin, Solutions of Ill-posed Problems, 1st ed. (V.H. Winston and Sons, Washington, 1977).

F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction, 1st ed. (SIAM, Philadelphia, 2001).
[CrossRef]

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Figures (10)

Fig. 1.
Fig. 1.

A typical double-beam optical scanning holography system.

Fig. 2.
Fig. 2.

The object and FZPs in the first experiment. Shown in Fig. 2(a) is the object, in which two elements (i.e., rectangles) are at z 1 and z 2 sections, respectively. Shown in Fig. 2(b) are the real parts of FZPs of a point source at z 1 and z 2.

Fig. 3.
Fig. 3.

The holograms containing two-sectional images of the object in the first experiment

Fig. 4.
Fig. 4.

Reconstructed sections by the conventional method on the hologram containing two sections only. Fig. 4(a) is the result of z 1 section and Fig. 4(b) is that of z 2 section.

Fig. 5.
Fig. 5.

Reconstructed sections by the inverse imaging method from the hologram with two sections. Fig. 5(a) is reconstructed section at z 1 and Fig. 5(b) is the one at z 2.

Fig. 6.
Fig. 6.

Reconstructed sections by Wiener filter. Fig. 6(a) is reconstructed section at z 1 and Fig. 6(b) is the one at z 2.

Fig. 7.
Fig. 7.

The object and FZPs in the three-section experiment. The object in Fig. 7(a) contains three elements at z 1=7mm, z 2=8mm and z 3=9mm sections. Real parts of FZPs of a point source at z 1, z 2 and z 3 are shown in Fig. 7(b).

Fig. 8.
Fig. 8.

The hologram containing three-sectional images of the object in the second experiment

Fig. 9.
Fig. 9.

Reconstructed sections from a hologram with three sections. Fig. 9(a), Fig. 9(b) and Fig. 9(c) show results of sections at z 1, z 2 and z 3 by the conventional method. Shown in Fig. 9(d), Fig. 9(e) and Fig. 9(f) are reconstructed sectional images at z 1, z 2 and z 3 by the inverse imaging method.

Fig. 10.
Fig. 10.

SNR with regard to the regularization parameter.

Equations (22)

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H ( k x , k y ; z ) osh = exp { j z 2 k 0 ( k x 2 + k y 2 ) } ,
h ( x , y ; z ) osh = j k 0 2 π z exp { j k 0 ( x 2 + y 2 ) 2 z } .
g c ( x , y ) = { ϕ ( x , y ; z ) 2 * h ( x , y ; z ) } dz ,
g c ( x , y ) = i = 1 n { ϕ ( x , y ; z i ) 2 * h ( x , y ; z i ) } .
g c ( x , y ) = ϕ ( x , y ; z 1 ) 2 * h ( x , y , z 1 ) + ϕ ( x , y ; z 2 ) 2 * h ( x , y ; z 2 ) .
h 1 ( x , y ) = h ( x , y ; z 1 ) = j k 0 2 π z 1 exp { j k 0 ( x 2 + y 2 ) 2 z 1 } ,
h 2 ( x , y ) = h ( x , y ; z 2 ) = j k 0 2 π z 2 exp { j k 0 ( x 2 + y 2 ) 2 z 2 } .
γ c = H 1 ψ 1 + H 2 ψ 2 .
γ c = [ H 1 H 2 ] [ ψ 1 ψ 2 ] = H ψ ,
g c ( x , y ) * h 1 * ( x , y )
= ϕ ( x , y ; z 1 ) 2 * h 1 ( x , y ) * h 1 * ( x , y ) + ϕ ( x , y ; z 2 ) 2 * h 2 ( x , y ) * h 1 * ( x , y )
= ϕ ( x , y ; z 1 ) 2 + ϕ ( x , y ; z 2 ) 2 * h 21 ( x , y ) ,
h 1 ( x , y ) * h 1 * ( x , y ) δ ( x , y ) ( except scaling ) ;
h 2 ( x , y ) * h 1 * ( x , y ) = h ( x , y ; z 2 z 1 ) = h 21 ( x , y ) .
H 1 * γ c = H 1 * H ψ .
Re [ H 1 * γ c ] = Re [ H 1 * H ] ψ ,
β c = A ψ + ν ,
f ( ψ ) = A ψ β c 2 + λ C ψ 2 ,
b k = q ( k 1 ) 2 q ( k 2 ) 2
p ( k ) = q ( k 1 ) + b k p ( k 1 )
a k = q ( k 1 ) 2 ( A T A + λ C T C ) p ( k ) 2
q ( k ) = q ( k 1 ) a k ( A T A + λ C T C ) p ( k ) .

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