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,” Applied Optics 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,” Optics Communications 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,” Applied Optics 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 Phase-space Filtering with Wigner Distribution Functions,” Applied Optics 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,” Applied Mathematics and 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)

H. Kim, S.-W. Min, B. Lee, and T.-C. Poon, “Optical Sectioning for Optical Scanning Holography Using Phase-space Filtering with Wigner Distribution Functions,” Applied Optics 47, 164–175 (2008).
[CrossRef]

2006 (3)

2004 (1)

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

2002 (3)

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]

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]

2001 (2)

B. W. Schilling and G. C. Templeton, “Three-dimensional Remote Sensing by Optical Scanning Holography,” Applied Optics 40, 5474–5481 (2001).
[CrossRef]

L. Vese, “A Study in the BV Space of a Denoising-deblurring Variational Problem,” Applied Mathematics and 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,” Optics Communications 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]

Arsenin, V.

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

Aubert, G.

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]

Banerjee, P. P.

P. P. Banerjee and R. M. Misra, “Dependence of Photorefractive Beam Fanning on Beam Parameters,” Optics Communications 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]

Blackledge, J. M.

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

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]

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]

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]

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.

H. Kim, S.-W. Min, B. Lee, and T.-C. Poon, “Optical Sectioning for Optical Scanning Holography Using Phase-space Filtering with Wigner Distribution Functions,” Applied Optics 47, 164–175 (2008).
[CrossRef]

Kim, T.

T. Kim, “Optical Sectioning by Optical Scanning Holography and a Wiener Filter,” Applied Optics 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]

Kornprobst, P.

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]

Kuo, C. J.

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

Kutay, M. A.

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).

Lee, B.

H. Kim, S.-W. Min, B. Lee, and T.-C. Poon, “Optical Sectioning for Optical Scanning Holography Using Phase-space Filtering with Wigner Distribution Functions,” Applied Optics 47, 164–175 (2008).
[CrossRef]

Martínez-Corral, M.

Min, S.-W.

H. Kim, S.-W. Min, B. Lee, and T.-C. Poon, “Optical Sectioning for Optical Scanning Holography Using Phase-space Filtering with Wigner Distribution Functions,” Applied Optics 47, 164–175 (2008).
[CrossRef]

Misra, R. M.

P. P. Banerjee and R. M. Misra, “Dependence of Photorefractive Beam Fanning on Beam Parameters,” Optics Communications 100, 166–172 (1993).
[CrossRef]

Natterer, F.

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

Ozaktas, H. M.

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).

Poon, T.-C.

H. Kim, S.-W. Min, B. Lee, and T.-C. Poon, “Optical Sectioning for Optical Scanning Holography Using Phase-space Filtering with Wigner Distribution Functions,” Applied Optics 47, 164–175 (2008).
[CrossRef]

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

B. D. Duncan and T.-C. Poon, “Gaussian Beam Analysis of Optical Scanning Holography,” J. Opt. Soc. Am. A 9, 229–236 (1992).
[CrossRef]

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]

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

Schilling, B. W.

B. W. Schilling and G. C. Templeton, “Three-dimensional Remote Sensing by Optical Scanning Holography,” Applied Optics 40, 5474–5481 (2001).
[CrossRef]

Stelzer, E.

Swoger, J.

Templeton, G. C.

B. W. Schilling and G. C. Templeton, “Three-dimensional Remote Sensing by Optical Scanning Holography,” Applied Optics 40, 5474–5481 (2001).
[CrossRef]

Tikhonov, A.

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

Vese, L.

L. Vese, “A Study in the BV Space of a Denoising-deblurring Variational Problem,” Applied Mathematics and Optimization 44, 131–161 (2001).
[CrossRef]

Vogel, C. R.

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

Wübbeling, F.

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

Zalevsky, Z.

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).

Zhong, W.

Applied Mathematics and Optimization (1)

L. Vese, “A Study in the BV Space of a Denoising-deblurring Variational Problem,” Applied Mathematics and Optimization 44, 131–161 (2001).
[CrossRef]

Applied Optics (4)

B. W. Schilling and G. C. Templeton, “Three-dimensional Remote Sensing by Optical Scanning Holography,” Applied Optics 40, 5474–5481 (2001).
[CrossRef]

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

T. Kim, “Optical Sectioning by Optical Scanning Holography and a Wiener Filter,” Applied Optics 45, 872–879 (2006).
[CrossRef] [PubMed]

H. Kim, S.-W. Min, B. Lee, and T.-C. Poon, “Optical Sectioning for Optical Scanning Holography Using Phase-space Filtering with Wigner Distribution Functions,” Applied Optics 47, 164–175 (2008).
[CrossRef]

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. Eng. (2)

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

Optics Communications (1)

P. P. Banerjee and R. M. Misra, “Dependence of Photorefractive Beam Fanning on Beam Parameters,” Optics Communications 100, 166–172 (1993).
[CrossRef]

Other (7)

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]

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]

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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)

Equations on this page are rendered with MathJax. Learn more.

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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