Abstract

Optical scanning holography (OSH) enables us to capture the three-dimensional information of an object, and a post-processing step known as sectional image reconstruction allows us to view its two-dimensional cross-section. Previous methods often produce reconstructed images that have blurry edges. In this paper, we argue that the hologram’s two-dimensional Fourier transform maps into a semi-spherical surface in the three-dimensional frequency domain of the object, a relationship akin to the Fourier diffraction theorem used in diffraction tomography. Thus, the sectional image reconstruction task is an ill-posed inverse problem, and here we make use of the total variation regularization with a nonnegative constraint and solve it with a gradient projection algorithm. Both simulated and experimental holograms are used to verify that edge-preserving reconstruction is achieved, and the axial distance between sections is reduced compared with previous regularization methods.

© 2010 Optical Society of America

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  1. 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]
  2. 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]
  3. U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer-Verlag, 2005).
  4. T.-C. Poon, Digital Holography and Three-Dimensional Display: Principles and Applications (Springer, 2006).
    [CrossRef]
  5. X. Zhang, E. Y. Lam, and T.-C. Poon, “Reconstruction of sectional images in holography using inverse imaging,” Opt. Express 16, 17215–17226 (2008).
    [CrossRef] [PubMed]
  6. T. Kim, “Optical sectioning by optical scanning holography and a Wiener filter,” Appl. Opt. 45, 872–879 (2006).
    [CrossRef] [PubMed]
  7. 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,” Appl. Opt. 47, D164–D175 (2008).
    [CrossRef] [PubMed]
  8. S.-J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE J. Sel. Top. Signal Process. 1, 606–617 (2007).
    [CrossRef]
  9. G. Indebetouw, “Properties of a scanning holographic microscopy: Improved resolution, extended depth-of-focus, and/or optical sectioning,” J. Mod. Opt. 49, 1479–1500 (2002).
    [CrossRef]
  10. R. K. Mueller, M. Kaveh, and G. Wade, “Reconstructive tomography and applications to ultrasonics,” Proc. IEEE 67, 567–587 (1979).
    [CrossRef]
  11. A. J. Devaney, “Geophysical diffraction tomography,” IEEE Trans. Geosci. Remote Sens. GE-22, 3–13 (1984).
    [CrossRef]
  12. A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336–350 (1982).
    [CrossRef] [PubMed]
  13. P. Guo and A. J. Devaney, “Comparison of reconstruction algorithms for optical diffraction tomography,” J. Opt. Soc. Am. A 22, 2338–2347 (2005).
    [CrossRef]
  14. S. J. LaRoque, E. Y. Sidky, and X. Pan, “Accurate image reconstruction from few-view and limited-angle data in diffraction tomography,” J. Opt. Soc. Am. A 25, 1772–1782 (2008).
    [CrossRef]
  15. G. Indebetouw, P. Klysubun, T. Kim, and T.-C. Poon, “Imaging properties of scanning holographic microscopy,” J. Opt. Soc. Am. A 17, 380–390 (2000).
    [CrossRef]
  16. T.-C. Poon, Optical Scanning Holography with MATLAB (Springer-Verlag, 2007).
    [CrossRef]
  17. E. Y. Lam, X. Zhang, H. Vo, T.-C. Poon, and G. Indebetouw, “Three-dimensional microscopy and sectional image reconstruction using optical scanning holography,” Appl. Opt. 48, H113–H119 (2009).
    [CrossRef] [PubMed]
  18. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005),
  19. G. Indebetouw and W. Zhong, “Scanning holographic microscopy of three-dimensional fluorescent specimens,” J. Opt. Soc. Am. A 23, 1699–1707 (2006).
    [CrossRef]
  20. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
    [CrossRef]
  21. C. L. Matson and H. Liu, “The Fourier diffraction theorem for turbid media,” in “Advances in Optical Imaging and Photon Migration,” J.Fujimoto and M.Patterson, eds., Vol. 21 of OSA Trends in Optics and Photonics (Optical Society of America, 1998) p. ATuD14.
  22. R. Bracewell, The Fourier Transform and Its Applications, 3rd ed. (McGraw-Hill, 1999),
  23. A. J. Devaney, “A computer simulation study of diffraction tomography,” IEEE Trans. Biomed. Eng. BME-30, 377–386 (1983).
    [CrossRef]
  24. Y. Zou and X. Pan, “Image reconstruction on PI-lines by use of filtered backprojection in helical cone-beam CT,” Phys. Med. Biol. 49, 2717–2731 (2004).
    [CrossRef] [PubMed]
  25. T. Chan and J. Shen, Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods (SIAM, 2005).
  26. E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
    [CrossRef]
  27. R. Gonzalez and R. Woods, Digital Image Processing, 3rd ed. (Pearson Prentice Hall, 2008).
  28. Y. Li and F. Santosa, “A computational algorithm for minimizing total variation in image restoration,” IEEE Trans. Image Process. 5, 987–995 (1996).
    [CrossRef] [PubMed]
  29. D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inverse Probl. 19, S165–S187 (2003).
    [CrossRef]
  30. M. K. Ng, H. Shen, E. Lam, and L. Zhang, “A total variation regularization based super-resolution reconstruction algorithm for digital video,” EURASIP Journal on Advances in Signal Processing 2007, 1–16 (2007). Article ID 74585.
    [CrossRef]
  31. E. Candès and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
    [CrossRef]
  32. E. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
    [CrossRef]
  33. X. Zhang, E. Y. Lam, and T.-C. Poon, “Fast iterative sectional image reconstruction in optical scanning holography,” in Digital Holography and Three-Dimensional Imaging (DH), OSA Technical Digest (CD) (Optical Society of America, 2009), paper DMA3.
  34. S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge Univ. Press, Cambridge, 2004).
  35. G. Indebetouw, “A posteriori quasi-sectioning of the three-dimensional reconstructions of scanning holographic microscopy,” J. Opt. Soc. Am. A 23, 2657–2661 (2006).
    [CrossRef]
  36. X. Zhang, E. Y. Lam, T. Kim, Y. S. Kim, and T.-C. Poon, “Blind sectional image reconstruction for optical scanning holography,” Opt. Lett. 34, 3098–3100 (2009).
    [CrossRef] [PubMed]

2009

2008

2007

M. K. Ng, H. Shen, E. Lam, and L. Zhang, “A total variation regularization based super-resolution reconstruction algorithm for digital video,” EURASIP Journal on Advances in Signal Processing 2007, 1–16 (2007). Article ID 74585.
[CrossRef]

S.-J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE J. Sel. Top. Signal Process. 1, 606–617 (2007).
[CrossRef]

T.-C. Poon, Optical Scanning Holography with MATLAB (Springer-Verlag, 2007).
[CrossRef]

2006

T.-C. Poon, Digital Holography and Three-Dimensional Display: Principles and Applications (Springer, 2006).
[CrossRef]

T. Kim, “Optical sectioning by optical scanning holography and a Wiener filter,” Appl. Opt. 45, 872–879 (2006).
[CrossRef] [PubMed]

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, “A posteriori quasi-sectioning of the three-dimensional reconstructions of scanning holographic microscopy,” J. Opt. Soc. Am. A 23, 2657–2661 (2006).
[CrossRef]

E. Candès and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[CrossRef]

E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[CrossRef]

2005

U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer-Verlag, 2005).

T. Chan and J. Shen, Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods (SIAM, 2005).

P. Guo and A. J. Devaney, “Comparison of reconstruction algorithms for optical diffraction tomography,” J. Opt. Soc. Am. A 22, 2338–2347 (2005).
[CrossRef]

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

2004

Y. Zou and X. Pan, “Image reconstruction on PI-lines by use of filtered backprojection in helical cone-beam CT,” Phys. Med. Biol. 49, 2717–2731 (2004).
[CrossRef] [PubMed]

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge Univ. Press, Cambridge, 2004).

2003

D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inverse Probl. 19, S165–S187 (2003).
[CrossRef]

2002

G. Indebetouw, “Properties of a scanning holographic microscopy: Improved resolution, extended depth-of-focus, 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]

2000

1999

R. Bracewell, The Fourier Transform and Its Applications, 3rd ed. (McGraw-Hill, 1999),

1998

C. L. Matson and H. Liu, “The Fourier diffraction theorem for turbid media,” in “Advances in Optical Imaging and Photon Migration,” J.Fujimoto and M.Patterson, eds., Vol. 21 of OSA Trends in Optics and Photonics (Optical Society of America, 1998) p. ATuD14.

1996

Y. Li and F. Santosa, “A computational algorithm for minimizing total variation in image restoration,” IEEE Trans. Image Process. 5, 987–995 (1996).
[CrossRef] [PubMed]

1985

1984

A. J. Devaney, “Geophysical diffraction tomography,” IEEE Trans. Geosci. Remote Sens. GE-22, 3–13 (1984).
[CrossRef]

1983

A. J. Devaney, “A computer simulation study of diffraction tomography,” IEEE Trans. Biomed. Eng. BME-30, 377–386 (1983).
[CrossRef]

1982

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336–350 (1982).
[CrossRef] [PubMed]

1979

R. K. Mueller, M. Kaveh, and G. Wade, “Reconstructive tomography and applications to ultrasonics,” Proc. IEEE 67, 567–587 (1979).
[CrossRef]

1969

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

Boyd, S.

S.-J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE J. Sel. Top. Signal Process. 1, 606–617 (2007).
[CrossRef]

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge Univ. Press, Cambridge, 2004).

Bracewell, R.

R. Bracewell, The Fourier Transform and Its Applications, 3rd ed. (McGraw-Hill, 1999),

Candès, E.

E. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
[CrossRef]

E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[CrossRef]

E. Candès and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[CrossRef]

Chan, T.

T. Chan and J. Shen, Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods (SIAM, 2005).

D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inverse Probl. 19, S165–S187 (2003).
[CrossRef]

Devaney, A. J.

P. Guo and A. J. Devaney, “Comparison of reconstruction algorithms for optical diffraction tomography,” J. Opt. Soc. Am. A 22, 2338–2347 (2005).
[CrossRef]

A. J. Devaney, “Geophysical diffraction tomography,” IEEE Trans. Geosci. Remote Sens. GE-22, 3–13 (1984).
[CrossRef]

A. J. Devaney, “A computer simulation study of diffraction tomography,” IEEE Trans. Biomed. Eng. BME-30, 377–386 (1983).
[CrossRef]

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336–350 (1982).
[CrossRef] [PubMed]

Gonzalez, R.

R. Gonzalez and R. Woods, Digital Image Processing, 3rd ed. (Pearson Prentice Hall, 2008).

Goodman, J. W.

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

Gorinevsky, D.

S.-J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE J. Sel. Top. Signal Process. 1, 606–617 (2007).
[CrossRef]

Guo, P.

Huisken, J.

Indebetouw, G.

Jueptner, W.

U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer-Verlag, 2005).

Kaveh, M.

R. K. Mueller, M. Kaveh, and G. Wade, “Reconstructive tomography and applications to ultrasonics,” Proc. IEEE 67, 567–587 (1979).
[CrossRef]

Kim, H.

Kim, S.-J.

S.-J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE J. Sel. Top. Signal Process. 1, 606–617 (2007).
[CrossRef]

Kim, T.

Kim, Y. S.

Klysubun, P.

Koh, K.

S.-J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE J. Sel. Top. Signal Process. 1, 606–617 (2007).
[CrossRef]

Lam, E.

M. K. Ng, H. Shen, E. Lam, and L. Zhang, “A total variation regularization based super-resolution reconstruction algorithm for digital video,” EURASIP Journal on Advances in Signal Processing 2007, 1–16 (2007). Article ID 74585.
[CrossRef]

Lam, E. Y.

LaRoque, S. J.

Lee, B.

Li, Y.

Y. Li and F. Santosa, “A computational algorithm for minimizing total variation in image restoration,” IEEE Trans. Image Process. 5, 987–995 (1996).
[CrossRef] [PubMed]

Liu, H.

C. L. Matson and H. Liu, “The Fourier diffraction theorem for turbid media,” in “Advances in Optical Imaging and Photon Migration,” J.Fujimoto and M.Patterson, eds., Vol. 21 of OSA Trends in Optics and Photonics (Optical Society of America, 1998) p. ATuD14.

Lustig, M.

S.-J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE J. Sel. Top. Signal Process. 1, 606–617 (2007).
[CrossRef]

Martínez-Corral, M.

Matson, C. L.

C. L. Matson and H. Liu, “The Fourier diffraction theorem for turbid media,” in “Advances in Optical Imaging and Photon Migration,” J.Fujimoto and M.Patterson, eds., Vol. 21 of OSA Trends in Optics and Photonics (Optical Society of America, 1998) p. ATuD14.

Min, S.-W.

Mueller, R. K.

R. K. Mueller, M. Kaveh, and G. Wade, “Reconstructive tomography and applications to ultrasonics,” Proc. IEEE 67, 567–587 (1979).
[CrossRef]

Ng, M. K.

M. K. Ng, H. Shen, E. Lam, and L. Zhang, “A total variation regularization based super-resolution reconstruction algorithm for digital video,” EURASIP Journal on Advances in Signal Processing 2007, 1–16 (2007). Article ID 74585.
[CrossRef]

Pan, X.

S. J. LaRoque, E. Y. Sidky, and X. Pan, “Accurate image reconstruction from few-view and limited-angle data in diffraction tomography,” J. Opt. Soc. Am. A 25, 1772–1782 (2008).
[CrossRef]

Y. Zou and X. Pan, “Image reconstruction on PI-lines by use of filtered backprojection in helical cone-beam CT,” Phys. Med. Biol. 49, 2717–2731 (2004).
[CrossRef] [PubMed]

Poon, T.-C.

E. Y. Lam, X. Zhang, H. Vo, T.-C. Poon, and G. Indebetouw, “Three-dimensional microscopy and sectional image reconstruction using optical scanning holography,” Appl. Opt. 48, H113–H119 (2009).
[CrossRef] [PubMed]

X. Zhang, E. Y. Lam, and T.-C. Poon, “Fast iterative sectional image reconstruction in optical scanning holography,” in Digital Holography and Three-Dimensional Imaging (DH), OSA Technical Digest (CD) (Optical Society of America, 2009), paper DMA3.

X. Zhang, E. Y. Lam, T. Kim, Y. S. Kim, and T.-C. Poon, “Blind sectional image reconstruction for optical scanning holography,” Opt. Lett. 34, 3098–3100 (2009).
[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,” Appl. Opt. 47, D164–D175 (2008).
[CrossRef] [PubMed]

X. Zhang, E. Y. Lam, and T.-C. Poon, “Reconstruction of sectional images in holography using inverse imaging,” Opt. Express 16, 17215–17226 (2008).
[CrossRef] [PubMed]

T.-C. Poon, Optical Scanning Holography with MATLAB (Springer-Verlag, 2007).
[CrossRef]

T.-C. Poon, Digital Holography and Three-Dimensional Display: Principles and Applications (Springer, 2006).
[CrossRef]

G. Indebetouw, P. Klysubun, T. Kim, and T.-C. Poon, “Imaging properties of scanning holographic microscopy,” J. Opt. Soc. Am. A 17, 380–390 (2000).
[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]

Romberg, J.

E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[CrossRef]

Santosa, F.

Y. Li and F. Santosa, “A computational algorithm for minimizing total variation in image restoration,” IEEE Trans. Image Process. 5, 987–995 (1996).
[CrossRef] [PubMed]

Schnars, U.

U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer-Verlag, 2005).

Shen, H.

M. K. Ng, H. Shen, E. Lam, and L. Zhang, “A total variation regularization based super-resolution reconstruction algorithm for digital video,” EURASIP Journal on Advances in Signal Processing 2007, 1–16 (2007). Article ID 74585.
[CrossRef]

Shen, J.

T. Chan and J. Shen, Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods (SIAM, 2005).

Sidky, E. Y.

Stelzer, E.

Strong, D.

D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inverse Probl. 19, S165–S187 (2003).
[CrossRef]

Swoger, J.

Tao, T.

E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[CrossRef]

E. Candès and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[CrossRef]

Vandenberghe, L.

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge Univ. Press, Cambridge, 2004).

Vo, H.

Wade, G.

R. K. Mueller, M. Kaveh, and G. Wade, “Reconstructive tomography and applications to ultrasonics,” Proc. IEEE 67, 567–587 (1979).
[CrossRef]

Wakin, M. B.

E. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
[CrossRef]

Wolf, E.

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

Woods, R.

R. Gonzalez and R. Woods, Digital Image Processing, 3rd ed. (Pearson Prentice Hall, 2008).

Zhang, L.

M. K. Ng, H. Shen, E. Lam, and L. Zhang, “A total variation regularization based super-resolution reconstruction algorithm for digital video,” EURASIP Journal on Advances in Signal Processing 2007, 1–16 (2007). Article ID 74585.
[CrossRef]

Zhang, X.

Zhong, W.

Zou, Y.

Y. Zou and X. Pan, “Image reconstruction on PI-lines by use of filtered backprojection in helical cone-beam CT,” Phys. Med. Biol. 49, 2717–2731 (2004).
[CrossRef] [PubMed]

Appl. Opt.

EURASIP Journal on Advances in Signal Processing

M. K. Ng, H. Shen, E. Lam, and L. Zhang, “A total variation regularization based super-resolution reconstruction algorithm for digital video,” EURASIP Journal on Advances in Signal Processing 2007, 1–16 (2007). Article ID 74585.
[CrossRef]

IEEE J. Sel. Top. Signal Process.

S.-J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE J. Sel. Top. Signal Process. 1, 606–617 (2007).
[CrossRef]

IEEE Signal Process. Mag.

E. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
[CrossRef]

IEEE Trans. Biomed. Eng.

A. J. Devaney, “A computer simulation study of diffraction tomography,” IEEE Trans. Biomed. Eng. BME-30, 377–386 (1983).
[CrossRef]

IEEE Trans. Geosci. Remote Sens.

A. J. Devaney, “Geophysical diffraction tomography,” IEEE Trans. Geosci. Remote Sens. GE-22, 3–13 (1984).
[CrossRef]

IEEE Trans. Image Process.

Y. Li and F. Santosa, “A computational algorithm for minimizing total variation in image restoration,” IEEE Trans. Image Process. 5, 987–995 (1996).
[CrossRef] [PubMed]

IEEE Trans. Inf. Theory

E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[CrossRef]

E. Candès and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[CrossRef]

Inverse Probl.

D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inverse Probl. 19, S165–S187 (2003).
[CrossRef]

J. Mod. Opt.

G. Indebetouw, “Properties of a scanning holographic microscopy: Improved resolution, extended depth-of-focus, and/or optical sectioning,” J. Mod. Opt. 49, 1479–1500 (2002).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Med. Biol.

Y. Zou and X. Pan, “Image reconstruction on PI-lines by use of filtered backprojection in helical cone-beam CT,” Phys. Med. Biol. 49, 2717–2731 (2004).
[CrossRef] [PubMed]

Proc. IEEE

R. K. Mueller, M. Kaveh, and G. Wade, “Reconstructive tomography and applications to ultrasonics,” Proc. IEEE 67, 567–587 (1979).
[CrossRef]

Ultrason. Imaging

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336–350 (1982).
[CrossRef] [PubMed]

Other

T.-C. Poon, Optical Scanning Holography with MATLAB (Springer-Verlag, 2007).
[CrossRef]

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

X. Zhang, E. Y. Lam, and T.-C. Poon, “Fast iterative sectional image reconstruction in optical scanning holography,” in Digital Holography and Three-Dimensional Imaging (DH), OSA Technical Digest (CD) (Optical Society of America, 2009), paper DMA3.

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge Univ. Press, Cambridge, 2004).

U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer-Verlag, 2005).

T.-C. Poon, Digital Holography and Three-Dimensional Display: Principles and Applications (Springer, 2006).
[CrossRef]

T. Chan and J. Shen, Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods (SIAM, 2005).

C. L. Matson and H. Liu, “The Fourier diffraction theorem for turbid media,” in “Advances in Optical Imaging and Photon Migration,” J.Fujimoto and M.Patterson, eds., Vol. 21 of OSA Trends in Optics and Photonics (Optical Society of America, 1998) p. ATuD14.

R. Bracewell, The Fourier Transform and Its Applications, 3rd ed. (McGraw-Hill, 1999),

R. Gonzalez and R. Woods, Digital Image Processing, 3rd ed. (Pearson Prentice Hall, 2008).

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Figures (12)

Fig. 1
Fig. 1

Schematic diagram of the optical scanning holography system.

Fig. 2
Fig. 2

2D slice of the 2D Fourier transform of a diffraction in the spatial frequency domain.

Fig. 3
Fig. 3

Flowchart of edge-preserving reconstruction method (“PDN’s Method” stands for the primal-dual Newton’s method and “GP Method” stands for the gradient projection method).

Fig. 4
Fig. 4

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

Fig. 5
Fig. 5

Holograms containing two-sectional images of the object in the first experiment. Sine hologram in (a) is the real part and cosine hologram in (b) is the imaginary part.

Fig. 6
Fig. 6

Reconstructed sections by the TV regularization method. (a) Section at z 1 = 14 mm and (b) at z 2 = 15 mm .

Fig. 7
Fig. 7

Reconstructed sections by the Tikhonov regularization method. (a) Section at z 1 = 14 mm and (b) at z 2 = 15 mm .

Fig. 8
Fig. 8

Cross-section of the middle row at z 1 .

Fig. 9
Fig. 9

Holograms measured by a physical OSH system [36]. (a) Real part of the recorded hologram. (b) Imaginary part of the recorded hologram.

Fig. 10
Fig. 10

Object in the second experiment scanned by a physical system.

Fig. 11
Fig. 11

Reconstructed sectional images by TV regularization and nonnegative constraint are shown in (a) and (b).

Fig. 12
Fig. 12

Reconstructed sectional images by inverse imaging using Tikhonov regularization are shown in (a) and (b).

Equations (13)

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H ( k x , k y ; z ) = exp { j z 2 k 0 ( k x 2 + k y 2 ) } ,
k z = ( k 0 2 k x 2 k y 2 ) 1 2 = k 0 ( 1 k x 2 + k y 2 k 0 2 ) 1 2 k 0 ( 1 1 2 k x 2 + k y 2 k 0 2 ) , provided k x 2 + k y 2 k 0 2 = k 0 k x 2 + k y 2 2 k 0 .
H ( k x , k y ; z ) = exp { j z 2 k 0 ( k x 2 + k y 2 ) } exp { j z ( k 0 k z ) } .
q ( x , y ) = F x y 1 { G ( k x , k y ; z ) H ( k x , k y ; z ) } d z .
q ( x , y ) G ( k x , k y ; z ) exp { j z ( k 0 k z ) } exp { j ( x k x + y k y ) } d k x d k y d z = ( G ( k x , k y ; z ) exp { j z k 0 } ) exp { j ( x k x + y k y + k z z ) } d k x d k y d z .
Q ( k x , k y ) = G ( k x , k y ; z ) H ( k x , k y ; z ) d z F x y { | g ( x , y ; z ) | 2 } exp { j z ( k z k 0 ) } d z = | g ( x , y ; z ) | 2 exp { j [ x k x + y k y z ( k z k 0 ) ] } d x d y d z .
Q ( k x , k y ) = G ̃ ( k x , k y , ( k z k 0 ) ) = G ̃ ( k x , k y , k 0 k 0 2 k x 2 k y 2 ) .
q ( x , y ) = | g ( x , y ; z ) | 2 h ( x , y ; z ) d z ,
q = i = 1 n H i g i = H g ,
J ( g ) = q H g 2 + α C g 2 ,
J TV ( g ) = q H g 2 + α g TV ,
v ( x , y ) TV = | v ( x , y ) | d x d y .
J ̂ TV ( g ) = q H g 2 + α g TV , subject to g 0 ,

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