Abstract

Conventional numerical reconstruction for digital holography using a filter applied in the spatial-frequency domain to extract the primary image may yield suboptimal image quality because of the loss in high-frequency components and interference from other undesirable terms of a hologram. We propose a new numerical reconstruction approach using a statistical technique. This approach reconstructs the complex field of the object from the real-valued hologram intensity data. Because holographic image reconstruction is an ill-posed problem, our statistical technique is based on penalized-likelihood estimation. We develop a Poisson statistical model for this problem and derive an optimization transfer algorithm that monotonically decreases the cost function at each iteration. Simulation results show that our statistical technique has the potential to improve image quality in digital holography relative to conventional reconstruction techniques.

© 2004 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).
  2. D. Gabor, “A new microscope principle,” Nature (London) 161, 777–778 (1948).
    [CrossRef]
  3. E. N. Leith, J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. 52, 1123–1130 (1962).
    [CrossRef]
  4. H. Chen, M. Shih, E. Arons, E. Leith, J. Lopez, D. Dilworth, P. C. Sun, “Electronic holographic imaging through living human tissue,” Appl. Opt. 33, 3630–3632 (1994).
    [CrossRef] [PubMed]
  5. E. Leith, H. Chen, Y. Chen, D. Dilworth, J. Lopez, R. Masri, J. Rudd, J. Valdmanis, “Electronic holography and speckle methods for imaging through tissue using femtosecond gated pulses,” Appl. Opt. 30, 4204–4210 (1991).
    [CrossRef] [PubMed]
  6. Y. Takaki, H. Ohzu, “Fast numerical reconstruction technique for high-resolution hybrid holographic microscopy,” Appl. Opt. 38, 2204–2211 (1999).
    [CrossRef]
  7. Y. Takaki, H. Kawai, H. Ohzu, “Hybrid holographic mi- croscopy free of conjugate and zero-order images,” Appl. Opt. 38, 4990–4996 (1999).
    [CrossRef]
  8. E. Cuche, F. Bevilacqua, C. Depeursinge, “Digital holography for quantitative phase-contrasting imaging,” Opt. Lett. 24, 291–293 (1999).
    [CrossRef]
  9. E. Cuche, P. Marquet, C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39, 4070–4075 (2000).
    [CrossRef]
  10. I. Yamaguchi, T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22, 1268–1270 (1997).
    [CrossRef] [PubMed]
  11. S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “Wave front reconstruction of Fresnel off-axis holograms with compensation of aberrations by means of phase-shifting digital holography,” Opt. Lasers Eng. 37, 331–340 (2002).
    [CrossRef]
  12. M. Liebling, T. Blu, E. Cuche, P. Marquet, C. Depeursinge, M. Unser, “A novel non-diffractive reconstruction method for digital holographic microscopy,” in Proceedings of the IEEE International Symposium on Biomedical Imaging (Institute of Electrical and Electronics Engineers, New York, 2002), pp. 625–628.
  13. C. A. Bouman, K. Sauer, “A unified approach statistical tomography using coordinate descent optimization,” IEEE Trans. Image Process. 5, 480–492 (1996).
    [CrossRef]
  14. H. Erdoğan, J. A. Fessler, “Monotonic algorithms for transmission tomography,” IEEE Trans. Med. Imaging 18, 801–814 (1999).
    [CrossRef] [PubMed]
  15. J. A. Conchello, “Superresolution and convergence properties of the expectation-maximization algorithm for maximum-likelihood deconvolution of incoherent images,” J. Opt. Soc. Am. A 15, 2609–2619 (1998).
    [CrossRef]
  16. T. J. Holmes, “Maximum-likelihood image restoration adapted for noncoherent optical imaging,” J. Opt. Soc. Am. A 5, 666–673 (1988).
    [CrossRef]
  17. M. Çetin, W. C. Karl, A. S. Willsky, “Edge-preserving image reconstruction for coherent imaging applications,” in Proceedings of the IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 2002), Vol. 2, pp. 481–484.
  18. B. W. Silverman, Density Estimation for Statistics and Data Analysis (Chapman & Hall, New York, 1986).
  19. J. A. Fessler, A. O. Hero, “Penalized maximum-likelihood image reconstruction using space-alternating generalized EM algorithms,” IEEE Trans. Image Process. 4, 1417–1429 (1995).
    [CrossRef] [PubMed]
  20. H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968).
  21. J. L. Marroquin, M. Tapia, “Parallel algorithms for phase unwrapping based on Markov random field models,” J. Opt. Soc. Am. A 12, 2578–2585 (1995).
    [CrossRef]
  22. W. Kim, M. H. Hayes, “Phase retrieval using two Fourier-transform intensities,” J. Opt. Soc. Am. A 7, 441–449 (1990).
    [CrossRef]
  23. S. Sotthivirat, J. A. Fessler, “Image recovery using partitioned-separable paraboloidal surrogate coordinate ascent algorithms,” IEEE Trans. Image Process. 11, 306–317 (2002).
    [CrossRef]
  24. E. Arons, E. Leith, “Coherence confocal-imaging system for enhanced depth discrimination in transmitted light,” Appl. Opt. 35, 2499–2506 (1996).
    [CrossRef]
  25. P.-C. Sun, E. N. Leith, “Broad-source image plane holography as a confocal imaging process,” Appl. Opt. 33, 597–602 (1994).
    [CrossRef] [PubMed]
  26. Y. Censor, “Finite series expansion reconstruction methods,” Proc. IEEE 71, 409–419 (1983).
    [CrossRef]
  27. D. L. Snyder, A. M. Hammoud, R. L. White, “Image recovery from data acquired with a charge-coupled-device camera,” J. Opt. Soc. Am. A 10, 1014–1023 (1993).
    [CrossRef] [PubMed]
  28. M. Cetin, W. C. Karl, A. S. Willsky, “Edge-preserving image reconstruction for coherent imaging applications,” in Proceedings of the IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 2002), Vol. 2, pp. 481–484.
  29. M. Çetin, W. C. Karl, “Feature-enhanced synthetic aperture radar image formation based on nonquadratic regularization,” IEEE Trans. Image Process. 10, 623–631 (2001).
    [CrossRef]
  30. K. Lange, “Convergence of EM image reconstruction algo-rithms with Gibbs smoothing,” IEEE Trans. Med. Imaging 9, 439–446 (1990).
    [CrossRef]
  31. J. A. Fessler, “Grouped coordinate descent algorithms for robust edge-preserving image restoration,” in Image Reconstruction and Restoration II, T. J. Schulz, ed., Proc. SPIE3170, 184–194 (1997).
    [CrossRef]
  32. P. J. Huber, Robust Statistics (Wiley, New York, 1981).
  33. J. A. Fessler, “Grouped-coordinate ascent algorithms for penalized-likelihood transmission image reconstruction,” IEEE Trans. Med. Imaging 16, 166–175 (1997).
    [CrossRef] [PubMed]
  34. A. R. De Pierro, “A modified expectation maximization algorithm for penalized likelihood estimation in emission tomography,” IEEE Trans. Med. Imaging 14, 132–137 (1995).
    [CrossRef] [PubMed]
  35. S. Sotthivirat, “Statistical image recovery techniques for optical imaging systems,” Ph.D. dissertation (University of Michigan, Ann Arbor, Mich., 2003).

2002

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “Wave front reconstruction of Fresnel off-axis holograms with compensation of aberrations by means of phase-shifting digital holography,” Opt. Lasers Eng. 37, 331–340 (2002).
[CrossRef]

S. Sotthivirat, J. A. Fessler, “Image recovery using partitioned-separable paraboloidal surrogate coordinate ascent algorithms,” IEEE Trans. Image Process. 11, 306–317 (2002).
[CrossRef]

2001

M. Çetin, W. C. Karl, “Feature-enhanced synthetic aperture radar image formation based on nonquadratic regularization,” IEEE Trans. Image Process. 10, 623–631 (2001).
[CrossRef]

2000

1999

1998

1997

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

J. A. Fessler, “Grouped-coordinate ascent algorithms for penalized-likelihood transmission image reconstruction,” IEEE Trans. Med. Imaging 16, 166–175 (1997).
[CrossRef] [PubMed]

1996

C. A. Bouman, K. Sauer, “A unified approach statistical tomography using coordinate descent optimization,” IEEE Trans. Image Process. 5, 480–492 (1996).
[CrossRef]

E. Arons, E. Leith, “Coherence confocal-imaging system for enhanced depth discrimination in transmitted light,” Appl. Opt. 35, 2499–2506 (1996).
[CrossRef]

1995

A. R. De Pierro, “A modified expectation maximization algorithm for penalized likelihood estimation in emission tomography,” IEEE Trans. Med. Imaging 14, 132–137 (1995).
[CrossRef] [PubMed]

J. A. Fessler, A. O. Hero, “Penalized maximum-likelihood image reconstruction using space-alternating generalized EM algorithms,” IEEE Trans. Image Process. 4, 1417–1429 (1995).
[CrossRef] [PubMed]

J. L. Marroquin, M. Tapia, “Parallel algorithms for phase unwrapping based on Markov random field models,” J. Opt. Soc. Am. A 12, 2578–2585 (1995).
[CrossRef]

1994

1993

1991

1990

W. Kim, M. H. Hayes, “Phase retrieval using two Fourier-transform intensities,” J. Opt. Soc. Am. A 7, 441–449 (1990).
[CrossRef]

K. Lange, “Convergence of EM image reconstruction algo-rithms with Gibbs smoothing,” IEEE Trans. Med. Imaging 9, 439–446 (1990).
[CrossRef]

1988

1983

Y. Censor, “Finite series expansion reconstruction methods,” Proc. IEEE 71, 409–419 (1983).
[CrossRef]

1962

1948

D. Gabor, “A new microscope principle,” Nature (London) 161, 777–778 (1948).
[CrossRef]

Arons, E.

Bevilacqua, F.

Blu, T.

M. Liebling, T. Blu, E. Cuche, P. Marquet, C. Depeursinge, M. Unser, “A novel non-diffractive reconstruction method for digital holographic microscopy,” in Proceedings of the IEEE International Symposium on Biomedical Imaging (Institute of Electrical and Electronics Engineers, New York, 2002), pp. 625–628.

Bouman, C. A.

C. A. Bouman, K. Sauer, “A unified approach statistical tomography using coordinate descent optimization,” IEEE Trans. Image Process. 5, 480–492 (1996).
[CrossRef]

Censor, Y.

Y. Censor, “Finite series expansion reconstruction methods,” Proc. IEEE 71, 409–419 (1983).
[CrossRef]

Cetin, M.

M. Cetin, W. C. Karl, A. S. Willsky, “Edge-preserving image reconstruction for coherent imaging applications,” in Proceedings of the IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 2002), Vol. 2, pp. 481–484.

Çetin, M.

M. Çetin, W. C. Karl, “Feature-enhanced synthetic aperture radar image formation based on nonquadratic regularization,” IEEE Trans. Image Process. 10, 623–631 (2001).
[CrossRef]

M. Çetin, W. C. Karl, A. S. Willsky, “Edge-preserving image reconstruction for coherent imaging applications,” in Proceedings of the IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 2002), Vol. 2, pp. 481–484.

Chen, H.

Chen, Y.

Conchello, J. A.

Cuche, E.

E. Cuche, P. Marquet, C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39, 4070–4075 (2000).
[CrossRef]

E. Cuche, F. Bevilacqua, C. Depeursinge, “Digital holography for quantitative phase-contrasting imaging,” Opt. Lett. 24, 291–293 (1999).
[CrossRef]

M. Liebling, T. Blu, E. Cuche, P. Marquet, C. Depeursinge, M. Unser, “A novel non-diffractive reconstruction method for digital holographic microscopy,” in Proceedings of the IEEE International Symposium on Biomedical Imaging (Institute of Electrical and Electronics Engineers, New York, 2002), pp. 625–628.

De Nicola, S.

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “Wave front reconstruction of Fresnel off-axis holograms with compensation of aberrations by means of phase-shifting digital holography,” Opt. Lasers Eng. 37, 331–340 (2002).
[CrossRef]

De Pierro, A. R.

A. R. De Pierro, “A modified expectation maximization algorithm for penalized likelihood estimation in emission tomography,” IEEE Trans. Med. Imaging 14, 132–137 (1995).
[CrossRef] [PubMed]

Depeursinge, C.

E. Cuche, P. Marquet, C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39, 4070–4075 (2000).
[CrossRef]

E. Cuche, F. Bevilacqua, C. Depeursinge, “Digital holography for quantitative phase-contrasting imaging,” Opt. Lett. 24, 291–293 (1999).
[CrossRef]

M. Liebling, T. Blu, E. Cuche, P. Marquet, C. Depeursinge, M. Unser, “A novel non-diffractive reconstruction method for digital holographic microscopy,” in Proceedings of the IEEE International Symposium on Biomedical Imaging (Institute of Electrical and Electronics Engineers, New York, 2002), pp. 625–628.

Dilworth, D.

Erdogan, H.

H. Erdoğan, J. A. Fessler, “Monotonic algorithms for transmission tomography,” IEEE Trans. Med. Imaging 18, 801–814 (1999).
[CrossRef] [PubMed]

Ferraro, P.

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “Wave front reconstruction of Fresnel off-axis holograms with compensation of aberrations by means of phase-shifting digital holography,” Opt. Lasers Eng. 37, 331–340 (2002).
[CrossRef]

Fessler, J. A.

S. Sotthivirat, J. A. Fessler, “Image recovery using partitioned-separable paraboloidal surrogate coordinate ascent algorithms,” IEEE Trans. Image Process. 11, 306–317 (2002).
[CrossRef]

H. Erdoğan, J. A. Fessler, “Monotonic algorithms for transmission tomography,” IEEE Trans. Med. Imaging 18, 801–814 (1999).
[CrossRef] [PubMed]

J. A. Fessler, “Grouped-coordinate ascent algorithms for penalized-likelihood transmission image reconstruction,” IEEE Trans. Med. Imaging 16, 166–175 (1997).
[CrossRef] [PubMed]

J. A. Fessler, A. O. Hero, “Penalized maximum-likelihood image reconstruction using space-alternating generalized EM algorithms,” IEEE Trans. Image Process. 4, 1417–1429 (1995).
[CrossRef] [PubMed]

J. A. Fessler, “Grouped coordinate descent algorithms for robust edge-preserving image restoration,” in Image Reconstruction and Restoration II, T. J. Schulz, ed., Proc. SPIE3170, 184–194 (1997).
[CrossRef]

Finizio, A.

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “Wave front reconstruction of Fresnel off-axis holograms with compensation of aberrations by means of phase-shifting digital holography,” Opt. Lasers Eng. 37, 331–340 (2002).
[CrossRef]

Gabor, D.

D. Gabor, “A new microscope principle,” Nature (London) 161, 777–778 (1948).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

Hammoud, A. M.

Hayes, M. H.

Hero, A. O.

J. A. Fessler, A. O. Hero, “Penalized maximum-likelihood image reconstruction using space-alternating generalized EM algorithms,” IEEE Trans. Image Process. 4, 1417–1429 (1995).
[CrossRef] [PubMed]

Holmes, T. J.

Huber, P. J.

P. J. Huber, Robust Statistics (Wiley, New York, 1981).

Karl, W. C.

M. Çetin, W. C. Karl, “Feature-enhanced synthetic aperture radar image formation based on nonquadratic regularization,” IEEE Trans. Image Process. 10, 623–631 (2001).
[CrossRef]

M. Cetin, W. C. Karl, A. S. Willsky, “Edge-preserving image reconstruction for coherent imaging applications,” in Proceedings of the IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 2002), Vol. 2, pp. 481–484.

M. Çetin, W. C. Karl, A. S. Willsky, “Edge-preserving image reconstruction for coherent imaging applications,” in Proceedings of the IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 2002), Vol. 2, pp. 481–484.

Kawai, H.

Kim, W.

Lange, K.

K. Lange, “Convergence of EM image reconstruction algo-rithms with Gibbs smoothing,” IEEE Trans. Med. Imaging 9, 439–446 (1990).
[CrossRef]

Leith, E.

Leith, E. N.

Liebling, M.

M. Liebling, T. Blu, E. Cuche, P. Marquet, C. Depeursinge, M. Unser, “A novel non-diffractive reconstruction method for digital holographic microscopy,” in Proceedings of the IEEE International Symposium on Biomedical Imaging (Institute of Electrical and Electronics Engineers, New York, 2002), pp. 625–628.

Lopez, J.

Marquet, P.

E. Cuche, P. Marquet, C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39, 4070–4075 (2000).
[CrossRef]

M. Liebling, T. Blu, E. Cuche, P. Marquet, C. Depeursinge, M. Unser, “A novel non-diffractive reconstruction method for digital holographic microscopy,” in Proceedings of the IEEE International Symposium on Biomedical Imaging (Institute of Electrical and Electronics Engineers, New York, 2002), pp. 625–628.

Marroquin, J. L.

Masri, R.

Ohzu, H.

Pierattini, G.

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “Wave front reconstruction of Fresnel off-axis holograms with compensation of aberrations by means of phase-shifting digital holography,” Opt. Lasers Eng. 37, 331–340 (2002).
[CrossRef]

Rudd, J.

Sauer, K.

C. A. Bouman, K. Sauer, “A unified approach statistical tomography using coordinate descent optimization,” IEEE Trans. Image Process. 5, 480–492 (1996).
[CrossRef]

Shih, M.

Silverman, B. W.

B. W. Silverman, Density Estimation for Statistics and Data Analysis (Chapman & Hall, New York, 1986).

Snyder, D. L.

Sotthivirat, S.

S. Sotthivirat, J. A. Fessler, “Image recovery using partitioned-separable paraboloidal surrogate coordinate ascent algorithms,” IEEE Trans. Image Process. 11, 306–317 (2002).
[CrossRef]

S. Sotthivirat, “Statistical image recovery techniques for optical imaging systems,” Ph.D. dissertation (University of Michigan, Ann Arbor, Mich., 2003).

Sun, P. C.

Sun, P.-C.

Takaki, Y.

Tapia, M.

Unser, M.

M. Liebling, T. Blu, E. Cuche, P. Marquet, C. Depeursinge, M. Unser, “A novel non-diffractive reconstruction method for digital holographic microscopy,” in Proceedings of the IEEE International Symposium on Biomedical Imaging (Institute of Electrical and Electronics Engineers, New York, 2002), pp. 625–628.

Upatnieks, J.

Valdmanis, J.

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968).

White, R. L.

Willsky, A. S.

M. Cetin, W. C. Karl, A. S. Willsky, “Edge-preserving image reconstruction for coherent imaging applications,” in Proceedings of the IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 2002), Vol. 2, pp. 481–484.

M. Çetin, W. C. Karl, A. S. Willsky, “Edge-preserving image reconstruction for coherent imaging applications,” in Proceedings of the IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 2002), Vol. 2, pp. 481–484.

Yamaguchi, I.

Zhang, T.

Appl. Opt.

IEEE Trans. Image Process.

S. Sotthivirat, J. A. Fessler, “Image recovery using partitioned-separable paraboloidal surrogate coordinate ascent algorithms,” IEEE Trans. Image Process. 11, 306–317 (2002).
[CrossRef]

M. Çetin, W. C. Karl, “Feature-enhanced synthetic aperture radar image formation based on nonquadratic regularization,” IEEE Trans. Image Process. 10, 623–631 (2001).
[CrossRef]

C. A. Bouman, K. Sauer, “A unified approach statistical tomography using coordinate descent optimization,” IEEE Trans. Image Process. 5, 480–492 (1996).
[CrossRef]

J. A. Fessler, A. O. Hero, “Penalized maximum-likelihood image reconstruction using space-alternating generalized EM algorithms,” IEEE Trans. Image Process. 4, 1417–1429 (1995).
[CrossRef] [PubMed]

IEEE Trans. Med. Imaging

H. Erdoğan, J. A. Fessler, “Monotonic algorithms for transmission tomography,” IEEE Trans. Med. Imaging 18, 801–814 (1999).
[CrossRef] [PubMed]

K. Lange, “Convergence of EM image reconstruction algo-rithms with Gibbs smoothing,” IEEE Trans. Med. Imaging 9, 439–446 (1990).
[CrossRef]

J. A. Fessler, “Grouped-coordinate ascent algorithms for penalized-likelihood transmission image reconstruction,” IEEE Trans. Med. Imaging 16, 166–175 (1997).
[CrossRef] [PubMed]

A. R. De Pierro, “A modified expectation maximization algorithm for penalized likelihood estimation in emission tomography,” IEEE Trans. Med. Imaging 14, 132–137 (1995).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nature (London)

D. Gabor, “A new microscope principle,” Nature (London) 161, 777–778 (1948).
[CrossRef]

Opt. Lasers Eng.

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “Wave front reconstruction of Fresnel off-axis holograms with compensation of aberrations by means of phase-shifting digital holography,” Opt. Lasers Eng. 37, 331–340 (2002).
[CrossRef]

Opt. Lett.

Proc. IEEE

Y. Censor, “Finite series expansion reconstruction methods,” Proc. IEEE 71, 409–419 (1983).
[CrossRef]

Other

M. Cetin, W. C. Karl, A. S. Willsky, “Edge-preserving image reconstruction for coherent imaging applications,” in Proceedings of the IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 2002), Vol. 2, pp. 481–484.

S. Sotthivirat, “Statistical image recovery techniques for optical imaging systems,” Ph.D. dissertation (University of Michigan, Ann Arbor, Mich., 2003).

J. A. Fessler, “Grouped coordinate descent algorithms for robust edge-preserving image restoration,” in Image Reconstruction and Restoration II, T. J. Schulz, ed., Proc. SPIE3170, 184–194 (1997).
[CrossRef]

P. J. Huber, Robust Statistics (Wiley, New York, 1981).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

M. Liebling, T. Blu, E. Cuche, P. Marquet, C. Depeursinge, M. Unser, “A novel non-diffractive reconstruction method for digital holographic microscopy,” in Proceedings of the IEEE International Symposium on Biomedical Imaging (Institute of Electrical and Electronics Engineers, New York, 2002), pp. 625–628.

H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968).

M. Çetin, W. C. Karl, A. S. Willsky, “Edge-preserving image reconstruction for coherent imaging applications,” in Proceedings of the IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 2002), Vol. 2, pp. 481–484.

B. W. Silverman, Density Estimation for Statistics and Data Analysis (Chapman & Hall, New York, 1986).

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Figures (8)

Fig. 1
Fig. 1

Diagram of digital holography.

Fig. 2
Fig. 2

Holographic reconstruction using a filtering method. IFFT, inverse fast Fourier transform.

Fig. 3
Fig. 3

Illustration of the marginal cost function h i ( l R ,   0 ) and surrogate functions as a function of l R . The solid curve is the original marginal cost function. The two other curves lying above the cost function are the surrogate functions. The function denoted by the dashed curve is called the paraboloidal-surrogate function, which has the same first derivative and the same point as those of the original cost function at l = l n .

Fig. 4
Fig. 4

Holographic reconstruction of a complex object. The top half of each pair represents the magnitude of the image, and the bottom half represents the phase of the image, except for the hologram data in (b). (a) Original image. (b) Two different holograms. (c) Conventional reconstruction using an apodizing Gaussian filter ( NRMSE = 40.0 % ) . (d) Half-size PL reconstruction using one hologram ( NRMSE = 17.5 % ) . Linear interpolation in the vertical direction to the same size as that of the original image is performed for display. (e) Full-size PL reconstruction using one hologram ( NRMSE = 17.3 % ) . (f) Full-size PL reconstruction using two holograms ( NRMSE = 14.1 % ) .

Fig. 5
Fig. 5

Profiles of the magnitude of the numerical reconstructed images across the second row of circles.

Fig. 6
Fig. 6

Profiles of the phase of the numerical reconstructed images across the second row of circles.

Fig. 7
Fig. 7

Contours of the marginal objective functions at one pixel when (a) using one hologram and (b) using two holograms for full-size reconstruction. The “×” mark indicates the optimal solution at 20 + i 110 , and the “○” marks indicate the updates of the estimates starting at 150 + i 150 .

Fig. 8
Fig. 8

PL reconstruction of a real object using the real object constraint: (a) original image, (b) and (c) hologram data, (d) conventional reconstruction using an apodizing Gaussian filter ( NRMSE = 43.8 % ) , (e) half-size PL reconstruction using one hologram ( NRMSE = 22.8 % ) , (f) full-size PL reconstruction using one hologram ( NRMSE = 21.1 % ) , (g) full-size PL reconstruction using two holograms ( NRMSE = 17.2 % ) .

Equations (106)

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

u o ( r ) = h ( r ;   r ) f true ( r ) d r ,
I ( r ) = | u o ( r ) + u ref ( r ) | 2 = | u o ( r ) | 2 + | u ref ( r ) | 2 + u o ( r ) u ref * ( r ) + u o * ( r ) u ref ( r ) ,
u ref   ( r ) = U ref exp ( - i 2 π r     α ) ,
α = sin   θ x λ ,   sin   θ y λ ,
E ( Y i ) = | u o ( r i ) + u ref ( r i ) | 2 + b i , i = 1 , , N ,
I ( ν ) = I o ( ν ) + | U ref | 2 δ ( ν ) + U ref U o ( ν - α ) + U ref U o * ( - ν - α ) ,
f true ( r ) f ( r ) = j = 1 P x j χ j ( r ) ,
u o ( r i ) = h ( r i ;   r ) i = 1 P x j χ j ( r ) d r = j = 1 P a ij x j = [ Ax ] i ,
a ij = h ( r i ;   r ) χ j ( r ) d r .
E ( Y i ) = | [ Ax ] i + u i | 2 + b i , i = 1 , , N .
Y i Poisson ( | [ Ax ] i + u i | 2 + b i ) , i = 1 , , N .
Φ ( x ) = L ( x ) + V ( x ) ,
L ( x ) = i = 1 N h i ( [ Ax ] i ) ,
h i ( l ) - y i log ( | l + u i | 2 +  b i ) + ( | l + u i | 2 + b i ) ,
V ( x ) = β i = 1 r ψ ( [ Cx ] i ) ,
ψ ( t ) = δ 2 [ | t / δ | - log ( 1 + | t / δ | ) ] ,
Cx = - 1 1 0 0 0 0 - 1 1 - 1 0 1 0 0 - 1 0 1  x 1 x 2 x 3 x 4 = x 2 - x 1 x 4 - x 3 x 3 - x 1 x 4 - x 2 .
x ˆ arg  minx   Φ ( x ) .
x n + 1 arg   minx   ϕ ( x ;   x n ) ,
Φ ( x n ) - Φ ( x ) ϕ ( x n ;   x n ) - ϕ ( x ;   x n ) x 0 .
( i ) ϕ ( x n ;   x n ) = Φ ( x n ) ,
( ii ) ϕ ( x ;   x n ) Φ ( x ) x C P ,
( iii ) x j   ϕ ( x ;   x n ) | x = x n  =  x j   Φ ( x ) | x = x n j ,
h i ( l R ,   l I ) = - y i       log α i R , n ( l R + u i R ) 2 + b i / 2 α i R , n + α i I , n ( l I + u i I ) 2 + b i / 2 α i I , n + [ ( l R + u i R ) 2 + b i / 2 ] + [ ( l I + u i I ) 2 + b i / 2 ] ,
α i R , n = ( l R , n + u i R ) 2 + b i / 2 k i n ,
α i I , n = ( l I , n + u i I ) 2 + b i / 2 k i n ,
k i n = | l i n + u i | 2 + b i , l i n = [ Ax n ] i .
h i ( l ) = h i ( l R ,   l I ) h i R , n ( l R ) + h i I , n ( l I ) ,
h i R , n ( l R ) - y i α i R , n       log ( l R + u i R ) 2 + b i / 2 α i R , n
+ ( l R + u i R ) 2 + b i / 2 ,
h i I , n ( l I ) - y i α i I , n       log ( l I + u i I ) 2 + b i / 2 α i I , n
+ ( l I + u i I ) 2 + b i / 2 .
q i R , n ( l R ) = h i R , n ( l i R , n ) + h ˙ i R , n ( l i R , n ) ( l R - l i R , n ) + 1 2 c i R , n ( l R - l i R , n ) 2 ,
( i ) q i R , n ( l i R , n ) = h i R , n ( l i R , n ) ,
( ii ) q i R , n ( l R ) h i R , n ( l R , n ) l R ,
( iii ) q ˙ i R , n ( l i R , n ) = h ˙ i R , n ( l i R , n ) .
c i R , n = maxlRR       h ˙ i R , n ( l R ) - h ˙ i R , n ( l i R , n ) l R - l i R , n ,
    c i R , n = 2 y i [ b i 2 + 2 b i ( l i R , n + u i R ) 2 ] 1 / 2 ( l i R , n + u i R ) 2 k i n ( b i 2 + b i { 2 ( l i R , n + u i R ) 2 + [ b i 2 + 2 b i ( l i R , n + u i R ) 2 ] 1 / 2 } ) + 2 .
L ( x ) = i = 1 N h i ( [ Ax ] i ) i = 1 N h i R , n ( [ Ax ] i R ) + h i I , n ( [ Ax ] i I ) i = 1 N q i R , n ( [ Ax ] i R ) + q i I , n ( [ Ax ] i I )
  Q ( x ;   x n ) .
q i R , n ( [ Ax ] i R ) = q i R , n j = 1 P p ij [ a ij ( x j - x j n ) ] R p ij + l i R , n j = 1 P p ij q i R , n [ a ij ( x j - x j n ) ] R p ij + l i R , n ,
Q ( x ; x n ) j = 1 P Q j n ( x j ) ,
Q j n ( x j ) Q j R , n ( x j ) + Q j I , n ( x j ) ,
Q j R , n ( x j ) i = 1 N p ij q i R , n [ a ij ( x j - x j n ) ] R p ij + l i R , n ,
x j n + 1 arg   minx j   Q j n ( x j )
= x j n - H j - 1 Q j n ( x j n ) , j = 1 , , P ,
Q j n ( x j n ) x j R   Q j n ( x j ) x j I   Q j n ( x j ) x j = x j n
= i = 1 N a ij R h ˙ i R , n ( l i R , n ) + a ij I h ˙ i I , n ( l i I , n ) i = 1 N - a ij I h ˙ i R , n ( l i R , n ) + a ij R h ˙ i I , n ( l i I , n ) = x j R   L ( x ) x j I   L ( x ) x = x n L ˙ j R , n L ˙ j I , n
    H j d j RR d j RI d j IR d j II = 2 ( x j R ) 2   Q j n ( x j ) 2 x j R x j I   Q j n ( x j ) 2 x j I x j R   Q j n ( x j ) 2 ( x j I ) 2   Q j n ( x j ) x j = x j n = i = 1 N 1 p ij   [ ( a ij R ) 2 c i R , n + ( a ij I ) 2 c i I , n ] i = 1 N a ij R a ij I p ij   ( - c i R , n + c i I , n ) i = 1 N a ij R a ij I p ij   ( - c i R , n + c i I , n ) i = 1 N 1 p ij   [ ( a ij I ) 2 c i R , n + ( a ij R ) 2 c i I , n ] .
x j R , n + 1 x j I , n + 1 = x j R , n - 1 det   H j   ( d j II L ˙ j R , n - d j RI L ˙ j I , n ) x j I , n - 1 det   H j   ( - d j RI L ˙ j R , n + d j RR L ˙ j I , n ) ,
det   H j = d j RR d j II - ( d j RI ) 2 .
V ( x ) = β R i = 1 r ψ ( [ C R x R ] i ) + β I i = 1 r ψ ( [ C I x I ] i ) ,
V ( x ) V ( x ;   x n ) j = 1 P S j n ( x j ) ,
V ( x ;   x n ) = β R i = 1 r φ ( [ C R x R ] i ;   [ C R x R , n ] i ) + β I i = 1 r φ ( [ C I x I ] i ;   [ C I x I , n ] i ) ,
φ ( t ;   s ) = ψ ( s ) + ψ ˙ ( s ) ( t - s ) + 1 2 ω ( s ) ( t - s ) 2 ,
ω ( s ) = ψ ˙ ( s ) s .
S j R , n ( x j R )
β R i = 1 r γ ij R φ c ij R ( x j - x j n ) R γ ij R + [ C R x R , n ] i ; [ C R x R , n ] i ,
S j I , n ( x j R )
β I i = 1 r γ ij I φ c ij I ( x j - x j n ) I γ ij I + [ C I x I , n ] i ; [ C I x I , n ] i ,
S j n ( x j )
= S j R , n ( x j R ) + S j I , n ( x j I ) ,
S j n ( x j n ) = x j R   S j R , n ( x j R ) x j I   S j I , n ( x j I ) x j = x j n = β R i = 1 r c ij R ψ ˙ ( [ C R x R , n ] i ) β I i = 1 r c ij I ψ ˙ ( [ C I x I , n ] i ) = x j R   V ( x ) x j I   V ( x ) x = x n V ˙ j R , n V ˙ j I , n .
2 S j n ( x j n ) = p j R , n 0 0 p j I , n ,
p j o , n = 2 S j o , n ( x j o ) 2 x j = x j n = β o i = 1 r       ( c ij o ) 2 γ ij o ω ( [ C o x o , n ] i ) .
ϕ j n ( x j ) = Q j n ( x j ) + S j n ( x j ) .
x j n + 1 = arg       minx j   ϕ j n ( x j ) .
x j R , n + 1 x j I , n + 1
= x j R , n x j I , n - 1 det H ˜ j
×  ( d j II + p j R , n ) ( L ˙ j R , n + V ˙ j R , n ) - d j RI ( L ˙ j I , n + V ˙ j I , n ) - d j RI ( L ˙ j R , n + V ˙ j R , n ) + ( d j RR + p j R , n ) ( L ˙ j I , n + V ˙ j I , n ) ,
 
H ˜ j = H j + p j R , n 0 0 p j I , n
det       H ˜ j = ( d j RR + p j R , n ) ( d j II + p j I , n ) - ( d j RI ) 2 .
a i = j = 1 P | a ij | , i = 1 , , N
l ˆ = j = 1 P a ij x ^ j , i = 1 , , N
k i n = | l ˆ + u i | 2 + b i , i = 1 , , N
h i = - 2 y i ( l ˆ + u i ) k i n + 2 ( l ˆ + u i ) , i = 1 , , N .
L ˙ j = i = 1 N a ij * h ˙ i
L ˙ j R = Re { L ˙ j } , L ˙ j I = Im { L ˙ j }
d j RR = i = 1 N       a i | a ij |   [ ( a ij R ) 2 c i R + ( a ij I ) 2 c i I ]
d j II = i = 1 N       a i | a ij |   [ ( a ij I ) 2 c i R + ( a ij R ) 2 c i I ]
d j RI = d j IR = i = 1 N       a i a ij R a ij I | a ij |   ( - c i R + c i I )
V ˙ j R = β R i = 1 r c ij R ψ ˙ ( [ C R x ^ R ] i )
V ˙ j I = β I i = 1 r c ij I ψ ˙ ( [ C I x ^ I ] i )
p j R = β R i = 1 r       ( c ij R ) 2 γ ij R   ω ( [ C R x ^ R ] i )
p j I = β I i = 1 r       ( c ij I ) 2 γ ij I   ω ( [ C I x ^ I ] i )
det   H j = ( d j RR + p j R ) ( d j II + p j I ) - ( d j RI ) 2
x ^ j R = x ^ j R - 1 det   H j   [ ( d j II + p j R ) ( L ˙ j R + V ˙ j R ) - d j RI ( L ˙ j I + V ˙ j I ) ]
x ^ j I = x ^ j I - 1 det   H j   [ - d j RI ( L ˙ j R + V ˙ j R ) + ( d j II + p j R ) ( L ˙ j I + V ˙ j I ) ]
u r 1 ( n 1 ,   n 2 ) = 200   exp - i   2 π 3   n 1 ,
u r 2 ( n 1 ,   n 2 ) = 150   exp - i   2 π 4   n 1 ,
PSNR 10   log 10 max i ( y i - b i ) 2 1 N i = 1 N  [ y i - E ( y i ) ] 2 .
NRMSE = x ˆ - x true x true × 100 % ,
( C 1 ) h ( m ) = q ( m ) for some m ,
( C 2 ) q ˙ ( l ) h ˙ ( l ) , l m ,
( C 3 ) q ˙ ( l ) h ˙ ( l ) , l m
q ( l ) = q ( m ) + m l q ˙ ( t ) d t h ( m ) + m l h ˙ ( t ) d t = h ( l ) .
q ( l ) = h ( m ) + l m [ - q ˙ ( t ) ] d t h ( m ) + l m [ - h ˙ ( t ) ] d t = h ( l ) .
c ( m ) = maxlm h ˙ ( l ) - h ˙ ( m ) l - m ,
q ( l ) = h ( m ) + h ˙ ( m ) ( l - m ) + 1 2 c ( m ) ( l - m ) 2
q ˙ ( l ) = h ˙ ( m ) + c ( m ) ( l - m ) h ˙ ( m ) + h ˙ ( l ) - h ˙ ( m ) l - m   ( l - m ) = h ˙ ( l ) .
h ˙ o ( l ;   l n ) = - 2 y ( l + u o ) [ ( l o , n + u o ) 2 + b / 2 ] k n [ ( l + u o ) 2 + b / 2 ] + 2 ( l + u o ) .
f ( l ) h ˙ o ( l ;   l n ) - h ˙ o ( l o , n ;   l n ) l - l o , n   = 2 y k n ( l + u o ) ( l o , n + u o ) - b / 2 ( l + u o ) 2 + b / 2 + 2 .
f ˙ ( l ) = 2 y k n - ( l o , n + u o ) ( l + u o ) 2 + b ( l + u o ) + ( b / 2 ) ( l o , n + u o ) [ ( l + u o ) 2 + b / 2 ] 2 = 0 .
l * = b + [ b 2 + 2 b ( l o , n + u o ) 2 ] 1 / 2 2 ( l o , n + u o ) - u o ,
f ( l * ) = 2 y [ b 2 + 2 b ( l o , n + u o ) 2 ] 1 / 2 ( l o , n + u o ) 2 k n ( b 2 + b { 2 ( l o , n + u o ) 2 + [ b 2 + 2 b ( l o , n + u o ) 2 ] 1 / 2 } ) + 2

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