Abstract

In this paper, we propose a new technique for high-quality reconstruction from single digital holographic acquisitions. The unknown complex object field is found as the solution of a nonlinear inverse problem that consists in the minimization of an energy functional. The latter includes total-variation (TV) regularization terms that constrain the spatial amplitude and phase distributions of the reconstructed data. The algorithm that we derive tolerates downsampling, which allows to acquire substantially fewer measurements for reconstruction compared to the state of the art. We demonstrate the effectiveness of our method through several experiments on simulated and real off-axis holograms.

© 2013 Optical Society of America

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References

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  1. E. Cuche, P. Marquet, and C. Depeursinge, “Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms,” Appl. Opt. 38(34), 6994–7001 (1999).
    [Crossref]
  2. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22(16), 1268–1270 (1997).
    [Crossref] [PubMed]
  3. G. Popescu, T. Ikeda, R. Dasari, and M. Feld, “Diffraction phase microscopy for quantifying cell structure and dynamics,” Opt. Lett. 31(6), 775–777 (2006).
    [Crossref] [PubMed]
  4. Y. Awatsuji, T. Tahara, A. Kaneko, T. Koyama, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Parallel two-step phase-shifting digital holography,” Appl. Opt. 47(19), D183–D189 (2008).
    [Crossref]
  5. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35(2), 227–246 (1972).
  6. V. Katkovnik, A. Migukin, and J. Astola, “3D wave field reconstruction from intensity-only data: variational inverse imaging techniques,” in 9th Euro-American Workshop on Information Optics (Helsinki, Finland, 2010).
  7. M. Cetin, W. Karl, and A. Willsky, “Edge-preserving image reconstruction for coherent imaging applications,” in Proceedings of ICIP 2002 International Conference on Image Processing (Rochester, NY, USA, 2002).
  8. X. Zhang and E. Lam, “Edge-preserving sectional image reconstruction in optical scanning holography,” J. Opt. Soc. Am. A 27(7), 1630–1637 (2010).
    [Crossref]
  9. D. Brady, K. Choi, D. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17(15), 13,040–13,049 (2009).
  10. M. Marim, M. Atlan, E. Angelini, and J. Olivo-Marin, “Compressed sensing with off-axis frequency-shifting holography,” Opt. Lett. 35(6), 871–873 (2010).
    [Crossref] [PubMed]
  11. Y. Rivenson, A. Stern, and B. Javidi, “Compressive fresnel holography,” J. Disp. Technol. 6(10), 506–509 (2010).
    [Crossref]
  12. S. Sotthivirat and J. Fessler, “Penalized-likelihood image reconstruction for digital holography,” J. Opt. Soc. Am. A 21(5), 737–750 (2004).
    [Crossref]
  13. M. Marim, E. Angelini, J.-C. Olivo-Marin, and M. Atlan, “Off-axis compressed holographic microscopy in low-light conditions,” Opt. Lett. 36(1), 79–81 (2011).
    [Crossref] [PubMed]
  14. N. Pavillon, C. Seelamantula, J. Kuhn, M. Unser, and C. Depeursinge, “Suppression of the zero-order term in off-axis digital holography through nonlinear filtering,” Appl. Opt. 48(34), H186–H195 (2009).
    [Crossref]
  15. T. Colomb, J. Kühn, F. Charrière, C. Depeursinge, P. Marquet, and N. Aspert, “Total aberrations compensation in digital holographic microscopy with a reference conjugated hologram,” Opt. Express 14(10), 4300–4306 (2006).
    [Crossref] [PubMed]
  16. M. Bigas, E. Cabruja, J. Forest, and J. Salvi, “Review of CMOS image sensors,” Microelectron. J. 37(5), 433–451 (2006).
    [Crossref]
  17. L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259–268 (1992).
    [Crossref]
  18. T. Chan and P. Mulet, “On the convergence of the lagged diffusivity fixed point method in total variation image restoration,” SIAM J. Numer. Anal. 36(2), 354–367 (1999).
    [Crossref]
  19. O. Scherzer, ed., Energy Minimization Methods (Springer, 2011).
  20. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw Hills Companies, Inc., 1996).
  21. M. Guizar-Sicairos, A. Diaz, M. Holler, M. Lucas, A. Menzel, R. Wepf, and O. Bunk, “Phase tomography from x-ray coherent diffractive imaging projections,” Opt. Express 19(22), 345–,357 (2011).
    [Crossref]
  22. K. Choi, R. Horisaki, J. Hahn, S. Lim, D. L. Marks, T. J. Schulz, and D. J. Brady, “Compressive holography of diffuse objects,” Appl. Opt. 49(34), H1–H10 (2010).
    [Crossref]

2011 (2)

M. Guizar-Sicairos, A. Diaz, M. Holler, M. Lucas, A. Menzel, R. Wepf, and O. Bunk, “Phase tomography from x-ray coherent diffractive imaging projections,” Opt. Express 19(22), 345–,357 (2011).
[Crossref]

M. Marim, E. Angelini, J.-C. Olivo-Marin, and M. Atlan, “Off-axis compressed holographic microscopy in low-light conditions,” Opt. Lett. 36(1), 79–81 (2011).
[Crossref] [PubMed]

2010 (4)

2009 (2)

2008 (1)

2006 (3)

2004 (1)

1999 (2)

T. Chan and P. Mulet, “On the convergence of the lagged diffusivity fixed point method in total variation image restoration,” SIAM J. Numer. Anal. 36(2), 354–367 (1999).
[Crossref]

E. Cuche, P. Marquet, and C. Depeursinge, “Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms,” Appl. Opt. 38(34), 6994–7001 (1999).
[Crossref]

1997 (1)

1992 (1)

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259–268 (1992).
[Crossref]

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35(2), 227–246 (1972).

Angelini, E.

Aspert, N.

Astola, J.

V. Katkovnik, A. Migukin, and J. Astola, “3D wave field reconstruction from intensity-only data: variational inverse imaging techniques,” in 9th Euro-American Workshop on Information Optics (Helsinki, Finland, 2010).

Atlan, M.

Awatsuji, Y.

Bigas, M.

M. Bigas, E. Cabruja, J. Forest, and J. Salvi, “Review of CMOS image sensors,” Microelectron. J. 37(5), 433–451 (2006).
[Crossref]

Brady, D.

D. Brady, K. Choi, D. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17(15), 13,040–13,049 (2009).

Brady, D. J.

Bunk, O.

M. Guizar-Sicairos, A. Diaz, M. Holler, M. Lucas, A. Menzel, R. Wepf, and O. Bunk, “Phase tomography from x-ray coherent diffractive imaging projections,” Opt. Express 19(22), 345–,357 (2011).
[Crossref]

Cabruja, E.

M. Bigas, E. Cabruja, J. Forest, and J. Salvi, “Review of CMOS image sensors,” Microelectron. J. 37(5), 433–451 (2006).
[Crossref]

Cetin, M.

M. Cetin, W. Karl, and A. Willsky, “Edge-preserving image reconstruction for coherent imaging applications,” in Proceedings of ICIP 2002 International Conference on Image Processing (Rochester, NY, USA, 2002).

Chan, T.

T. Chan and P. Mulet, “On the convergence of the lagged diffusivity fixed point method in total variation image restoration,” SIAM J. Numer. Anal. 36(2), 354–367 (1999).
[Crossref]

Charrière, F.

Choi, K.

K. Choi, R. Horisaki, J. Hahn, S. Lim, D. L. Marks, T. J. Schulz, and D. J. Brady, “Compressive holography of diffuse objects,” Appl. Opt. 49(34), H1–H10 (2010).
[Crossref]

D. Brady, K. Choi, D. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17(15), 13,040–13,049 (2009).

Colomb, T.

Cuche, E.

Dasari, R.

Depeursinge, C.

Diaz, A.

M. Guizar-Sicairos, A. Diaz, M. Holler, M. Lucas, A. Menzel, R. Wepf, and O. Bunk, “Phase tomography from x-ray coherent diffractive imaging projections,” Opt. Express 19(22), 345–,357 (2011).
[Crossref]

Fatemi, E.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259–268 (1992).
[Crossref]

Feld, M.

Fessler, J.

Forest, J.

M. Bigas, E. Cabruja, J. Forest, and J. Salvi, “Review of CMOS image sensors,” Microelectron. J. 37(5), 433–451 (2006).
[Crossref]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35(2), 227–246 (1972).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw Hills Companies, Inc., 1996).

Guizar-Sicairos, M.

M. Guizar-Sicairos, A. Diaz, M. Holler, M. Lucas, A. Menzel, R. Wepf, and O. Bunk, “Phase tomography from x-ray coherent diffractive imaging projections,” Opt. Express 19(22), 345–,357 (2011).
[Crossref]

Hahn, J.

Holler, M.

M. Guizar-Sicairos, A. Diaz, M. Holler, M. Lucas, A. Menzel, R. Wepf, and O. Bunk, “Phase tomography from x-ray coherent diffractive imaging projections,” Opt. Express 19(22), 345–,357 (2011).
[Crossref]

Horisaki, R.

K. Choi, R. Horisaki, J. Hahn, S. Lim, D. L. Marks, T. J. Schulz, and D. J. Brady, “Compressive holography of diffuse objects,” Appl. Opt. 49(34), H1–H10 (2010).
[Crossref]

D. Brady, K. Choi, D. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17(15), 13,040–13,049 (2009).

Ikeda, T.

Javidi, B.

Y. Rivenson, A. Stern, and B. Javidi, “Compressive fresnel holography,” J. Disp. Technol. 6(10), 506–509 (2010).
[Crossref]

Kaneko, A.

Karl, W.

M. Cetin, W. Karl, and A. Willsky, “Edge-preserving image reconstruction for coherent imaging applications,” in Proceedings of ICIP 2002 International Conference on Image Processing (Rochester, NY, USA, 2002).

Katkovnik, V.

V. Katkovnik, A. Migukin, and J. Astola, “3D wave field reconstruction from intensity-only data: variational inverse imaging techniques,” in 9th Euro-American Workshop on Information Optics (Helsinki, Finland, 2010).

Koyama, T.

Kubota, T.

Kuhn, J.

Kühn, J.

Lam, E.

Lim, S.

K. Choi, R. Horisaki, J. Hahn, S. Lim, D. L. Marks, T. J. Schulz, and D. J. Brady, “Compressive holography of diffuse objects,” Appl. Opt. 49(34), H1–H10 (2010).
[Crossref]

D. Brady, K. Choi, D. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17(15), 13,040–13,049 (2009).

Lucas, M.

M. Guizar-Sicairos, A. Diaz, M. Holler, M. Lucas, A. Menzel, R. Wepf, and O. Bunk, “Phase tomography from x-ray coherent diffractive imaging projections,” Opt. Express 19(22), 345–,357 (2011).
[Crossref]

Marim, M.

Marks, D.

D. Brady, K. Choi, D. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17(15), 13,040–13,049 (2009).

Marks, D. L.

Marquet, P.

Matoba, O.

Menzel, A.

M. Guizar-Sicairos, A. Diaz, M. Holler, M. Lucas, A. Menzel, R. Wepf, and O. Bunk, “Phase tomography from x-ray coherent diffractive imaging projections,” Opt. Express 19(22), 345–,357 (2011).
[Crossref]

Migukin, A.

V. Katkovnik, A. Migukin, and J. Astola, “3D wave field reconstruction from intensity-only data: variational inverse imaging techniques,” in 9th Euro-American Workshop on Information Optics (Helsinki, Finland, 2010).

Mulet, P.

T. Chan and P. Mulet, “On the convergence of the lagged diffusivity fixed point method in total variation image restoration,” SIAM J. Numer. Anal. 36(2), 354–367 (1999).
[Crossref]

Nishio, K.

Olivo-Marin, J.

Olivo-Marin, J.-C.

Osher, S.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259–268 (1992).
[Crossref]

Pavillon, N.

Popescu, G.

Rivenson, Y.

Y. Rivenson, A. Stern, and B. Javidi, “Compressive fresnel holography,” J. Disp. Technol. 6(10), 506–509 (2010).
[Crossref]

Rudin, L.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259–268 (1992).
[Crossref]

Salvi, J.

M. Bigas, E. Cabruja, J. Forest, and J. Salvi, “Review of CMOS image sensors,” Microelectron. J. 37(5), 433–451 (2006).
[Crossref]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35(2), 227–246 (1972).

Schulz, T. J.

Seelamantula, C.

Sotthivirat, S.

Stern, A.

Y. Rivenson, A. Stern, and B. Javidi, “Compressive fresnel holography,” J. Disp. Technol. 6(10), 506–509 (2010).
[Crossref]

Tahara, T.

Unser, M.

Ura, S.

Wepf, R.

M. Guizar-Sicairos, A. Diaz, M. Holler, M. Lucas, A. Menzel, R. Wepf, and O. Bunk, “Phase tomography from x-ray coherent diffractive imaging projections,” Opt. Express 19(22), 345–,357 (2011).
[Crossref]

Willsky, A.

M. Cetin, W. Karl, and A. Willsky, “Edge-preserving image reconstruction for coherent imaging applications,” in Proceedings of ICIP 2002 International Conference on Image Processing (Rochester, NY, USA, 2002).

Yamaguchi, I.

Zhang, T.

Zhang, X.

Appl. Opt. (4)

J. Disp. Technol. (1)

Y. Rivenson, A. Stern, and B. Javidi, “Compressive fresnel holography,” J. Disp. Technol. 6(10), 506–509 (2010).
[Crossref]

J. Opt. Soc. Am. A (2)

Microelectron. J. (1)

M. Bigas, E. Cabruja, J. Forest, and J. Salvi, “Review of CMOS image sensors,” Microelectron. J. 37(5), 433–451 (2006).
[Crossref]

Opt. Express (3)

M. Guizar-Sicairos, A. Diaz, M. Holler, M. Lucas, A. Menzel, R. Wepf, and O. Bunk, “Phase tomography from x-ray coherent diffractive imaging projections,” Opt. Express 19(22), 345–,357 (2011).
[Crossref]

D. Brady, K. Choi, D. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17(15), 13,040–13,049 (2009).

T. Colomb, J. Kühn, F. Charrière, C. Depeursinge, P. Marquet, and N. Aspert, “Total aberrations compensation in digital holographic microscopy with a reference conjugated hologram,” Opt. Express 14(10), 4300–4306 (2006).
[Crossref] [PubMed]

Opt. Lett. (4)

Optik (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35(2), 227–246 (1972).

Phys. D (1)

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259–268 (1992).
[Crossref]

SIAM J. Numer. Anal. (1)

T. Chan and P. Mulet, “On the convergence of the lagged diffusivity fixed point method in total variation image restoration,” SIAM J. Numer. Anal. 36(2), 354–367 (1999).
[Crossref]

Other (4)

O. Scherzer, ed., Energy Minimization Methods (Springer, 2011).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw Hills Companies, Inc., 1996).

V. Katkovnik, A. Migukin, and J. Astola, “3D wave field reconstruction from intensity-only data: variational inverse imaging techniques,” in 9th Euro-American Workshop on Information Optics (Helsinki, Finland, 2010).

M. Cetin, W. Karl, and A. Willsky, “Edge-preserving image reconstruction for coherent imaging applications,” in Proceedings of ICIP 2002 International Conference on Image Processing (Rochester, NY, USA, 2002).

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Figures (7)

Fig. 1
Fig. 1 Full-sized oracle maps and corresponding hologram acquisitions used in the synthetic experiments. From (A) to (D): Pentagon, Man, Airplane, and Airport maps defining the objects of Table 1. From (E) to (I): intensity holograms of size 512 × 512 associated with the objects no. 1, 2, 3, 4, and 5.

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Fig. 2
Fig. 2 Reconstruction of the phase-only Man object from the synthetic hologram #5 without downsampling (D = 1) and with downsampling (D = 2).

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Fig. 3
Fig. 3 Reconstruction from the phase-only hologram Neuron.

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Fig. 4
Fig. 4 Reconstruction from the phase-only hologram Epithelial.

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Fig. 5
Fig. 5 Reconstructed amplitudes from the hologram USAF 5-4.

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Fig. 6
Fig. 6 Reconstructed phases from the hologram USAF 5-4.

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Fig. 7
Fig. 7 Reconstructed amplitudes and phases from the fully sampled hologram USAF 9-8.

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Tables (3)

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Table 1 Reconstruction Quality in Synthetic Experiments

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Table 2 Effective Field of View in Synthetic Experiments

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Table 3 Object-Dependent Parameters Used for Optical Acquisition

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Equations (25)

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I ( x ) = | o ( x ; 0 ) + r ( x ; 0 ) | 2 .
I ( x ) = | ( 𝒫 d ℋ Ψ d ) ( x ) + r ( x ; 0 ) | 2 .
I [ m ] = | ( P d ℋ Ψ d ) ( x ) + r ( x ; 0 ) | 2 | x = m Δ x ,
I ( x ) = | ( 𝒫 d ℋ Ψ d ) ( x ) + α r ¯ ( x ) | 2 , ∀ x ∈ m Δ x with w [ m ] = 1.
𝒟 ( Ψ d , α ) = ∑ m ∈ ℤ 2 w [ m ] ( | ( 𝒫 d ℋ Ψ d ) ( x ) + α r ¯ ( x ) | 2 − I ( x ) ) 2 | x = m Δ x < ε .
ε = exp ( − ε SNR ln ( 10 ) / 10 ) ∑ m ∈ ℤ 2 w [ m ] I 2 ( x ) | x = m Δ x ,
TV ϕ ( Ψ d ) = ∫ ℝ 2 ‖ ∇ ( arg u Ψ d ) ( x ) ‖ d x ,
TV ρ ( Ψ d ) = ∫ ℝ 2 ‖ ∇ | Ψ d | ( x ) ‖ d x ,
Ψ ˜ d = arg min Ψ d [ TV ϕ ( Ψ d ) + γ TV ρ ( Ψ d ) ]                 s . t .         𝒟 ˜ ( Ψ d ) < ε ,
𝒟 ˜ ( Ψ d ) = arg min α 𝒟 ( Ψ d , α ) .
Ψ ˜ d = arg min Ψ d [ TV ϕ ( Ψ d ) + γ TV ρ ( Ψ d ) ]           s . t .                   𝒟 ˜ ( Ψ d ) < ε .
𝒟 ( Ψ d , α ) = ∑ i w i ( | ( A Ψ d ) i + α r ¯ i | 2 − y i ) 2 ,
TV ϕ ( Ψ d ) = ∑ i [ ( R 1 ( arg u Ψ d ) ) i 2 + ( R 2 ( arg u Ψ d ) ) i 2 + ν ] 1 / 2 ,
TV ρ ( Ψ d ) = ∑ i [ ( R 1 | { Ψ d } | ) i 2 + ( R 2 | { Ψ d } | ) i 2 + ν ] 1 / 2 ,
H ^ ( ω ) = { 1 , ‖ ω ‖ ≤ Δ ω o / 2 0 , ‖ ω ‖ > Δ ω o / 2.
P ^ d ( ω ) = exp ( i k d ( 1 − λ 2 ‖ ω ‖ 2 ) 1 / 2 ) .
I ˜ ( x ) = | ( 𝒫 d ℋ Ψ d ) ( x ) + α r ¯ ( x ) | 2
ε S N R < 10 log 10 ( ∑ m ∈ ℤ 2 w [ m ] I ( x ) 2 | x = m Δ x ∑ m ∈ ℤ 2 w [ m ] ( I ˜ ( x ) − I ( x ) ) 2 | x = m Δ x ) ,
∑ m ∈ ℤ 2 w [ m ] ( I ˜ ( x ) − I ( x ) ) 2 | x = m Δ x < exp ( − ε SNR ln ( 10 ) / 10 ) ∑ m ∈ ℤ 2 w [ m ] I 2 ( x ) | x = m Δ x .
( R 1 , 2 ( arg u Ψ d ) ) i = { v i , v i < π v i − 2 π , v i ≥ π .
𝒞 ( χ ) = ∑ m ∈ ℤ 2 ( | a [ m ] + χ b [ m ] | 2 − c [ m ] ) 2 ,
𝒞 ( χ ) = A χ 4 + B χ 3 + C χ 2 + D χ + E ,
A = ∑ m ∈ ℤ 2 | b [ m ] | 4 , B = ∑ m ∈ ℤ 2 4 Re { a [ m ] b * [ m ] } | b [ m ] | 2 , C = ∑ m ∈ ℤ 2 2 | b [ m ] | 2 ( | a [ m ] | 2 − c [ m ] ) + 4 ( Re { a [ m ] b * [ m ] } ) 2 , D = ∑ m ∈ ℤ 2 4 Re { a [ m ] b * [ m ] } ( | a [ m ] | 2 − c [ m ] ) , E = ∑ m ∈ ℤ 2 ( | a [ m ] | 2 − c [ m ] ) 2 .
A χ 4 + B χ 3 + C χ 2 + D χ + ( E − ε ) .
4 A χ 3 + 3 B χ 2 + 2 C χ + D

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