Abstract

This work evaluates the importance of approximate Fourier phase information in the phase retrieval problem. The main discovery is that a rough phase estimate (up to π/2 rad) allows development of very efficient algorithms whose reconstruction time is an order of magnitude faster than that of the current method of choice—the hybrid input–output (HIO) algorithm. Moreover, a heuristic explanation is provided of why continuous optimization methods like gradient descent or Newton-type algorithms fail when applied to the phase retrieval problem and how the approximate phase information can remedy this situation. Numerical simulations are presented to demonstrate the validity of our analysis and success of our reconstruction method even in cases where the HIO algorithm fails, namely, complex-valued signals without tight support information.

© 2011 Optical Society of America

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References

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  1. R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990).
    [Crossref]
  2. J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
    [Crossref]
  3. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [Crossref] [PubMed]
  4. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  5. J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987).
    [Crossref]
  6. L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268(1992).
    [Crossref]
  7. E. Osherovich, M. Zibulevsky, and I. Yavneh, “Signal reconstruction from the modulus of its Fourier transform,” Tech. Rep. CS-2009-09 (Technion, 2008).
  8. E. Osherovich, M. Zibulevsky, and I. Yavneh, “Fast reconstruction method for diffraction imaging,” in “Advances in Visual Computing,” Vol. 5876 of Lecture Notes in Computer Science (Springer, 2009), pp. 1063–1072.
    [Crossref]
  9. E. Osherovich, M. Zibulevsky, and I. Yavneh, “Simultaneous deconvolution and phase retrieval from noisy data,” Tech. Rep. CS-2010-10 (Technion—Israel Institute of Technology, 2010).
  10. E. Osherovich, M. Zibulevsky, and I. Yavneh, “Numerical solution of inverse problems in optics: Phase retrieval, holography, deblurring, image reconstruction from its defocused versions, and combinations thereof,” Tech. Rep. CS-2010-15(Technion—Israel Institute of Technology, 2010).
  11. M. Nieto-Vesperinas, “A study of the performance of nonlinear least-square optimization methods in the problem of phase retrieval,” J. Mod. Opt. 33, 713–722 (1986).
    [Crossref]
  12. M. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. 30, 140–154(1982).
    [Crossref]
  13. D. P. Bertsekas, Nonlinear Programming, 2nd ed. (Athena Scientific, 1999).

1999 (1)

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

1992 (1)

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

1990 (1)

1987 (1)

1986 (1)

M. Nieto-Vesperinas, “A study of the performance of nonlinear least-square optimization methods in the problem of phase retrieval,” J. Mod. Opt. 33, 713–722 (1986).
[Crossref]

1982 (2)

M. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. 30, 140–154(1982).
[Crossref]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[Crossref] [PubMed]

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, 237–246 (1972).

Bertsekas, D. P.

D. P. Bertsekas, Nonlinear Programming, 2nd ed. (Athena Scientific, 1999).

Charalambous, P.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

Fatemi, E.

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

Fienup, J. R.

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, 237–246 (1972).

Hayes, M.

M. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. 30, 140–154(1982).
[Crossref]

Kirz, J.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

Miao, J.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

Millane, R. P.

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, “A study of the performance of nonlinear least-square optimization methods in the problem of phase retrieval,” J. Mod. Opt. 33, 713–722 (1986).
[Crossref]

Osher, S.

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

Osherovich, E.

E. Osherovich, M. Zibulevsky, and I. Yavneh, “Signal reconstruction from the modulus of its Fourier transform,” Tech. Rep. CS-2009-09 (Technion, 2008).

E. Osherovich, M. Zibulevsky, and I. Yavneh, “Numerical solution of inverse problems in optics: Phase retrieval, holography, deblurring, image reconstruction from its defocused versions, and combinations thereof,” Tech. Rep. CS-2010-15(Technion—Israel Institute of Technology, 2010).

E. Osherovich, M. Zibulevsky, and I. Yavneh, “Simultaneous deconvolution and phase retrieval from noisy data,” Tech. Rep. CS-2010-10 (Technion—Israel Institute of Technology, 2010).

E. Osherovich, M. Zibulevsky, and I. Yavneh, “Fast reconstruction method for diffraction imaging,” in “Advances in Visual Computing,” Vol. 5876 of Lecture Notes in Computer Science (Springer, 2009), pp. 1063–1072.
[Crossref]

Rudin, L. I.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268(1992).
[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, 237–246 (1972).

Sayre, D.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

Yavneh, I.

E. Osherovich, M. Zibulevsky, and I. Yavneh, “Simultaneous deconvolution and phase retrieval from noisy data,” Tech. Rep. CS-2010-10 (Technion—Israel Institute of Technology, 2010).

E. Osherovich, M. Zibulevsky, and I. Yavneh, “Signal reconstruction from the modulus of its Fourier transform,” Tech. Rep. CS-2009-09 (Technion, 2008).

E. Osherovich, M. Zibulevsky, and I. Yavneh, “Numerical solution of inverse problems in optics: Phase retrieval, holography, deblurring, image reconstruction from its defocused versions, and combinations thereof,” Tech. Rep. CS-2010-15(Technion—Israel Institute of Technology, 2010).

E. Osherovich, M. Zibulevsky, and I. Yavneh, “Fast reconstruction method for diffraction imaging,” in “Advances in Visual Computing,” Vol. 5876 of Lecture Notes in Computer Science (Springer, 2009), pp. 1063–1072.
[Crossref]

Zibulevsky, M.

E. Osherovich, M. Zibulevsky, and I. Yavneh, “Fast reconstruction method for diffraction imaging,” in “Advances in Visual Computing,” Vol. 5876 of Lecture Notes in Computer Science (Springer, 2009), pp. 1063–1072.
[Crossref]

E. Osherovich, M. Zibulevsky, and I. Yavneh, “Numerical solution of inverse problems in optics: Phase retrieval, holography, deblurring, image reconstruction from its defocused versions, and combinations thereof,” Tech. Rep. CS-2010-15(Technion—Israel Institute of Technology, 2010).

E. Osherovich, M. Zibulevsky, and I. Yavneh, “Signal reconstruction from the modulus of its Fourier transform,” Tech. Rep. CS-2009-09 (Technion, 2008).

E. Osherovich, M. Zibulevsky, and I. Yavneh, “Simultaneous deconvolution and phase retrieval from noisy data,” Tech. Rep. CS-2010-10 (Technion—Israel Institute of Technology, 2010).

Appl. Opt. (1)

IEEE Trans. Acoust. Speech Signal Process. (1)

M. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. 30, 140–154(1982).
[Crossref]

J. Mod. Opt. (1)

M. Nieto-Vesperinas, “A study of the performance of nonlinear least-square optimization methods in the problem of phase retrieval,” J. Mod. Opt. 33, 713–722 (1986).
[Crossref]

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

Nature (1)

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

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, 237–246 (1972).

Physica D (1)

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

Other (5)

E. Osherovich, M. Zibulevsky, and I. Yavneh, “Signal reconstruction from the modulus of its Fourier transform,” Tech. Rep. CS-2009-09 (Technion, 2008).

E. Osherovich, M. Zibulevsky, and I. Yavneh, “Fast reconstruction method for diffraction imaging,” in “Advances in Visual Computing,” Vol. 5876 of Lecture Notes in Computer Science (Springer, 2009), pp. 1063–1072.
[Crossref]

E. Osherovich, M. Zibulevsky, and I. Yavneh, “Simultaneous deconvolution and phase retrieval from noisy data,” Tech. Rep. CS-2010-10 (Technion—Israel Institute of Technology, 2010).

E. Osherovich, M. Zibulevsky, and I. Yavneh, “Numerical solution of inverse problems in optics: Phase retrieval, holography, deblurring, image reconstruction from its defocused versions, and combinations thereof,” Tech. Rep. CS-2010-15(Technion—Israel Institute of Technology, 2010).

D. P. Bertsekas, Nonlinear Programming, 2nd ed. (Athena Scientific, 1999).

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Figures (7)

Fig. 1
Fig. 1 Current reconstruction methods.

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Fig. 2
Fig. 2 Three possible scenarios (in the Fourier domain).

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Fig. 3
Fig. 3 Test images (object plane intensity).

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Fig. 4
Fig. 4 Reconstruction success rate of our method as a function of phase uncertainty interval.

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Fig. 5
Fig. 5 Reconstruction results after 1000 iterations with phase uncertainty interval of 1.57   rad . Top row, loose support; bottom row, tight support.

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Fig. 6
Fig. 6 Error behavior in the case of loose support.

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Fig. 7
Fig. 7 Error behavior in the case of tight support.

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Equations (16)

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

x ^ ≡ F x .
x t k + 1 = { y t k , t   ∉   ∨ x t k − γ y t k , t ∈ ∨ .
f ( x ) = 1 2 ‖ | F x | − | z ^ | ‖ 2 .
∇ f ( x ) = x − F − 1 ( | z ^ | ∘ F x | F x | ) ,
x o = 0 , ∀     o ∈ O ,
x s ≥ 0 , ∀     s   ∉   O .
〈 ∇ f ( x ) , ( w − x ) 〉 ≥ 0 ,
〈 ∇ f ( x ) , ( w − x ) 〉 = λ 〈 ∇ f ( x ) , z 〉 + ( γ − 1 ) 〈 ∇ f ( x ) , x 〉 .
〈 ∇ f ( x ) , z 〉 = 〈 F ∇ f ( x ) , F z 〉
= ∑ i ( | x ^ i | − | z ^ i | ) | z ^ i | cos ⁡ α i .
0 = 〈 ∇ f ( x ) , x 〉 = 〈 F ∇ f ( x ) , F x 〉
= ∑ i ( | x ^ i | − | z ^ i | ) | x ^ i | .
∑ i ( | x ^ i | − | z ^ i | ) | z ^ i | cos ⁡ α i = ∑ i ∈ B ( | x ^ i | − | z ^ i | ) | z ^ i | cos ⁡ α i + ∑ i ∈ S ( | x ^ i | − | z ^ i | ) | z ^ i | cos ⁡ α i + ∑ i ∈ A ( | x ^ i | − | z ^ i | ) | z ^ i | cos ⁡ α i ,
min x d ( F x , C ) subject to x O = 0 ,
min x ‖ | F x | − | z ^ | ‖ 2 subject to x O = 0 , F x ∈ C .
min x ‖ | F x | − | z ^ | ‖ 2 + μ d ( F x , C ) subject to x O = 0 , α ≤ ∠ ( F x ) ≤ β .

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