Abstract

Here, the possibility of a noniterative solution to the phase retrieval problem is explored. A new look is taken at the phase retrieval problem that reveals that knowledge of a diffraction pattern’s frequency components is enough to recover the image without projective iterations. This occurs when the image is formed using Gaussian bases that give the convenience of a continuous Fourier transform existing in a compact form where square pixels do not. The Gaussian bases are appropriate when circular apertures are used to detect the diffraction pattern because of their optical transfer functions, as discussed briefly. An algorithm is derived that is capable of recovering an image formed by Gaussian bases from only the Fourier transform’s modulus, without background constraints. A practical example is shown.

© 2013 Optical Society of America

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References

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  1. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–250 (1972).
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    [CrossRef]
  5. J. S. Wu, U. Weierstall, and J. Spence, “Iterative phase retrieval without support,” Opt. Lett. 29, 2737–2739 (2004).
    [CrossRef]
  6. S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
    [CrossRef]
  7. J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662–1669 (1998).
    [CrossRef]
  8. S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 011301 (2007).
    [CrossRef]
  9. V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A 20, 40–55 (2003).
    [CrossRef]
  10. H. H. Bauschke, P. L. Combettes, and D. R. Luke, “Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization,” J. Opt. Soc. Am. A 19, 1334–1345 (2002).
    [CrossRef]
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    [CrossRef]
  12. D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21, 37–50 (2005).
    [CrossRef]
  13. R. Trahan and D. Hyland, “Mitigating the effect of noise in the hybrid input–output method of phase retrieval,” Appl. Opt. 52, 3031–3037 (2013).
    [CrossRef]
  14. S. Babaoe-Kafaki, “A quadratic hybridization of Polak–Ribière–Polyak and Fletcher–Reeves conjugate gradient methods,” J. Optim. Theory Appl. 154, 916–932 (2012).
    [CrossRef]
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    [CrossRef]
  16. E. Polak, “Computational methods in optimization; a unified approach,” Math. Program. 3, 131–133 (1972).
  17. E. Polak and G. Ribiére, “Note sur la convergence de méthodes de directions conjuguées,” Rev. Fr. Inf. Rech. Oper. 3, 35–43 (1969).
  18. G. Liu, “Fourier phase retrieval algorithm with noise constraints,” Sig. Process. 21, 339–347 (1990).
    [CrossRef]
  19. G. Liu, “Object reconstruction from noisy holograms: multiplicative noise model,” Opt. Commun. 79, 402–406 (1990).
    [CrossRef]
  20. R. Bates and D. Mnyama, “The status of practical Fourier phase retrieval,” Advances in Electronics and Electron Physics (Academic, 1986), Vol. 67, pp. 1–64.
  21. M. Kohl, A. A. Minkevich, and T. Baumback, “Improved success rate and stability for phase retrieval by including randomized overrelaxation in the hybrid input output algorithm,” Opt. Express 20, 17093–17106 (2012).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  28. H. Altwaijry and D. Hyland, “Detection and characterization of near Earth asteroids using stellar occultation,” in AAS/AIAA Astrodynamics Specialist Conference, Hilton Head, South Carolina, 2013.
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    [CrossRef]
  30. A. R. Smith, “A pixel is not a little square,” Microsoft Technical Memo 6 (Microsoft, 1995).
  31. R. Fernando, GPU Gems: Programming Techniques, Tips and Tricks for Real-Time Graphics (Pearson Higher Education, 2004).
  32. M. Pharr and R. Fernando, GPU Gems 2: Programming Techniques for High-Performance Graphics and General-Purpose Computation (Addison-Wesley, 2005).
  33. P. S. Heckbert, “Survey of texture mapping,” IEEE Comp. Grap. Appl. 6, 56–67 (1986).
    [CrossRef]
  34. D. Shreiner, G. Sellers, J. M. Kessenish, and B. M. Licea-Kane, OpenGL Programming Guide (Addison-Wesley, 2013).
  35. E. Hartman and J. Keeler, “Layered neural networks with Gaussian hidden units as universal approximations,” Neural Comput. 2, 210–215 (1990).
    [CrossRef]
  36. M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge University, 1997).
  37. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964).

2013 (1)

2012 (2)

M. Kohl, A. A. Minkevich, and T. Baumback, “Improved success rate and stability for phase retrieval by including randomized overrelaxation in the hybrid input output algorithm,” Opt. Express 20, 17093–17106 (2012).
[CrossRef]

S. Babaoe-Kafaki, “A quadratic hybridization of Polak–Ribière–Polyak and Fletcher–Reeves conjugate gradient methods,” J. Optim. Theory Appl. 154, 916–932 (2012).
[CrossRef]

2011 (1)

2008 (1)

2007 (3)

S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 011301 (2007).
[CrossRef]

G. Williams, M. Pfeifer, I. Vartanyants, and I. Robinson, “Effectiveness of iterative algorithms in recovering phase in the presence of noise,” Acta Crystallogr. A 63, 36–42 (2007).
[CrossRef]

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
[CrossRef]

2005 (1)

D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21, 37–50 (2005).
[CrossRef]

2004 (1)

2003 (3)

V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A 20, 40–55 (2003).
[CrossRef]

H. H. Bauschke, P. L. Combettes, and D. R. Luke, “Hybrid projection–reflection method for phase retrieval,” J. Opt. Soc. Am. A 20, 1025–2034 (2003).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[CrossRef]

2002 (1)

1998 (1)

1990 (3)

G. Liu, “Fourier phase retrieval algorithm with noise constraints,” Sig. Process. 21, 339–347 (1990).
[CrossRef]

G. Liu, “Object reconstruction from noisy holograms: multiplicative noise model,” Opt. Commun. 79, 402–406 (1990).
[CrossRef]

E. Hartman and J. Keeler, “Layered neural networks with Gaussian hidden units as universal approximations,” Neural Comput. 2, 210–215 (1990).
[CrossRef]

1986 (2)

1982 (1)

1978 (1)

1972 (2)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–250 (1972).

E. Polak, “Computational methods in optimization; a unified approach,” Math. Program. 3, 131–133 (1972).

1969 (1)

E. Polak and G. Ribiére, “Note sur la convergence de méthodes de directions conjuguées,” Rev. Fr. Inf. Rech. Oper. 3, 35–43 (1969).

1964 (1)

R. Fletcher and C. Reeves, “Function minimization by conjugate gradients,” Comput. J. 7, 149–154 (1964).
[CrossRef]

1957 (1)

R. Hanbury Brown and R. Q. Twiss, “Interferometry of the intensity fluctuations in light,” Proc. R. Soc. B 242, 300–324 (1957).
[CrossRef]

1956 (1)

R. Hanbury Brown and R. Q. Twiss, “A test of a new type of Stellar Interferometer on Sirius,” Nature 178, 1046–1048 (1956).
[CrossRef]

1954 (1)

R. Hanbury Brown and R. Q. Twiss, “A new type of interferometer for use in radio astronomy,” Philos. Mag. 45(7), 663–682 (1954).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964).

Altwaijry, H.

H. Altwaijry and D. Hyland, “Detection and characterization of near Earth asteroids using stellar occultation,” in AAS/AIAA Astrodynamics Specialist Conference, Hilton Head, South Carolina, 2013.

Babaoe-Kafaki, S.

S. Babaoe-Kafaki, “A quadratic hybridization of Polak–Ribière–Polyak and Fletcher–Reeves conjugate gradient methods,” J. Optim. Theory Appl. 154, 916–932 (2012).
[CrossRef]

Bates, R.

R. Bates and D. Mnyama, “The status of practical Fourier phase retrieval,” Advances in Electronics and Electron Physics (Academic, 1986), Vol. 67, pp. 1–64.

Baumback, T.

Bauschke, H. H.

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge University, 1997).

Chapman, H. N.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[CrossRef]

J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662–1669 (1998).
[CrossRef]

Chen, W.

Chen, X.

Combettes, P. L.

Elser, V.

Falat, I.

Fernando, R.

R. Fernando, GPU Gems: Programming Techniques, Tips and Tricks for Real-Time Graphics (Pearson Higher Education, 2004).

M. Pharr and R. Fernando, GPU Gems 2: Programming Techniques for High-Performance Graphics and General-Purpose Computation (Addison-Wesley, 2005).

Fienup, J. R.

Fletcher, R.

R. Fletcher and C. Reeves, “Function minimization by conjugate gradients,” Comput. J. 7, 149–154 (1964).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–250 (1972).

Gur, E.

Hanbury Brown, R.

R. Hanbury Brown and R. Q. Twiss, “Interferometry of the intensity fluctuations in light,” Proc. R. Soc. B 242, 300–324 (1957).
[CrossRef]

R. Hanbury Brown and R. Q. Twiss, “A test of a new type of Stellar Interferometer on Sirius,” Nature 178, 1046–1048 (1956).
[CrossRef]

R. Hanbury Brown and R. Q. Twiss, “A new type of interferometer for use in radio astronomy,” Philos. Mag. 45(7), 663–682 (1954).

Hartman, E.

E. Hartman and J. Keeler, “Layered neural networks with Gaussian hidden units as universal approximations,” Neural Comput. 2, 210–215 (1990).
[CrossRef]

Hau-Riege, S. P.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[CrossRef]

He, H.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Heckbert, P. S.

P. S. Heckbert, “Survey of texture mapping,” IEEE Comp. Grap. Appl. 6, 56–67 (1986).
[CrossRef]

Howells, M. R.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Hyland, D.

R. Trahan and D. Hyland, “Mitigating the effect of noise in the hybrid input–output method of phase retrieval,” Appl. Opt. 52, 3031–3037 (2013).
[CrossRef]

H. Altwaijry and D. Hyland, “Detection and characterization of near Earth asteroids using stellar occultation,” in AAS/AIAA Astrodynamics Specialist Conference, Hilton Head, South Carolina, 2013.

Keeler, J.

E. Hartman and J. Keeler, “Layered neural networks with Gaussian hidden units as universal approximations,” Neural Comput. 2, 210–215 (1990).
[CrossRef]

Kessenish, J. M.

D. Shreiner, G. Sellers, J. M. Kessenish, and B. M. Licea-Kane, OpenGL Programming Guide (Addison-Wesley, 2013).

Kohl, M.

Licea-Kane, B. M.

D. Shreiner, G. Sellers, J. M. Kessenish, and B. M. Licea-Kane, OpenGL Programming Guide (Addison-Wesley, 2013).

Liu, G.

G. Liu, “Object reconstruction from noisy holograms: multiplicative noise model,” Opt. Commun. 79, 402–406 (1990).
[CrossRef]

G. Liu, “Fourier phase retrieval algorithm with noise constraints,” Sig. Process. 21, 339–347 (1990).
[CrossRef]

Luke, D. R.

Marchesini, S.

S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 011301 (2007).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Miao, J.

Minkevich, A. A.

Mnyama, D.

R. Bates and D. Mnyama, “The status of practical Fourier phase retrieval,” Advances in Electronics and Electron Physics (Academic, 1986), Vol. 67, pp. 1–64.

Noy, A.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Osten, W.

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
[CrossRef]

Pedrini, G.

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
[CrossRef]

Pfeifer, M.

G. Williams, M. Pfeifer, I. Vartanyants, and I. Robinson, “Effectiveness of iterative algorithms in recovering phase in the presence of noise,” Acta Crystallogr. A 63, 36–42 (2007).
[CrossRef]

Pharr, M.

M. Pharr and R. Fernando, GPU Gems 2: Programming Techniques for High-Performance Graphics and General-Purpose Computation (Addison-Wesley, 2005).

Polak, E.

E. Polak, “Computational methods in optimization; a unified approach,” Math. Program. 3, 131–133 (1972).

E. Polak and G. Ribiére, “Note sur la convergence de méthodes de directions conjuguées,” Rev. Fr. Inf. Rech. Oper. 3, 35–43 (1969).

Reeves, C.

R. Fletcher and C. Reeves, “Function minimization by conjugate gradients,” Comput. J. 7, 149–154 (1964).
[CrossRef]

Ribiére, G.

E. Polak and G. Ribiére, “Note sur la convergence de méthodes de directions conjuguées,” Rev. Fr. Inf. Rech. Oper. 3, 35–43 (1969).

Robinson, I.

G. Williams, M. Pfeifer, I. Vartanyants, and I. Robinson, “Effectiveness of iterative algorithms in recovering phase in the presence of noise,” Acta Crystallogr. A 63, 36–42 (2007).
[CrossRef]

Sarafis, V.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–250 (1972).

Sayre, D.

Sellers, G.

D. Shreiner, G. Sellers, J. M. Kessenish, and B. M. Licea-Kane, OpenGL Programming Guide (Addison-Wesley, 2013).

Shreiner, D.

D. Shreiner, G. Sellers, J. M. Kessenish, and B. M. Licea-Kane, OpenGL Programming Guide (Addison-Wesley, 2013).

Smith, A. R.

A. R. Smith, “A pixel is not a little square,” Microsoft Technical Memo 6 (Microsoft, 1995).

Spence, J.

J. S. Wu, U. Weierstall, and J. Spence, “Iterative phase retrieval without support,” Opt. Lett. 29, 2737–2739 (2004).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964).

Trahan, R.

Twiss, R. Q.

R. Hanbury Brown and R. Q. Twiss, “Interferometry of the intensity fluctuations in light,” Proc. R. Soc. B 242, 300–324 (1957).
[CrossRef]

R. Hanbury Brown and R. Q. Twiss, “A test of a new type of Stellar Interferometer on Sirius,” Nature 178, 1046–1048 (1956).
[CrossRef]

R. Hanbury Brown and R. Q. Twiss, “A new type of interferometer for use in radio astronomy,” Philos. Mag. 45(7), 663–682 (1954).

Vacha, F.

Vacha, M.

Vartanyants, I.

G. Williams, M. Pfeifer, I. Vartanyants, and I. Robinson, “Effectiveness of iterative algorithms in recovering phase in the presence of noise,” Acta Crystallogr. A 63, 36–42 (2007).
[CrossRef]

Wackerman, C. C.

Weierstall, U.

J. S. Wu, U. Weierstall, and J. Spence, “Iterative phase retrieval without support,” Opt. Lett. 29, 2737–2739 (2004).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Williams, G.

G. Williams, M. Pfeifer, I. Vartanyants, and I. Robinson, “Effectiveness of iterative algorithms in recovering phase in the presence of noise,” Acta Crystallogr. A 63, 36–42 (2007).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge University, 1997).

Wu, J. S.

Zalevsky, Z.

Zhang, F.

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
[CrossRef]

Acta Crystallogr. A (1)

G. Williams, M. Pfeifer, I. Vartanyants, and I. Robinson, “Effectiveness of iterative algorithms in recovering phase in the presence of noise,” Acta Crystallogr. A 63, 36–42 (2007).
[CrossRef]

Appl. Opt. (3)

Comput. J. (1)

R. Fletcher and C. Reeves, “Function minimization by conjugate gradients,” Comput. J. 7, 149–154 (1964).
[CrossRef]

IEEE Comp. Grap. Appl. (1)

P. S. Heckbert, “Survey of texture mapping,” IEEE Comp. Grap. Appl. 6, 56–67 (1986).
[CrossRef]

Inverse Probl. (1)

D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21, 37–50 (2005).
[CrossRef]

J. Opt. Soc. Am. A (5)

J. Optim. Theory Appl. (1)

S. Babaoe-Kafaki, “A quadratic hybridization of Polak–Ribière–Polyak and Fletcher–Reeves conjugate gradient methods,” J. Optim. Theory Appl. 154, 916–932 (2012).
[CrossRef]

Math. Program. (1)

E. Polak, “Computational methods in optimization; a unified approach,” Math. Program. 3, 131–133 (1972).

Nature (1)

R. Hanbury Brown and R. Q. Twiss, “A test of a new type of Stellar Interferometer on Sirius,” Nature 178, 1046–1048 (1956).
[CrossRef]

Neural Comput. (1)

E. Hartman and J. Keeler, “Layered neural networks with Gaussian hidden units as universal approximations,” Neural Comput. 2, 210–215 (1990).
[CrossRef]

Opt. Commun. (1)

G. Liu, “Object reconstruction from noisy holograms: multiplicative noise model,” Opt. Commun. 79, 402–406 (1990).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Optik (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–250 (1972).

Philos. Mag. (1)

R. Hanbury Brown and R. Q. Twiss, “A new type of interferometer for use in radio astronomy,” Philos. Mag. 45(7), 663–682 (1954).

Phys. Rev. A (1)

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
[CrossRef]

Phys. Rev. B (1)

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Proc. R. Soc. B (1)

R. Hanbury Brown and R. Q. Twiss, “Interferometry of the intensity fluctuations in light,” Proc. R. Soc. B 242, 300–324 (1957).
[CrossRef]

Rev. Fr. Inf. Rech. Oper. (1)

E. Polak and G. Ribiére, “Note sur la convergence de méthodes de directions conjuguées,” Rev. Fr. Inf. Rech. Oper. 3, 35–43 (1969).

Rev. Sci. Instrum. (1)

S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 011301 (2007).
[CrossRef]

Sig. Process. (1)

G. Liu, “Fourier phase retrieval algorithm with noise constraints,” Sig. Process. 21, 339–347 (1990).
[CrossRef]

Other (8)

R. Bates and D. Mnyama, “The status of practical Fourier phase retrieval,” Advances in Electronics and Electron Physics (Academic, 1986), Vol. 67, pp. 1–64.

H. Altwaijry and D. Hyland, “Detection and characterization of near Earth asteroids using stellar occultation,” in AAS/AIAA Astrodynamics Specialist Conference, Hilton Head, South Carolina, 2013.

A. R. Smith, “A pixel is not a little square,” Microsoft Technical Memo 6 (Microsoft, 1995).

R. Fernando, GPU Gems: Programming Techniques, Tips and Tricks for Real-Time Graphics (Pearson Higher Education, 2004).

M. Pharr and R. Fernando, GPU Gems 2: Programming Techniques for High-Performance Graphics and General-Purpose Computation (Addison-Wesley, 2005).

D. Shreiner, G. Sellers, J. M. Kessenish, and B. M. Licea-Kane, OpenGL Programming Guide (Addison-Wesley, 2013).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge University, 1997).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964).

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Figures (11)

Fig. 1.
Fig. 1.

Comparison of various sampling methods’ representations of the function cos(x).

Fig. 2.
Fig. 2.

Example image with arbitrary geometry formed with Gaussian bases.

Fig. 3.
Fig. 3.

Comparison of the jinc(x) function and a unit Gaussian approximation.

Fig. 4.
Fig. 4.

Sample image for the algorithm demonstration.

Fig. 5.
Fig. 5.

Sample image spectrum analysis showing the progressive development of the four translated images. Each image has different line styles connecting the four Gaussians in the order that Algorithm 1 identified the Gaussians.

Fig. 6.
Fig. 6.

Pleiades star cluster used as an example of GRB phase retrieval; this image serves as both of the inputs to the phase retrieval algorithm.

Fig. 7.
Fig. 7.

Squared Fourier modulus of the Pleiades star cluster image represented as a 1024×1024 array.

Fig. 8.
Fig. 8.

Spectrum analysis of the Fourier transform of the Pleiades image.

Fig. 9.
Fig. 9.

Estimated image of the Pleiades resulting from the application of Algorithm 1 to the spectrum analysis in Fig. 8. The continuous image is point sampled for display.

Fig. 10.
Fig. 10.

512×512 pixel result from the HIO method with a rectangular support region.

Fig. 11.
Fig. 11.

Topmost star in the (a) GRB and (b) HIO images as shown in Figs. 9 and 10, respectively.

Tables (1)

Tables Icon

Algorithm 1. Image Reconstruction from Spectrum Analysis.

Equations (9)

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I^(x,y)=I(i,j),i=round(x),j=round(y),
I^(x,y)=I(i,j)(i+x)(j+y)+I(i+,j)(xi)(j+y)+I(i,j+)(i+x)(yj)+I(i+,j+)(xi)(yj),
U(ω)=CJ1(ω)ω,
I(ω)=C2[J1(ω)ω]2=C2[jinc(ω)]2.
I(θ̲)=j=1NAjexp(12σj2(θ̲θ̲j)2)
J(u̲)=2πj=1NAjσjexp(2πiu̲·θ̲j)exp(2σj2π2u̲·u̲),
J(u̲)=I(θ̲)exp(2πiu̲·θ̲)dθ̲.
|J2(u̲)|=|2πj=1Nk=1N[AjAkσjσkexp(2π2σj2σk2u̲·u̲)exp(2πiu̲·(θ̲jθ̲k))]|.
J^(θ̲)=F{|J2(u̲)|}(θ̲)=2πj=1Nk=1NAjAkσjσkσj2+σk2exp((θ̲+θ̲jθ̲k)22(σj2+σk2)).

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