Abstract

The problem of recovering a complex signal from the magnitudes of any number of its fractional Fourier transforms at any set of fractional orders is addressed. This problem corresponds to the problem of phase retrieval from the transverse intensity profiles of an optical field at arbitrary locations in an optical system involving arbitrary concatenations of lenses and sections of free space. The dependence of the results on the number of orders, their spread, and the noise is investigated. Generally, increasing the number of orders improves the results, but with diminishing return beyond a certain point. Selecting the measurement planes such that their fractional orders are well separated or spread as much as possible also leads to better results.

© 2005 Optical Society of America

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2005 (3)

2004 (2)

T. E. Gureyev, A. Pogany, D. M. Paganin, S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[CrossRef]

L. J. Allen, W. McBride, M. P. Oxley, “Exit wave reconstruction using soft x-rays,” Opt. Commun. 233, 77–82 (2004).
[CrossRef]

2003 (6)

T. E. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220, 49–58 (2003).
[CrossRef]

T. Alieva, M. J. Bastiaans, L. Stankovic, “Signal reconstruction from two close fractional Fourier power spectra,” IEEE Trans. Signal Process. 51, 112–123 (2003).
[CrossRef]

M. Bastiaans, K. Wolf, “Phase reconstruction from intensity measurements in linear systems,” J. Opt. Soc. Am. A 20, 1046–1049 (2003).
[CrossRef]

B. Hennelly, J. T. Sheridan, “Fractional Fourier transform-based image encryption: phase retrieval algorithm,” Opt. Commun. 226, 61–80 (2003).
[CrossRef]

X. Liu, K.-H. Brenner, “Reconstruction of two-dimensional complex amplitudes from intensity measurements,” Opt. Commun. 225, 19–30 (2003).
[CrossRef]

D. Dragoman, “Redundancy of phase-space distribution functions in complex field recovery problems,” Appl. Opt. 42, 1932–1937 (2003).
[CrossRef] [PubMed]

2002 (1)

2001 (2)

L. J. Allen, M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[CrossRef]

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev E 63, 037602-1–037602-4 (2001).
[CrossRef]

2000 (2)

C. Candan, M. Alper Kutay, H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

T. Alieva, M. J. Bastiaans, “On fractional Fourier transform moments,” IEEE Signal Process. Lett. 7, 321–323 (2000).
[CrossRef]

1999 (2)

M. A. Kutay, H. Özaktaş, H. M. Ozaktas, O. Arikan, “The fractional Fourier domain decomposition,” Signal Process. 77, 105–109 (1999).
[CrossRef]

W.-X. Cong, N.-X. Chen, B.-Y. Gu, “Phase retrieval in the Fresnel transform system: a recursive algorithm,” J. Opt. Soc. Am. A 16, 1827–1830 (1999).
[CrossRef]

1998 (5)

1997 (1)

1996 (3)

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdağı, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

Z. Zalevsky, D. Mendlovic, R. G. Dorsch, “Gerchberg–Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
[CrossRef] [PubMed]

K. A. Nugent, T. E. Gureyev, D. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

1995 (4)

D. F. McAlister, M. Beck, L. Clarke, A. Mayer, M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20, 1181–1183 (1995).
[CrossRef] [PubMed]

O. Aytür, H. M. Ozaktas, “Non-orthogonal domains in phase space of quantum optics and their relation to fractional Fourier transforms,” Opt. Commun. 120, 166–170 (1995).
[CrossRef]

H. M. Ozaktas, O. Aytür, “Fractional Fourier domains,” Signal Process. 46, 119–124 (1995).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
[CrossRef]

1994 (5)

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
[CrossRef]

H. M. Ozaktas, B. Barshan, D. Mendlovic, “Convolution and filtering in fractional Fourier domains,” Opt. Rev. 1, 15–16 (1994).
[CrossRef]

G. Yang, B. Dong, B. Gu, J. Zhuang, O. K. Ersoy, “Gerchberg–Saxton and Yang–Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. 33, 209–218 (1994).
[CrossRef] [PubMed]

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

1993 (3)

1992 (2)

W. Coene, G. Janssen, M. Op. de Beeck, D. Van Dyck, “Phase retrieval through focus variation for ultra-resolution in field-emission transmission electron microscopy,” Phys. Rev. Lett. 69, 3743–3746 (1992).
[CrossRef] [PubMed]

V. Yu. Ivanov, V. P. Sivokon, M. A. Vorontsov, “Phase retrieval from a set of intensity measurements: theory and experiment,” J. Opt. Soc. Am. A 9, 1515–1524 (1992).
[CrossRef]

1990 (1)

1987 (1)

1986 (1)

1982 (1)

1981 (2)

G. Yang, B. Gu, “On the amplitude-phase retrieval problem in the optical system,” Acta Phys. Sin. 30, 410–413 (1981).

B. Gu, G. Yang, “On the phase retrieval problem in optical and electronic microscopy,” Acta Opt. Sin. 1, 517–522 (1981).

1973 (3)

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973).
[CrossRef]

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: I. Test calculations,” J. Phys. D 6, 2200–2216 (1973).
[CrossRef]

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: II. Sources of error,” J. Phys. D 6, 2217–2224 (1973).
[CrossRef]

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Alieva, T.

T. Alieva, M. J. Bastiaans, L. Stankovic, “Signal reconstruction from two close fractional Fourier power spectra,” IEEE Trans. Signal Process. 51, 112–123 (2003).
[CrossRef]

T. Alieva, M. J. Bastiaans, “On fractional Fourier transform moments,” IEEE Signal Process. Lett. 7, 321–323 (2000).
[CrossRef]

Allen, L. J.

L. J. Allen, W. McBride, M. P. Oxley, “Exit wave reconstruction using soft x-rays,” Opt. Commun. 233, 77–82 (2004).
[CrossRef]

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev E 63, 037602-1–037602-4 (2001).
[CrossRef]

L. J. Allen, M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[CrossRef]

Almeida, L. B.

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

Alper Kutay, M.

C. Candan, M. Alper Kutay, H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

Arikan, O.

M. A. Kutay, H. Özaktaş, H. M. Ozaktas, O. Arikan, “The fractional Fourier domain decomposition,” Signal Process. 77, 105–109 (1999).
[CrossRef]

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdağı, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

Atli, H.

M. G. Ertosun, H. Atli, H. M. Ozaktas, B. Barshan, “Complex signal recovery from two fractional Fourier transform intensities: order and noise dependence,” Opt. Commun. 244, 61–70 (2005).
[CrossRef]

Aytür, O.

O. Aytür, H. M. Ozaktas, “Non-orthogonal domains in phase space of quantum optics and their relation to fractional Fourier transforms,” Opt. Commun. 120, 166–170 (1995).
[CrossRef]

H. M. Ozaktas, O. Aytür, “Fractional Fourier domains,” Signal Process. 46, 119–124 (1995).
[CrossRef]

Barnea, Z.

K. A. Nugent, T. E. Gureyev, D. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Barshan, B.

M. G. Ertosun, H. Atli, H. M. Ozaktas, B. Barshan, “Complex signal recovery from two fractional Fourier transform intensities: order and noise dependence,” Opt. Commun. 244, 61–70 (2005).
[CrossRef]

H. M. Ozaktas, B. Barshan, D. Mendlovic, “Convolution and filtering in fractional Fourier domains,” Opt. Rev. 1, 15–16 (1994).
[CrossRef]

H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
[CrossRef]

Bastiaans, M.

Bastiaans, M. J.

T. Alieva, M. J. Bastiaans, L. Stankovic, “Signal reconstruction from two close fractional Fourier power spectra,” IEEE Trans. Signal Process. 51, 112–123 (2003).
[CrossRef]

T. Alieva, M. J. Bastiaans, “On fractional Fourier transform moments,” IEEE Signal Process. Lett. 7, 321–323 (2000).
[CrossRef]

Bauschke, H. H.

Beck, M.

Bozdagi, G.

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdağı, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

Brenner, K.-H.

X. Liu, K.-H. Brenner, “Reconstruction of two-dimensional complex amplitudes from intensity measurements,” Opt. Commun. 225, 19–30 (2003).
[CrossRef]

Candan, C.

C. Candan, M. Alper Kutay, H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

Chapman, H. N.

Chen, N.-X.

W.-X. Cong, N.-X. Chen, B.-Y. Gu, “Phase retrieval in the Fresnel transform system: a recursive algorithm,” J. Opt. Soc. Am. A 16, 1827–1830 (1999).
[CrossRef]

W.-X. Cong, N.-X. Chen, B.-Y. Gu, “Recursive algorithm for phase retrieval in the fractional Fourier transform domain,” Appl. Opt. 37, 6906–6910 (1998).
[CrossRef]

W.-X. Cong, N.-X. Chen, B.-Y. Gu, “A new method for phase retrieval in the optical system,” Chin. Phys. Lett. 15, 24–26 (1998).
[CrossRef]

W.-X. Cong, N.-X. Chen, B.-Y. Gu, “A recursive method for phase retrieval in Fourier transform domain,” Chin. Sci. Bull. 43, 40–44 (1998).
[CrossRef]

Clarke, L.

Coene, W.

W. Coene, G. Janssen, M. Op. de Beeck, D. Van Dyck, “Phase retrieval through focus variation for ultra-resolution in field-emission transmission electron microscopy,” Phys. Rev. Lett. 69, 3743–3746 (1992).
[CrossRef] [PubMed]

Combettes, P. L.

Cong, W.-X.

W.-X. Cong, N.-X. Chen, B.-Y. Gu, “Phase retrieval in the Fresnel transform system: a recursive algorithm,” J. Opt. Soc. Am. A 16, 1827–1830 (1999).
[CrossRef]

W.-X. Cong, N.-X. Chen, B.-Y. Gu, “A new method for phase retrieval in the optical system,” Chin. Phys. Lett. 15, 24–26 (1998).
[CrossRef]

W.-X. Cong, N.-X. Chen, B.-Y. Gu, “A recursive method for phase retrieval in Fourier transform domain,” Chin. Sci. Bull. 43, 40–44 (1998).
[CrossRef]

W.-X. Cong, N.-X. Chen, B.-Y. Gu, “Recursive algorithm for phase retrieval in the fractional Fourier transform domain,” Appl. Opt. 37, 6906–6910 (1998).
[CrossRef]

Cookson, D.

K. A. Nugent, T. E. Gureyev, D. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Dainty, J. C.

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, 1987), pp. 231–275.

de Beeck, M. Op.

W. Coene, G. Janssen, M. Op. de Beeck, D. Van Dyck, “Phase retrieval through focus variation for ultra-resolution in field-emission transmission electron microscopy,” Phys. Rev. Lett. 69, 3743–3746 (1992).
[CrossRef] [PubMed]

Dong, B.

Dorsch, R. G.

Dragoman, D.

Ersoy, O. K.

Ertosun, M. G.

M. G. Ertosun, H. Atli, H. M. Ozaktas, B. Barshan, “Complex signal recovery from two fractional Fourier transform intensities: order and noise dependence,” Opt. Commun. 244, 61–70 (2005).
[CrossRef]

Faulkner, H. M. L.

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev E 63, 037602-1–037602-4 (2001).
[CrossRef]

Fienup, J. R.

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

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, 1987), pp. 231–275.

George, N.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Gu, B.

Gu, B.-Y.

W.-X. Cong, N.-X. Chen, B.-Y. Gu, “Phase retrieval in the Fresnel transform system: a recursive algorithm,” J. Opt. Soc. Am. A 16, 1827–1830 (1999).
[CrossRef]

W.-X. Cong, N.-X. Chen, B.-Y. Gu, “A recursive method for phase retrieval in Fourier transform domain,” Chin. Sci. Bull. 43, 40–44 (1998).
[CrossRef]

W.-X. Cong, N.-X. Chen, B.-Y. Gu, “A new method for phase retrieval in the optical system,” Chin. Phys. Lett. 15, 24–26 (1998).
[CrossRef]

W.-X. Cong, N.-X. Chen, B.-Y. Gu, “Recursive algorithm for phase retrieval in the fractional Fourier transform domain,” Appl. Opt. 37, 6906–6910 (1998).
[CrossRef]

Gureyev, T. E.

T. E. Gureyev, A. Pogany, D. M. Paganin, S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[CrossRef]

T. E. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220, 49–58 (2003).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Hennelly, B.

B. Hennelly, J. T. Sheridan, “Fractional Fourier transform-based image encryption: phase retrieval algorithm,” Opt. Commun. 226, 61–80 (2003).
[CrossRef]

Ivanov, V. Yu.

Janssen, G.

W. Coene, G. Janssen, M. Op. de Beeck, D. Van Dyck, “Phase retrieval through focus variation for ultra-resolution in field-emission transmission electron microscopy,” Phys. Rev. Lett. 69, 3743–3746 (1992).
[CrossRef] [PubMed]

Kutay, M. A.

M. A. Kutay, H. Özaktaş, H. M. Ozaktas, O. Arikan, “The fractional Fourier domain decomposition,” Signal Process. 77, 105–109 (1999).
[CrossRef]

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdağı, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

İ. Ş. Yetik, M. A. Kutay, H. Özaktaş, H. M. Ozaktas, “Continuous and discrete fractional Fourier domain decomposition,” in Proceedings of the 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing, (Institute of Electrical and Electronics Engineers, 2000), pp. I:93–96.
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Liu, X.

X. Liu, K.-H. Brenner, “Reconstruction of two-dimensional complex amplitudes from intensity measurements,” Opt. Commun. 225, 19–30 (2003).
[CrossRef]

Lohmann, A. W.

Luke, D. R.

Mayer, A.

McAlister, D. F.

McBride, W.

L. J. Allen, W. McBride, M. P. Oxley, “Exit wave reconstruction using soft x-rays,” Opt. Commun. 233, 77–82 (2004).
[CrossRef]

Mendlovic, D.

Miao, J.

Millane, R. P.

Misell, D. L.

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973).
[CrossRef]

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: I. Test calculations,” J. Phys. D 6, 2200–2216 (1973).
[CrossRef]

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: II. Sources of error,” J. Phys. D 6, 2217–2224 (1973).
[CrossRef]

Nugent, K. A.

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev E 63, 037602-1–037602-4 (2001).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Onural, L.

Oxley, M. P.

L. J. Allen, W. McBride, M. P. Oxley, “Exit wave reconstruction using soft x-rays,” Opt. Commun. 233, 77–82 (2004).
[CrossRef]

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev E 63, 037602-1–037602-4 (2001).
[CrossRef]

L. J. Allen, M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[CrossRef]

Ozaktas, H. M.

M. G. Ertosun, H. Atli, H. M. Ozaktas, B. Barshan, “Complex signal recovery from two fractional Fourier transform intensities: order and noise dependence,” Opt. Commun. 244, 61–70 (2005).
[CrossRef]

C. Candan, M. Alper Kutay, H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

M. A. Kutay, H. Özaktaş, H. M. Ozaktas, O. Arikan, “The fractional Fourier domain decomposition,” Signal Process. 77, 105–109 (1999).
[CrossRef]

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdağı, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

O. Aytür, H. M. Ozaktas, “Non-orthogonal domains in phase space of quantum optics and their relation to fractional Fourier transforms,” Opt. Commun. 120, 166–170 (1995).
[CrossRef]

H. M. Ozaktas, O. Aytür, “Fractional Fourier domains,” Signal Process. 46, 119–124 (1995).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
[CrossRef]

H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
[CrossRef]

H. M. Ozaktas, B. Barshan, D. Mendlovic, “Convolution and filtering in fractional Fourier domains,” Opt. Rev. 1, 15–16 (1994).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation: II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

İ. Ş. Yetik, M. A. Kutay, H. Özaktaş, H. M. Ozaktas, “Continuous and discrete fractional Fourier domain decomposition,” in Proceedings of the 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing, (Institute of Electrical and Electronics Engineers, 2000), pp. I:93–96.
[CrossRef]

Özaktas, H.

M. A. Kutay, H. Özaktaş, H. M. Ozaktas, O. Arikan, “The fractional Fourier domain decomposition,” Signal Process. 77, 105–109 (1999).
[CrossRef]

İ. Ş. Yetik, M. A. Kutay, H. Özaktaş, H. M. Ozaktas, “Continuous and discrete fractional Fourier domain decomposition,” in Proceedings of the 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing, (Institute of Electrical and Electronics Engineers, 2000), pp. I:93–96.
[CrossRef]

Paganin, D.

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev E 63, 037602-1–037602-4 (2001).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Paganin, D. M.

T. E. Gureyev, A. Pogany, D. M. Paganin, S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[CrossRef]

Pogany, A.

T. E. Gureyev, A. Pogany, D. M. Paganin, S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[CrossRef]

Raymer, M. G.

Rolleston, R.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Sayre, D.

Sheridan, J. T.

B. Hennelly, J. T. Sheridan, “Fractional Fourier transform-based image encryption: phase retrieval algorithm,” Opt. Commun. 226, 61–80 (2003).
[CrossRef]

Sivokon, V. P.

Stankovic, L.

T. Alieva, M. J. Bastiaans, L. Stankovic, “Signal reconstruction from two close fractional Fourier power spectra,” IEEE Trans. Signal Process. 51, 112–123 (2003).
[CrossRef]

Testorf, M.

Van Dyck, D.

W. Coene, G. Janssen, M. Op. de Beeck, D. Van Dyck, “Phase retrieval through focus variation for ultra-resolution in field-emission transmission electron microscopy,” Phys. Rev. Lett. 69, 3743–3746 (1992).
[CrossRef] [PubMed]

Vorontsov, M. A.

Wilkins, S. W.

T. E. Gureyev, A. Pogany, D. M. Paganin, S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[CrossRef]

Wolf, K.

Yang, G.

Yetik, I. S.

İ. Ş. Yetik, M. A. Kutay, H. Özaktaş, H. M. Ozaktas, “Continuous and discrete fractional Fourier domain decomposition,” in Proceedings of the 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing, (Institute of Electrical and Electronics Engineers, 2000), pp. I:93–96.
[CrossRef]

Zalevsky, Z.

Z. Zalevsky, D. Mendlovic, R. G. Dorsch, “Gerchberg–Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
[CrossRef] [PubMed]

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Zhang, Y.

Zhuang, J.

Acta Opt. Sin. (1)

B. Gu, G. Yang, “On the phase retrieval problem in optical and electronic microscopy,” Acta Opt. Sin. 1, 517–522 (1981).

Acta Phys. Sin. (1)

G. Yang, B. Gu, “On the amplitude-phase retrieval problem in the optical system,” Acta Phys. Sin. 30, 410–413 (1981).

Appl. Opt. (7)

Chin. Phys. Lett. (1)

W.-X. Cong, N.-X. Chen, B.-Y. Gu, “A new method for phase retrieval in the optical system,” Chin. Phys. Lett. 15, 24–26 (1998).
[CrossRef]

Chin. Sci. Bull. (1)

W.-X. Cong, N.-X. Chen, B.-Y. Gu, “A recursive method for phase retrieval in Fourier transform domain,” Chin. Sci. Bull. 43, 40–44 (1998).
[CrossRef]

IEEE Signal Process. Lett. (1)

T. Alieva, M. J. Bastiaans, “On fractional Fourier transform moments,” IEEE Signal Process. Lett. 7, 321–323 (2000).
[CrossRef]

IEEE Trans. Signal Process (1)

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdağı, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

IEEE Trans. Signal Process. (3)

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

T. Alieva, M. J. Bastiaans, L. Stankovic, “Signal reconstruction from two close fractional Fourier power spectra,” IEEE Trans. Signal Process. 51, 112–123 (2003).
[CrossRef]

C. Candan, M. Alper Kutay, H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

J. Opt. Soc. Am. A (14)

R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990).
[CrossRef]

W.-X. Cong, N.-X. Chen, B.-Y. Gu, “Phase retrieval in the Fresnel transform system: a recursive algorithm,” J. Opt. Soc. Am. A 16, 1827–1830 (1999).
[CrossRef]

V. Yu. Ivanov, V. P. Sivokon, M. A. Vorontsov, “Phase retrieval from a set of intensity measurements: theory and experiment,” J. Opt. Soc. Am. A 9, 1515–1524 (1992).
[CrossRef]

M. Bastiaans, K. Wolf, “Phase reconstruction from intensity measurements in linear systems,” J. Opt. Soc. Am. A 20, 1046–1049 (2003).
[CrossRef]

R. Rolleston, N. George, “Stationary phase approximations in Fresnel-zone magnitude-only reconstructions,” J. Opt. Soc. Am. A 4, 148–153 (1987).
[CrossRef]

J. Miao, D. Sayre, 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]

B. Dong, Y. Zhang, B. Gu, G. Yang, “Numerical investigation of phase retrieval in a fractional Fourier transform,” J. Opt. Soc. Am. A 14, 2709–2714 (1997).
[CrossRef]

Y. Zhang, B. Dong, B. Gu, G. Yang, “Beam shaping in the fractional Fourier transform domain,” J. Opt. Soc. Am. A 15, 1114–1120 (1998).
[CrossRef]

H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation: II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
[CrossRef]

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional order Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
[CrossRef]

H. H. Bauschke, P. L. Combettes, 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]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
[CrossRef]

J. Phys. D (3)

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973).
[CrossRef]

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: I. Test calculations,” J. Phys. D 6, 2200–2216 (1973).
[CrossRef]

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: II. Sources of error,” J. Phys. D 6, 2217–2224 (1973).
[CrossRef]

Opt. Commun. (8)

T. E. Gureyev, A. Pogany, D. M. Paganin, S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[CrossRef]

T. E. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220, 49–58 (2003).
[CrossRef]

B. Hennelly, J. T. Sheridan, “Fractional Fourier transform-based image encryption: phase retrieval algorithm,” Opt. Commun. 226, 61–80 (2003).
[CrossRef]

M. G. Ertosun, H. Atli, H. M. Ozaktas, B. Barshan, “Complex signal recovery from two fractional Fourier transform intensities: order and noise dependence,” Opt. Commun. 244, 61–70 (2005).
[CrossRef]

O. Aytür, H. M. Ozaktas, “Non-orthogonal domains in phase space of quantum optics and their relation to fractional Fourier transforms,” Opt. Commun. 120, 166–170 (1995).
[CrossRef]

X. Liu, K.-H. Brenner, “Reconstruction of two-dimensional complex amplitudes from intensity measurements,” Opt. Commun. 225, 19–30 (2003).
[CrossRef]

L. J. Allen, W. McBride, M. P. Oxley, “Exit wave reconstruction using soft x-rays,” Opt. Commun. 233, 77–82 (2004).
[CrossRef]

L. J. Allen, M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[CrossRef]

Opt. Lett. (2)

Opt. Rev. (1)

H. M. Ozaktas, B. Barshan, D. Mendlovic, “Convolution and filtering in fractional Fourier domains,” Opt. Rev. 1, 15–16 (1994).
[CrossRef]

Optik (Stuttgart) (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Phys. Rev E (1)

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev E 63, 037602-1–037602-4 (2001).
[CrossRef]

Phys. Rev. Lett. (3)

W. Coene, G. Janssen, M. Op. de Beeck, D. Van Dyck, “Phase retrieval through focus variation for ultra-resolution in field-emission transmission electron microscopy,” Phys. Rev. Lett. 69, 3743–3746 (1992).
[CrossRef] [PubMed]

K. A. Nugent, T. E. Gureyev, D. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Signal Process. (2)

H. M. Ozaktas, O. Aytür, “Fractional Fourier domains,” Signal Process. 46, 119–124 (1995).
[CrossRef]

M. A. Kutay, H. Özaktaş, H. M. Ozaktas, O. Arikan, “The fractional Fourier domain decomposition,” Signal Process. 77, 105–109 (1999).
[CrossRef]

Other (4)

İ. Ş. Yetik, M. A. Kutay, H. Özaktaş, H. M. Ozaktas, “Continuous and discrete fractional Fourier domain decomposition,” in Proceedings of the 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing, (Institute of Electrical and Electronics Engineers, 2000), pp. I:93–96.
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

H. Stark, ed., Image Recovery: Theory and Application (Academic, 1987).

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, 1987), pp. 231–275.

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Figures (3)

Fig. 1
Fig. 1

(a) Magnitude and (b) phase of f(u).

Fig. 2
Fig. 2

Final recovery error. The right-hand plots contain the same data as the left-hand plots, with the horizontal axis scaled to correspond to the actual time of computation: Time = M × Number of iterations. (a), (b), variation A; (c), (d), variation B.

Fig. 3
Fig. 3

Final recovery error in the presence of noise for variation A. (a) Final error after 10 iterations. (b) Final error after 100 iterations.

Equations (4)

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

f a ( u ) = - exp { - i [ π sgn ( α ) / 4 - α / 2 ] } sin α 1 / 2 × exp [ i π ( cot α u 2 - 2 csc α u u + cot α u 2 ) ] f ( u ) d u ,
[ ( 1 / 8 ) - 4 4 ϕ k ( u ) - ϕ ( u ) 2 d u ] 1 / 2 .
SNR = - 4 4 f ( u ) 2 d u - 4 4 σ 2 d u ,
[ ( 1 / 4 ) - 2 2 ϕ k ( u ) - ϕ ( u ) 2 d u ] 1 / 2 .

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