Abstract

We develop and test a nonlinear optimization algorithm for solving the problem of phase retrieval with transverse translation diversity, where the diverse far-field intensity measurements are taken after translating the object relative to a known illumination pattern. Analytical expressions for the gradient of a squared-error metric with respect to the object, illumination and translations allow joint optimization of the object and system parameters. This approach achieves superior reconstructions, with respect to a previously reported technique [H. M. L. Faulkner and J. M. Rodenburg, Phys. Rev. Lett. 93, 023903 (2004)], when the system parameters are inaccurately known or in the presence of noise. Applicability of this method for samples that are smaller than the illumination pattern is explored.

© 2008 Optical Society of America

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    [CrossRef] [PubMed]
  2. 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]
  3. P. S. Idell, J. R. Fienup, and R. S. Goodman, "Image synthesis from nonimaged laser-speckle patterns," Opt. Lett. 12, 858-860 (1987).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  5. 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 (London) 400, 342-344 (1999).
    [CrossRef]
  6. S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101 (2003).
    [CrossRef]
  7. J. R. Fienup, "Lensless coherent imaging by phase retrieval with an illumination pattern constraint," Opt. Express 14, 498-508 (2006).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  14. M. Guizar-Sicairos and J. R. Fienup, "Holography with extended reference by autocorrelation linear differential operation," Opt. Express 15, 17592-17612 (2007).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  20. J. M. Rodenburg and H. M. L. Faulkner, "A phase retrieval algorithm for shifting illumination," Appl. Phys. Lett. 85, 4795-4797 (2004).
    [CrossRef]
  21. J. M. Rodenburg, A. C. Hurst, and A. G. Cullis, "Transmission microscopy without lenses for objects of unlimited size," Ultramicroscopy 107, 227-231 (2007).
    [CrossRef]
  22. J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, "Hard-x-ray lensless imaging of extended objects," Phys. Rev. Lett. 98, 034801 (2007).
    [CrossRef] [PubMed]
  23. H. M. L. Faulkner and J. M. Rodenburg, "Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy," Ultramicroscopy 103, 153-164 (2005).
    [CrossRef] [PubMed]
  24. R. A. Gonsalves and R. Childlaw, "Wavefront sensing by phase retrieval," Proc. SPIE 207, 32-39 (1979).
  25. R. A. Gonsalves, "Phase retrieval and diversity in adaptive optics," Opt. Eng. 21, 829-832 (1982).
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    [CrossRef]
  27. R. G. Paxman, T. J. Schulz, and J. R. Fienup, "Joint estimation of object and aberrations by using phase diversity," J. Opt. Soc. Am. A 9, 1072-1085 (1992).
    [CrossRef]
  28. R. G. Paxman, "Diversity imaging," in Signal Recovery and Synthesis, 2001 OSA Technical Digest Series (Optical Society of America, 2001), paper SWA1.
  29. G. R. Brady and J. R. Fienup, "Nonlinear optimization algorithm for retreiving the full complex pupil function," Opt. Express 14, 474-486 (2006).
    [CrossRef] [PubMed]
  30. M. R. Bolcar and J. R. Fienup, "Method of phase diversity in multi-aperture systems utilizing individual subaperture control," Proc. SPIE 5896-58960G (2005).
    [CrossRef]
  31. B. J. Thelen, M. F. Reiley, and R. G. Paxman, "Fine-resolution multispectral imaging using wavelength diversity," in Signal Recovery and Synthesis, Vol. 11 of 1995 OSA Technical Digest Series (Optical Society of America 1995), pp. 44-46, paper RTuD3.
  32. H. R. Ingleby and D. R. McGaughey "Parallel multiframe blind deconvolution using wavelength diversity," Proc. SPIE 5562, 58-64 (2004).
    [CrossRef]
  33. W. Hoppe, "Beugung im inhomogenen Primärstrahlwellenfeld. III. Amplituden - und Phasenbestimmung bei unperiodischen Objekten," Acta Crystallogr. Sect. A 25, 508-514 (1969).
    [CrossRef]
  34. J. M. Rodenburg and R. H. T. Bates, "The theory of super-resolution electron microscopy viaWigner-distribution deconvolution," Phil. Trans. R. Soc. Lond. A 339, 521-553 (1992).
    [CrossRef]
  35. N. Nakajima, "Phase retrieval from Fresnel zone intensity measurements by use of Gaussian filtering," Appl. Opt. 37, 6219-6226 (1998).
    [CrossRef]
  36. S. T. Thurman, The Institute of Optics, University of Rochester, Rochester, New York, USA, R. T. DeRosa and J. R. Fienup are preparing a manuscript to be called "Amplitude metrics for field retrieval with hard-edge and uniformly-illuminated apertures."
  37. J. R. Fienup, "Invariant error metrics for image reconstruction," Appl. Opt. 36, 8352-8357 (1997).
    [CrossRef]
  38. M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, "Efficient subpixel image registration algorithms," Opt. Lett. 33, 156-158 (2008).
    [CrossRef] [PubMed]
  39. Image modified from an original courtesy of Universidad Autónoma de Nuevo León, Mexico.

2008 (2)

2007 (4)

S. G. Podorov, K. M. Pavlov, and D. M. Paganin, "A non-iterative reconstruction method for direct and unambiguous coherent diffractive imaging," Opt. Express 15, 9954-9962 (2007).
[CrossRef] [PubMed]

M. Guizar-Sicairos and J. R. Fienup, "Holography with extended reference by autocorrelation linear differential operation," Opt. Express 15, 17592-17612 (2007).
[CrossRef] [PubMed]

J. M. Rodenburg, A. C. Hurst, and A. G. Cullis, "Transmission microscopy without lenses for objects of unlimited size," Ultramicroscopy 107, 227-231 (2007).
[CrossRef]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, "Hard-x-ray lensless imaging of extended objects," Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

2006 (4)

2005 (2)

M. R. Bolcar and J. R. Fienup, "Method of phase diversity in multi-aperture systems utilizing individual subaperture control," Proc. SPIE 5896-58960G (2005).
[CrossRef]

H. M. L. Faulkner and J. M. Rodenburg, "Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy," Ultramicroscopy 103, 153-164 (2005).
[CrossRef] [PubMed]

2004 (3)

H. M. L. Faulkner and J. M. Rodenburg, "Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm," Phys. Rev. Lett. 93, 023903 (2004).
[CrossRef] [PubMed]

J. M. Rodenburg and H. M. L. Faulkner, "A phase retrieval algorithm for shifting illumination," Appl. Phys. Lett. 85, 4795-4797 (2004).
[CrossRef]

H. R. Ingleby and D. R. McGaughey "Parallel multiframe blind deconvolution using wavelength diversity," Proc. SPIE 5562, 58-64 (2004).
[CrossRef]

2003 (3)

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

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, 1025-1034 (2003).
[CrossRef]

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 (London) 400, 342-344 (1999).
[CrossRef]

1998 (1)

1997 (2)

1992 (2)

R. G. Paxman, T. J. Schulz, and J. R. Fienup, "Joint estimation of object and aberrations by using phase diversity," J. Opt. Soc. Am. A 9, 1072-1085 (1992).
[CrossRef]

J. M. Rodenburg and R. H. T. Bates, "The theory of super-resolution electron microscopy viaWigner-distribution deconvolution," Phil. Trans. R. Soc. Lond. A 339, 521-553 (1992).
[CrossRef]

1988 (2)

1987 (2)

1986 (1)

1982 (2)

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

R. A. Gonsalves, "Phase retrieval and diversity in adaptive optics," Opt. Eng. 21, 829-832 (1982).

1979 (1)

R. A. Gonsalves and R. Childlaw, "Wavefront sensing by phase retrieval," Proc. SPIE 207, 32-39 (1979).

1969 (1)

W. Hoppe, "Beugung im inhomogenen Primärstrahlwellenfeld. III. Amplituden - und Phasenbestimmung bei unperiodischen Objekten," Acta Crystallogr. Sect. A 25, 508-514 (1969).
[CrossRef]

1962 (1)

Bates, R. H. T.

J. M. Rodenburg and R. H. T. Bates, "The theory of super-resolution electron microscopy viaWigner-distribution deconvolution," Phil. Trans. R. Soc. Lond. A 339, 521-553 (1992).
[CrossRef]

Bauschke, H. H.

Bolcar, M. R.

M. R. Bolcar and J. R. Fienup, "Method of phase diversity in multi-aperture systems utilizing individual subaperture control," Proc. SPIE 5896-58960G (2005).
[CrossRef]

Brady, G. R.

Bunk, O.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, "Hard-x-ray lensless imaging of extended objects," Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Cederquist, J. N.

Chapman, H. N.

H. N. Chapman et al., "High-resolution ab initio three-dimensional x-ray diffraction microscopy," J. Opt. Soc. Am. A 23, 1179-1200 (2006).
[CrossRef]

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

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 (London) 400, 342-344 (1999).
[CrossRef]

Childlaw, R.

R. A. Gonsalves and R. Childlaw, "Wavefront sensing by phase retrieval," Proc. SPIE 207, 32-39 (1979).

Combettes, P. L.

Cullis, A. G.

J. M. Rodenburg, A. C. Hurst, and A. G. Cullis, "Transmission microscopy without lenses for objects of unlimited size," Ultramicroscopy 107, 227-231 (2007).
[CrossRef]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, "Hard-x-ray lensless imaging of extended objects," Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

David, C.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, "Hard-x-ray lensless imaging of extended objects," Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Dobson, B. R.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, "Hard-x-ray lensless imaging of extended objects," Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Eberhardt, W.

O. Hellwig, S. Eisebitt, W. Eberhardt, W. F. Schlotter, J. Lüning, and J. Stöhr, "Magnetic imaging with soft x-ray spectroholography," J. Appl. Phys. 99, 08H307 (2006).
[CrossRef]

Eisebitt, S.

O. Hellwig, S. Eisebitt, W. Eberhardt, W. F. Schlotter, J. Lüning, and J. Stöhr, "Magnetic imaging with soft x-ray spectroholography," J. Appl. Phys. 99, 08H307 (2006).
[CrossRef]

Elser, V.

Faulkner, H. M. L.

H. M. L. Faulkner and J. M. Rodenburg, "Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy," Ultramicroscopy 103, 153-164 (2005).
[CrossRef] [PubMed]

J. M. Rodenburg and H. M. L. Faulkner, "A phase retrieval algorithm for shifting illumination," Appl. Phys. Lett. 85, 4795-4797 (2004).
[CrossRef]

H. M. L. Faulkner and J. M. Rodenburg, "Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm," Phys. Rev. Lett. 93, 023903 (2004).
[CrossRef] [PubMed]

Fienup, J. R.

M. Guizar-Sicairos and J. R. Fienup, "Phase retrieval with Fourier-weighted projections," J. Opt. Soc. Am. A 25, 701-709 (2008).
[CrossRef]

M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, "Efficient subpixel image registration algorithms," Opt. Lett. 33, 156-158 (2008).
[CrossRef] [PubMed]

M. Guizar-Sicairos and J. R. Fienup, "Holography with extended reference by autocorrelation linear differential operation," Opt. Express 15, 17592-17612 (2007).
[CrossRef] [PubMed]

J. R. Fienup, "Lensless coherent imaging by phase retrieval with an illumination pattern constraint," Opt. Express 14, 498-508 (2006).
[CrossRef] [PubMed]

G. R. Brady and J. R. Fienup, "Nonlinear optimization algorithm for retreiving the full complex pupil function," Opt. Express 14, 474-486 (2006).
[CrossRef] [PubMed]

M. R. Bolcar and J. R. Fienup, "Method of phase diversity in multi-aperture systems utilizing individual subaperture control," Proc. SPIE 5896-58960G (2005).
[CrossRef]

J. R. Fienup, "Invariant error metrics for image reconstruction," Appl. Opt. 36, 8352-8357 (1997).
[CrossRef]

R. G. Paxman, T. J. Schulz, and J. R. Fienup, "Joint estimation of object and aberrations by using phase diversity," J. Opt. Soc. Am. A 9, 1072-1085 (1992).
[CrossRef]

J. N. Cederquist, J. R. Fienup, J. C. Marron, and R. G. Paxman, "Phase retrieval from experimental far-field speckle data," Opt. Lett. 13, 619-621 (1988).
[CrossRef] [PubMed]

R. G. Paxman and J. R. Fienup, "Optical misalignment sensing and image reconstruction using phase diversity," J. Opt. Soc. Am. A 5, 914-923 (1988).
[CrossRef]

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]

P. S. Idell, J. R. Fienup, and R. S. Goodman, "Image synthesis from nonimaged laser-speckle patterns," Opt. Lett. 12, 858-860 (1987).
[CrossRef] [PubMed]

J. R. Fienup and C. C. Wackerman, "Phase-retrieval stagnation problems and solutions," J. Opt. Soc. Am. A 3, 1897-1907 (1986).
[CrossRef]

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

Gonsalves, R. A.

R. A. Gonsalves, "Phase retrieval and diversity in adaptive optics," Opt. Eng. 21, 829-832 (1982).

R. A. Gonsalves and R. Childlaw, "Wavefront sensing by phase retrieval," Proc. SPIE 207, 32-39 (1979).

Goodman, R. S.

Guizar-Sicairos, M.

Hau-Riege, S. P.

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

Hellwig, O.

O. Hellwig, S. Eisebitt, W. Eberhardt, W. F. Schlotter, J. Lüning, and J. Stöhr, "Magnetic imaging with soft x-ray spectroholography," J. Appl. Phys. 99, 08H307 (2006).
[CrossRef]

Hoppe, W.

W. Hoppe, "Beugung im inhomogenen Primärstrahlwellenfeld. III. Amplituden - und Phasenbestimmung bei unperiodischen Objekten," Acta Crystallogr. Sect. A 25, 508-514 (1969).
[CrossRef]

Howells, M. R.

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

Hurst, A. C.

J. M. Rodenburg, A. C. Hurst, and A. G. Cullis, "Transmission microscopy without lenses for objects of unlimited size," Ultramicroscopy 107, 227-231 (2007).
[CrossRef]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, "Hard-x-ray lensless imaging of extended objects," Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Idell, P. S.

Ingleby, H. R.

H. R. Ingleby and D. R. McGaughey "Parallel multiframe blind deconvolution using wavelength diversity," Proc. SPIE 5562, 58-64 (2004).
[CrossRef]

Jefimovs, K.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, "Hard-x-ray lensless imaging of extended objects," Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Johnson, I.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, "Hard-x-ray lensless imaging of extended objects," Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Kawanami, H.

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 (London) 400, 342-344 (1999).
[CrossRef]

Leith, E. N.

Luke, D. R.

Lüning, J.

O. Hellwig, S. Eisebitt, W. Eberhardt, W. F. Schlotter, J. Lüning, and J. Stöhr, "Magnetic imaging with soft x-ray spectroholography," J. Appl. Phys. 99, 08H307 (2006).
[CrossRef]

Marchesini, S.

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

Marron, J. C.

McGaughey, D. R.

H. R. Ingleby and D. R. McGaughey "Parallel multiframe blind deconvolution using wavelength diversity," Proc. SPIE 5562, 58-64 (2004).
[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 (London) 400, 342-344 (1999).
[CrossRef]

Nakajima, N.

Noy, A.

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

Paganin, D. M.

Pavlov, K. M.

Paxman, R. G.

Pfeiffer, F.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, "Hard-x-ray lensless imaging of extended objects," Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Podorov, S. G.

Rodenburg, J. M.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, "Hard-x-ray lensless imaging of extended objects," Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

J. M. Rodenburg, A. C. Hurst, and A. G. Cullis, "Transmission microscopy without lenses for objects of unlimited size," Ultramicroscopy 107, 227-231 (2007).
[CrossRef]

H. M. L. Faulkner and J. M. Rodenburg, "Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy," Ultramicroscopy 103, 153-164 (2005).
[CrossRef] [PubMed]

H. M. L. Faulkner and J. M. Rodenburg, "Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm," Phys. Rev. Lett. 93, 023903 (2004).
[CrossRef] [PubMed]

J. M. Rodenburg and H. M. L. Faulkner, "A phase retrieval algorithm for shifting illumination," Appl. Phys. Lett. 85, 4795-4797 (2004).
[CrossRef]

J. M. Rodenburg and R. H. T. Bates, "The theory of super-resolution electron microscopy viaWigner-distribution deconvolution," Phil. Trans. R. Soc. Lond. A 339, 521-553 (1992).
[CrossRef]

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 (London) 400, 342-344 (1999).
[CrossRef]

Schlotter, W. F.

O. Hellwig, S. Eisebitt, W. Eberhardt, W. F. Schlotter, J. Lüning, and J. Stöhr, "Magnetic imaging with soft x-ray spectroholography," J. Appl. Phys. 99, 08H307 (2006).
[CrossRef]

Schulz, T. J.

Spence, J. C. H.

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

Stöhr, J.

O. Hellwig, S. Eisebitt, W. Eberhardt, W. F. Schlotter, J. Lüning, and J. Stöhr, "Magnetic imaging with soft x-ray spectroholography," J. Appl. Phys. 99, 08H307 (2006).
[CrossRef]

Takahashi, T.

Takajo, H.

Thurman, S. T.

Ueda, R.

Upatnieks, J.

Wackerman, C. C.

Weierstall, U.

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

Acta Crystallogr. Sect. A (1)

W. Hoppe, "Beugung im inhomogenen Primärstrahlwellenfeld. III. Amplituden - und Phasenbestimmung bei unperiodischen Objekten," Acta Crystallogr. Sect. A 25, 508-514 (1969).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

J. M. Rodenburg and H. M. L. Faulkner, "A phase retrieval algorithm for shifting illumination," Appl. Phys. Lett. 85, 4795-4797 (2004).
[CrossRef]

J. Appl. Phys. (1)

O. Hellwig, S. Eisebitt, W. Eberhardt, W. F. Schlotter, J. Lüning, and J. Stöhr, "Magnetic imaging with soft x-ray spectroholography," J. Appl. Phys. 99, 08H307 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (9)

Nature (London) (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 (London) 400, 342-344 (1999).
[CrossRef]

Opt. Eng. (1)

R. A. Gonsalves, "Phase retrieval and diversity in adaptive optics," Opt. Eng. 21, 829-832 (1982).

Opt. Express (4)

Opt. Lett. (3)

Phil. Trans. R. Soc. Lond. A (1)

J. M. Rodenburg and R. H. T. Bates, "The theory of super-resolution electron microscopy viaWigner-distribution deconvolution," Phil. Trans. R. Soc. Lond. A 339, 521-553 (1992).
[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. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Phys. Rev. Lett. (2)

H. M. L. Faulkner and J. M. Rodenburg, "Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm," Phys. Rev. Lett. 93, 023903 (2004).
[CrossRef] [PubMed]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, "Hard-x-ray lensless imaging of extended objects," Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Proc. SPIE (3)

M. R. Bolcar and J. R. Fienup, "Method of phase diversity in multi-aperture systems utilizing individual subaperture control," Proc. SPIE 5896-58960G (2005).
[CrossRef]

R. A. Gonsalves and R. Childlaw, "Wavefront sensing by phase retrieval," Proc. SPIE 207, 32-39 (1979).

H. R. Ingleby and D. R. McGaughey "Parallel multiframe blind deconvolution using wavelength diversity," Proc. SPIE 5562, 58-64 (2004).
[CrossRef]

Ultramicroscopy (2)

H. M. L. Faulkner and J. M. Rodenburg, "Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy," Ultramicroscopy 103, 153-164 (2005).
[CrossRef] [PubMed]

J. M. Rodenburg, A. C. Hurst, and A. G. Cullis, "Transmission microscopy without lenses for objects of unlimited size," Ultramicroscopy 107, 227-231 (2007).
[CrossRef]

Other (5)

J. W. Goodman, Introduction to Fourier Optics, 3rd Ed. (Roberts & Company, Englewood, 2005).

B. J. Thelen, M. F. Reiley, and R. G. Paxman, "Fine-resolution multispectral imaging using wavelength diversity," in Signal Recovery and Synthesis, Vol. 11 of 1995 OSA Technical Digest Series (Optical Society of America 1995), pp. 44-46, paper RTuD3.

R. G. Paxman, "Diversity imaging," in Signal Recovery and Synthesis, 2001 OSA Technical Digest Series (Optical Society of America, 2001), paper SWA1.

S. T. Thurman, The Institute of Optics, University of Rochester, Rochester, New York, USA, R. T. DeRosa and J. R. Fienup are preparing a manuscript to be called "Amplitude metrics for field retrieval with hard-edge and uniformly-illuminated apertures."

Image modified from an original courtesy of Universidad Autónoma de Nuevo León, Mexico.

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Figures (5)

Fig. 1.
Fig. 1.

Reconstruction examples using single-measurement phase retrieval with a support constraint. (a) Amplitude of object-space field, f(x,y). The amplitude of two reconstructions that exhibit the twin-image problem, to different extents, are shown in (b) and (c), their final support errors are ES =0.0162 and ES =0.0092, respectively. A 172×210 portion of the 256×256 array is shown in (a)–(c).

Fig. 2.
Fig. 2.

Object, o(x,y), (a) amplitude and (b) phase. (c) Amplitude of p(x,y). (d) Circles indicate the twelve positions of p(x,y) that were used for the reconstruction. (e) Amplitude and (f) phase of the reconstruction using the PIE. Phase is shown from -0.4 to 0.4 radians in (b) and (f). A 172×210 portion of the 256×256 array is shown in (a)–(f).

Fig. 3.
Fig. 3.

Reconstruction results. Amplitude and phase after 500 iterations are shown in (a) and (b) for the PIE and (d) and (e) for the nonlinear optimization approach. (c) and (f) show the amplitude of the initial estimate of p(x,y) and the result after nonlinear optimization respectively. (g) Normalized invariant error metric, E, vs. iteration number. (h) Shift error, Δr vs. iteration number for the nonlinear optimization algorithm. Phase is shown from -0.4 to 0.4 radians in (b) and (e). A 172×210 portion of the 256×256 array is shown in (a)–(f).

Fig. 4.
Fig. 4.

Reconstructions from noisy data. Recovered amplitude and phase with the PIE, (a) and (b) respectively, and the nonlinear optimization algorithm, (c) and (d). (e) Cut through a measured Fourier intensity pattern (before and after applying noise). (f) Normalized invariant error metric, E, vs. iteration number for reconstructions from noisy data. Phase is shown from -0.4 to 0.4 radians in (b) and (d). A 172×210 portion of the 256×256 array is shown in (a)–(d).

Fig. 5.
Fig. 5.

(a) Object o(x,y). (b) Circles indicate the five positions of p(x,y) that were used for the nonlinear optimization reconstruction. (c) Reconstruction after 100 iterations of the nonlinear optimization algorithm. (d) Example reconstruction from a single intensity measurement using phase retrieval with support [dashed line in (b)] and nonnegativity constraints. A 165×165 portion of the 256×256 array is shown in (a)–(d).

Equations (29)

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F ( u , v ) = F ( u , v ) exp [ i θ ( u , v ) ] = DFT { f ( x , y ) } = 1 MN x , y f ( x , y ) exp [ i 2 π ( ux M + vy N ) ] ,
f n ( x , y ) = o ( x x n , y y n ) p ( x , y ) ,
I n ( u , v ) = F n ( u , v ) 2 .
F * ( u , v ) = F ( u , v ) exp [ i θ ( u , v ) ] = DFT { f * ( x , y ) } ,
E S 2 = ( x , y ) S f ̂ ( x , y ) 2 x , y f ̂ ( x , y ) 2 ,
h ̂ n ( x , y ) = o ̂ n ( x , y ) p ( x + x n , y + y n ) = f ̂ n ( x + x n , y + y n ) ,
H ̂ n ( u , v ) = H ̂ n ( u , v ) exp [ i ϕ ̂ n ( u , v ) ] = DFT { h ̂ n ( x , y ) } ,
G n ( u , v ) = I n ( u , v ) exp [ i ϕ ̂ n ( u , v ) ] .
o ̂ n + 1 ( x , y ) = o ̂ n ( x , y ) + p ( x + x n , y + y n ) max x , y p ( x , y ) p * ( x + x n , y + y n ) p ( x + x n , y + y n ) 2 + α β [ g n ( x , y ) h ̂ n ( x , y ) ] ,
f ̂ n + 1 ( x + x n , y + y n ) f ̂ n ( x + x n , y + y n )
= p ( x + x n , y + y n ) max x , y p ( x , y ) β [ g n ( x , y ) f ̂ n ( x + x n , y + y n ) ] ,
p ( x + x n , y + y n ) 2 p ( x + x n , y + y n ) 2 + α 1 .
[ f ̂ n R ( x + x n , y + y n ) + i f ̂ n I ( x + x n , y + y n ) ] { u , v [ F ̂ n ( u , v ) I n ( u , v ) ] 2 }
= 2 [ g n ( x , y ) f ̂ n ( x + x n , y + y n ) ] ,
p ( x + x n , y + y n ) max x , y p ( x , y ) β .
( Δ r ) 2 = min a , b 1 q n = 1 q ( x n x ̂ n a ) 2 + ( y n y ̂ n b ) 2 ,
( Δ r ) 2 = ( x n x ̂ n ) 2 ( x n x ̂ n ) 2 + ( y n y ̂ n ) 2 ( y n y ̂ n ) 2 ,
x n = 1 q n = 1 q x n .
ε = n = 1 q u , v W n ( u , v ) { [ F ̂ n ( u , v ) 2 + δ ] γ [ I n ( u , v ) + δ ] γ } 2 ,
F ̂ n = DFT { f ̂ n ( x , y ) } = DFT { o ̂ ( x x ̂ n , y y ̂ n ) p ̂ ( x , y ) } ,
ε o ̂ R ( x , y ) + i ε o ̂ I ( x , y ) = 4 n = 1 q p ̂ * ( x + x ̂ n , y + y ̂ n ) IDFT { W n [ ( F ̂ n 2 + δ ) γ ( I n + δ ) γ ]
× γ ( F ̂ n 2 + δ ) γ 1 F ̂ n exp [ i 2 π ( u x ̂ n M + v y ̂ n N ) ] } ,
ε p ̂ R ( x , y ) + i ε p ̂ I ( x , y ) = 4 n = 1 q o ̂ * ( x x ̂ n , y y ̂ n ) IDFT { W n [ ( F ̂ n 2 + δ ) γ ( I n + δ ) γ ]
× γ ( F ̂ n 2 + δ ) γ 1 F ̂ n } .
ε x ̂ n = 8 π M u , v W n [ ( F ̂ n 2 + δ ) γ ( I n + δ ) γ ] γ ( F ̂ n 2 + δ ) γ 1
× Im [ F ̂ n * DFT ( p ̂ ( x , y ) IDFT { u O ̂ ( u , v ) exp [ i 2 π ( u x ̂ n M + v y ̂ n N ) ] } ) ]
ε y ̂ n = 8 π N u , v W n [ ( F ̂ n 2 + δ ) γ ( I n + δ ) γ ] γ ( F ̂ n 2 + δ ) γ 1
× Im [ F ̂ n * DFT ( p ̂ ( x , y ) IDFT { v O ̂ ( u , v ) exp [ i 2 π ( u x ̂ n M + v y ̂ n N ) ] } ) ] .
E 2 = 1 q n = 1 q [ min ρ n , x , y x , y ρ n f ̂ n ( x x , y y ) f n ( x , y ) 2 x , y f n ( x , y ) 2 ] .

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