Abstract

Ptychography is a lensless coherent diffractive imaging that uses intensity measurements of multiple diffraction patterns collected with a localized illumination probe from overlapping regions of an object. An iterative algorithm is proposed that is targeted on optimal processing noisy measurements. The noise suppression is enabled by two instruments: first, the maximum-likelihood technique formulated for Poissonian (photon-counting) measurements, and, second, sparse approximation of the phase and magnitude of the object and probe. It is shown that the maximum-likelihood estimate of the wavefield at the sensor plane for noisy measurements is essentially different from the famous Gerchberg–Saxton–Fienup solution, where the magnitude of the estimate is replaced by the square root of the intensity measurement. The simulation experiments demonstrate the state-of-the-art performance of the proposed algorithm both numerically and visually.

© 2013 Optical Society of America

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  1. W. Hoppe, “Trace structure analysis, ptychography, phase tomography,” Ultramicroscopy 10, 187–198 (1982).
    [CrossRef]
  2. J. M. Rodenburg, “Ptychography and related diffractive imaging methods,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Elsevier, 2008), pp. 87–184.
  3. 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]
  4. M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010).
    [CrossRef]
  5. F. Hüe, J. M. Rodenburg, A. M. Maiden, F. Sweeney, and P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B 82, 121415 (2010).
    [CrossRef]
  6. 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]
  7. A. Schropp, P. Boye, A. Goldschmidt, S. Hönig, R. Hoppe, J. Patommel, C. Rakete, D. Samberg, S. Stephan, S. Schöder, M. Burghammer, and C. G. Schroer, “Non-destructive and quantitative imaging of a nano-structured microchip by ptychographic hard x-ray scanning microscopy,” J. Microsc. 241, 9–12 (2011).
    [CrossRef]
  8. J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004).
    [CrossRef]
  9. A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
    [CrossRef]
  10. P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
    [CrossRef]
  11. A. M. Maiden, M. J. Humphry, F. Zhang, and J. M. Rodenburg, “Superresolution imaging via ptychography,” J. Opt. Soc. Am. A 28, 604–612 (2011).
    [CrossRef]
  12. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  13. R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
    [CrossRef]
  14. J. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [CrossRef]
  15. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef]
  16. V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A 20, 40–55 (2003).
    [CrossRef]
  17. Y. Censor, N. Cohen, T. Kotzer, and J. Shamir, “Summed squared distance error reduction by simultaneous multiprojections and applications,” Appl. Math. Comput. 126, 157–179 (2002).
    [CrossRef]
  18. S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 011301 (2007).
    [CrossRef]
  19. P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys. 14, 063004 (2012).
    [CrossRef]
  20. V. Katkovnik and J. Astola, “High-accuracy wavefield reconstruction: decoupled inverse imaging with sparse modeling of phase and amplitude,” J. Opt. Soc. Am. A 29, 44–54 (2012).
    [CrossRef]
  21. V. Katkovnik and J. Astola, “Phase retrieval via spatial light modulator phase modulation in 4f optical setup: numerical inverse imaging with sparse regularization for phase and amplitude,” J. Opt. Soc. Am. A 29, 105–116 (2012).
    [CrossRef]
  22. M. Elad, Sparse and Redundant Representations: from Theory to Applications in Signal and Image Processing (Springer, 2010).
  23. K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
    [CrossRef]
  24. A. Danielyan, V. Katkovnik, and K. Egiazarian, “BM3D frames and variational image deblurring,” IEEE Trans. Image Process. 21, 1715–1728 (2012).
    [CrossRef]
  25. E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
    [CrossRef]
  26. D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
    [CrossRef]
  27. F. Facchinei and C. Kanzow, “Generalized Nash equilibrium problems,” Quart. J. Oper. Res. 5, 173–210 (2007).
    [CrossRef]

2012 (4)

2011 (2)

A. Schropp, P. Boye, A. Goldschmidt, S. Hönig, R. Hoppe, J. Patommel, C. Rakete, D. Samberg, S. Stephan, S. Schöder, M. Burghammer, and C. G. Schroer, “Non-destructive and quantitative imaging of a nano-structured microchip by ptychographic hard x-ray scanning microscopy,” J. Microsc. 241, 9–12 (2011).
[CrossRef]

A. M. Maiden, M. J. Humphry, F. Zhang, and J. M. Rodenburg, “Superresolution imaging via ptychography,” J. Opt. Soc. Am. A 28, 604–612 (2011).
[CrossRef]

2010 (2)

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010).
[CrossRef]

F. Hüe, J. M. Rodenburg, A. M. Maiden, F. Sweeney, and P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B 82, 121415 (2010).
[CrossRef]

2009 (2)

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[CrossRef]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[CrossRef]

2007 (5)

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]

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

F. Facchinei and C. Kanzow, “Generalized Nash equilibrium problems,” Quart. J. Oper. Res. 5, 173–210 (2007).
[CrossRef]

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[CrossRef]

2006 (2)

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[CrossRef]

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[CrossRef]

2004 (1)

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

2003 (1)

2002 (1)

Y. Censor, N. Cohen, T. Kotzer, and J. Shamir, “Summed squared distance error reduction by simultaneous multiprojections and applications,” Appl. Math. Comput. 126, 157–179 (2002).
[CrossRef]

1982 (2)

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

W. Hoppe, “Trace structure analysis, ptychography, phase tomography,” Ultramicroscopy 10, 187–198 (1982).
[CrossRef]

1978 (1)

1974 (1)

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

1972 (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, 237–246 (1972).

Astola, J.

Boye, P.

A. Schropp, P. Boye, A. Goldschmidt, S. Hönig, R. Hoppe, J. Patommel, C. Rakete, D. Samberg, S. Stephan, S. Schöder, M. Burghammer, and C. G. Schroer, “Non-destructive and quantitative imaging of a nano-structured microchip by ptychographic hard x-ray scanning microscopy,” J. Microsc. 241, 9–12 (2011).
[CrossRef]

Bunk, O.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010).
[CrossRef]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[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]

Burghammer, M.

A. Schropp, P. Boye, A. Goldschmidt, S. Hönig, R. Hoppe, J. Patommel, C. Rakete, D. Samberg, S. Stephan, S. Schöder, M. Burghammer, and C. G. Schroer, “Non-destructive and quantitative imaging of a nano-structured microchip by ptychographic hard x-ray scanning microscopy,” J. Microsc. 241, 9–12 (2011).
[CrossRef]

Candes, E. J.

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[CrossRef]

Censor, Y.

Y. Censor, N. Cohen, T. Kotzer, and J. Shamir, “Summed squared distance error reduction by simultaneous multiprojections and applications,” Appl. Math. Comput. 126, 157–179 (2002).
[CrossRef]

Cohen, N.

Y. Censor, N. Cohen, T. Kotzer, and J. Shamir, “Summed squared distance error reduction by simultaneous multiprojections and applications,” Appl. Math. Comput. 126, 157–179 (2002).
[CrossRef]

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]

Dabov, K.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[CrossRef]

Danielyan, A.

A. Danielyan, V. Katkovnik, and K. Egiazarian, “BM3D frames and variational image deblurring,” IEEE Trans. Image Process. 21, 1715–1728 (2012).
[CrossRef]

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]

Dierolf, M.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010).
[CrossRef]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[CrossRef]

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]

Donoho, D. L.

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[CrossRef]

Egiazarian, K.

A. Danielyan, V. Katkovnik, and K. Egiazarian, “BM3D frames and variational image deblurring,” IEEE Trans. Image Process. 21, 1715–1728 (2012).
[CrossRef]

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[CrossRef]

Elad, M.

M. Elad, Sparse and Redundant Representations: from Theory to Applications in Signal and Image Processing (Springer, 2010).

Elser, V.

Facchinei, F.

F. Facchinei and C. Kanzow, “Generalized Nash equilibrium problems,” Quart. J. Oper. Res. 5, 173–210 (2007).
[CrossRef]

Faulkner, H. M. L.

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

Fienup, J.

Fienup, J. R.

Foi, A.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

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

Goldschmidt, A.

A. Schropp, P. Boye, A. Goldschmidt, S. Hönig, R. Hoppe, J. Patommel, C. Rakete, D. Samberg, S. Stephan, S. Schöder, M. Burghammer, and C. G. Schroer, “Non-destructive and quantitative imaging of a nano-structured microchip by ptychographic hard x-ray scanning microscopy,” J. Microsc. 241, 9–12 (2011).
[CrossRef]

Guizar-Sicairos, M.

P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys. 14, 063004 (2012).
[CrossRef]

Hönig, S.

A. Schropp, P. Boye, A. Goldschmidt, S. Hönig, R. Hoppe, J. Patommel, C. Rakete, D. Samberg, S. Stephan, S. Schöder, M. Burghammer, and C. G. Schroer, “Non-destructive and quantitative imaging of a nano-structured microchip by ptychographic hard x-ray scanning microscopy,” J. Microsc. 241, 9–12 (2011).
[CrossRef]

Hoppe, R.

A. Schropp, P. Boye, A. Goldschmidt, S. Hönig, R. Hoppe, J. Patommel, C. Rakete, D. Samberg, S. Stephan, S. Schöder, M. Burghammer, and C. G. Schroer, “Non-destructive and quantitative imaging of a nano-structured microchip by ptychographic hard x-ray scanning microscopy,” J. Microsc. 241, 9–12 (2011).
[CrossRef]

Hoppe, W.

W. Hoppe, “Trace structure analysis, ptychography, phase tomography,” Ultramicroscopy 10, 187–198 (1982).
[CrossRef]

Hüe, F.

F. Hüe, J. M. Rodenburg, A. M. Maiden, F. Sweeney, and P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B 82, 121415 (2010).
[CrossRef]

Humphry, M. J.

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]

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]

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]

Kanzow, C.

F. Facchinei and C. Kanzow, “Generalized Nash equilibrium problems,” Quart. J. Oper. Res. 5, 173–210 (2007).
[CrossRef]

Katkovnik, V.

A. Danielyan, V. Katkovnik, and K. Egiazarian, “BM3D frames and variational image deblurring,” IEEE Trans. Image Process. 21, 1715–1728 (2012).
[CrossRef]

V. Katkovnik and J. Astola, “Phase retrieval via spatial light modulator phase modulation in 4f optical setup: numerical inverse imaging with sparse regularization for phase and amplitude,” J. Opt. Soc. Am. A 29, 105–116 (2012).
[CrossRef]

V. Katkovnik and J. Astola, “High-accuracy wavefield reconstruction: decoupled inverse imaging with sparse modeling of phase and amplitude,” J. Opt. Soc. Am. A 29, 44–54 (2012).
[CrossRef]

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[CrossRef]

Kewish, C. M.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010).
[CrossRef]

Kotzer, T.

Y. Censor, N. Cohen, T. Kotzer, and J. Shamir, “Summed squared distance error reduction by simultaneous multiprojections and applications,” Appl. Math. Comput. 126, 157–179 (2002).
[CrossRef]

Maiden, A. M.

A. M. Maiden, M. J. Humphry, F. Zhang, and J. M. Rodenburg, “Superresolution imaging via ptychography,” J. Opt. Soc. Am. A 28, 604–612 (2011).
[CrossRef]

F. Hüe, J. M. Rodenburg, A. M. Maiden, F. Sweeney, and P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B 82, 121415 (2010).
[CrossRef]

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[CrossRef]

Marchesini, S.

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

Menzel, A.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010).
[CrossRef]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[CrossRef]

Midgley, P. A.

F. Hüe, J. M. Rodenburg, A. M. Maiden, F. Sweeney, and P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B 82, 121415 (2010).
[CrossRef]

Patommel, J.

A. Schropp, P. Boye, A. Goldschmidt, S. Hönig, R. Hoppe, J. Patommel, C. Rakete, D. Samberg, S. Stephan, S. Schöder, M. Burghammer, and C. G. Schroer, “Non-destructive and quantitative imaging of a nano-structured microchip by ptychographic hard x-ray scanning microscopy,” J. Microsc. 241, 9–12 (2011).
[CrossRef]

Pfeiffer, F.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010).
[CrossRef]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[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]

Rakete, C.

A. Schropp, P. Boye, A. Goldschmidt, S. Hönig, R. Hoppe, J. Patommel, C. Rakete, D. Samberg, S. Stephan, S. Schöder, M. Burghammer, and C. G. Schroer, “Non-destructive and quantitative imaging of a nano-structured microchip by ptychographic hard x-ray scanning microscopy,” J. Microsc. 241, 9–12 (2011).
[CrossRef]

Rodenburg, J. M.

A. M. Maiden, M. J. Humphry, F. Zhang, and J. M. Rodenburg, “Superresolution imaging via ptychography,” J. Opt. Soc. Am. A 28, 604–612 (2011).
[CrossRef]

F. Hüe, J. M. Rodenburg, A. M. Maiden, F. Sweeney, and P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B 82, 121415 (2010).
[CrossRef]

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[CrossRef]

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]

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, “Ptychography and related diffractive imaging methods,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Elsevier, 2008), pp. 87–184.

Romberg, J.

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[CrossRef]

Samberg, D.

A. Schropp, P. Boye, A. Goldschmidt, S. Hönig, R. Hoppe, J. Patommel, C. Rakete, D. Samberg, S. Stephan, S. Schöder, M. Burghammer, and C. G. Schroer, “Non-destructive and quantitative imaging of a nano-structured microchip by ptychographic hard x-ray scanning microscopy,” J. Microsc. 241, 9–12 (2011).
[CrossRef]

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, 237–246 (1972).

Schneider, P.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010).
[CrossRef]

Schöder, S.

A. Schropp, P. Boye, A. Goldschmidt, S. Hönig, R. Hoppe, J. Patommel, C. Rakete, D. Samberg, S. Stephan, S. Schöder, M. Burghammer, and C. G. Schroer, “Non-destructive and quantitative imaging of a nano-structured microchip by ptychographic hard x-ray scanning microscopy,” J. Microsc. 241, 9–12 (2011).
[CrossRef]

Schroer, C. G.

A. Schropp, P. Boye, A. Goldschmidt, S. Hönig, R. Hoppe, J. Patommel, C. Rakete, D. Samberg, S. Stephan, S. Schöder, M. Burghammer, and C. G. Schroer, “Non-destructive and quantitative imaging of a nano-structured microchip by ptychographic hard x-ray scanning microscopy,” J. Microsc. 241, 9–12 (2011).
[CrossRef]

Schropp, A.

A. Schropp, P. Boye, A. Goldschmidt, S. Hönig, R. Hoppe, J. Patommel, C. Rakete, D. Samberg, S. Stephan, S. Schöder, M. Burghammer, and C. G. Schroer, “Non-destructive and quantitative imaging of a nano-structured microchip by ptychographic hard x-ray scanning microscopy,” J. Microsc. 241, 9–12 (2011).
[CrossRef]

Shamir, J.

Y. Censor, N. Cohen, T. Kotzer, and J. Shamir, “Summed squared distance error reduction by simultaneous multiprojections and applications,” Appl. Math. Comput. 126, 157–179 (2002).
[CrossRef]

Stephan, S.

A. Schropp, P. Boye, A. Goldschmidt, S. Hönig, R. Hoppe, J. Patommel, C. Rakete, D. Samberg, S. Stephan, S. Schöder, M. Burghammer, and C. G. Schroer, “Non-destructive and quantitative imaging of a nano-structured microchip by ptychographic hard x-ray scanning microscopy,” J. Microsc. 241, 9–12 (2011).
[CrossRef]

Sweeney, F.

F. Hüe, J. M. Rodenburg, A. M. Maiden, F. Sweeney, and P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B 82, 121415 (2010).
[CrossRef]

Tao, T.

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[CrossRef]

Thibault, P.

P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys. 14, 063004 (2012).
[CrossRef]

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010).
[CrossRef]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[CrossRef]

Wepf, R.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010).
[CrossRef]

Zhang, F.

Appl. Math. Comput. (1)

Y. Censor, N. Cohen, T. Kotzer, and J. Shamir, “Summed squared distance error reduction by simultaneous multiprojections and applications,” Appl. Math. Comput. 126, 157–179 (2002).
[CrossRef]

Appl. Opt. (1)

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]

IEEE Trans. Image Process. (2)

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[CrossRef]

A. Danielyan, V. Katkovnik, and K. Egiazarian, “BM3D frames and variational image deblurring,” IEEE Trans. Image Process. 21, 1715–1728 (2012).
[CrossRef]

IEEE Trans. Inf. Theory (2)

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[CrossRef]

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[CrossRef]

J. Microsc. (1)

A. Schropp, P. Boye, A. Goldschmidt, S. Hönig, R. Hoppe, J. Patommel, C. Rakete, D. Samberg, S. Stephan, S. Schöder, M. Burghammer, and C. G. Schroer, “Non-destructive and quantitative imaging of a nano-structured microchip by ptychographic hard x-ray scanning microscopy,” J. Microsc. 241, 9–12 (2011).
[CrossRef]

J. Opt. Soc. Am. A (4)

Nature (1)

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010).
[CrossRef]

New J. Phys. (1)

P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys. 14, 063004 (2012).
[CrossRef]

Opt. Acta (1)

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Opt. Lett. (1)

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, 237–246 (1972).

Phys. Rev. B (1)

F. Hüe, J. M. Rodenburg, A. M. Maiden, F. Sweeney, and P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B 82, 121415 (2010).
[CrossRef]

Phys. Rev. Lett. (1)

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]

Quart. J. Oper. Res. (1)

F. Facchinei and C. Kanzow, “Generalized Nash equilibrium problems,” Quart. J. Oper. Res. 5, 173–210 (2007).
[CrossRef]

Rev. Sci. Instrum. (1)

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

Ultramicroscopy (4)

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[CrossRef]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[CrossRef]

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]

W. Hoppe, “Trace structure analysis, ptychography, phase tomography,” Ultramicroscopy 10, 187–198 (1982).
[CrossRef]

Other (2)

J. M. Rodenburg, “Ptychography and related diffractive imaging methods,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Elsevier, 2008), pp. 87–184.

M. Elad, Sparse and Redundant Representations: from Theory to Applications in Signal and Image Processing (Springer, 2010).

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Figures (7)

Fig. 1.
Fig. 1.

Schematic of ptychographical imaging.

Fig. 2.
Fig. 2.

E0 criteria calculated for object (solid) and probe (dashed) versus the photon count. Results for three algorithms are compared: e-PIE (“squares”, blue in color); basic (no filtering) SP-PI (“o”, green in color); SP-PI (“stars”, red in color). The best fitting of object and probe complex-valued functions is enabled by the SP-PI algorithm with BM3D and observation filtering.

Fig. 3.
Fig. 3.

RMSE calculated for modulus (solid) and phase (dashed) of object. Three algorithms are compared: e-PIE (“squares”, blue in color); basic (no filtering) SP-PI (“o”, green in color); SP-PI (“stars”, red in color). The best accuracy is achieved by the SP-PI algorithm with BM3D and observation filtering.

Fig. 4.
Fig. 4.

E0 criteria calculated for object versus iteration number. Three algorithms are compared: e-PIE (“squares”, blue in color); basic (no filtering) SP-PI (“o”, green in color); SP-PI (“stars”, red in color). The best convergence rate is obtained by the SP-PI algorithm with BM3D and observation filtering.

Fig. 5.
Fig. 5.

Complex object reconstruction. Very noisy observations, χ=.01. e-PIE algorithm: first row, images of modulus and phase reconstructions; second row, cross sections of the true image (solid, red in color) and the reconstruction (dashed, blue in color).

Fig. 6.
Fig. 6.

Complex object reconstruction. Very noisy observations, χ=.01. SP-PI algorithm: first row, images of modulus and phase reconstructions; second row, cross sections of the true image (solid, red in color) and the reconstruction (dashed, blue in color).

Fig. 7.
Fig. 7.

Complex probe reconstruction used for initialization of all algorithms. Noiseless observations, χ=10,000, 200 iterations. SP-PI algorithm: first row, images of modulus and phase reconstructions; second row, cross sections of the true image (solid, red in color) and the reconstruction (dashed, blue in color).

Equations (63)

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ψj(r)=o(r+rj)p(r).
uj(ω)=F{ψj(r)}=F{o(r+rj)p(r)},
Ij(ω)=|uj(ω)|2=|F{o(r+rj)p(r)}|2.
[Δx,Δy]=λzΔl[1Ms,1Ns],
uj[l]=DFT{ψj[k]},
ψj[k]=o[k+kj]p[k],
uj=Aψj,
uj=A·(ojp)=
A·diag(p)Gjoo=
A·diag(Gjoo)p.
Ij=|uj|2,IjRns,j=1,,J.
p(Ij[l]=L)=exp(Ij[l]χ)(Ij[l]χ)LL!.
I^j[l]=Poisson{Ij[l]χ},
I^j[l]=Poisson{Ij[l]χ}/χ.
I^j[l]=Ij[l]+σj,lεj,l,
I^j=Ij+σjεj,
Y=Ψθ,θ=ΦY.
mod(o)=Ψa,oθa,o,angle(o)=Ψφ,oθφ,o,
θa,o=Φa,o·mod(o),θφ,o=Φφ,o·angle(o),
L1(o,p,Θ)=χ2j=1J(I^j|uj|2)./|uj|22+
1γoovo22,
vo=(Ψa,oθa,o)exp(iΨφ,oθφ,o),i=1,
L2(o,p,Θ)=τa,o·θa,olp+τφ,o·θφ,olp+
12θa,oΦa,omod(o)22+12θφ,oΦφ,oangle(o)22.
(ot+1,pt+1)=argmino,pL1(o,p,Θt),
Θt+1=argminΘL2(ot+1,pt+1,Θ).
(o*,p*)=argmino,pL1(o,p,Θ*),Θ*=argminΘL2(o*,p*,Θ).
L1(o,p,w,Θ)=χ2j=1J(I^j|wj|2)·/|wj|22+1γ1j=1Jujwj22+1γoovo22,vo=(Ψa,oθa,o)exp(i·Ψφ,oθφ,o).
(ot+1,pt+1,wt+1)=argmino,p,wL1(o,p,w,Θt),
Θt+1=argminΘL2(ot+1,pt+1,Θ).
ot+1=argminoL1(o,pt,wt,Θt),
pt+1=argminpL1(ot+1,p,wt,Θt),
wt+1=argminwL1(ot+1,pt+1,w,Θt),
Θt+1=argminΘL2(ot+1,pt+1,Θ).
o[k]=vo[k]/γo+1γ1j=1Jψ^j[kkj]p*[kkj]1/γo+1γ1j=1J|p[kkj]|2,ψ^j[k]=DFT1{wj},
p[k]=j=1Jψ^j[k]o*[k+kj]j=1J|o[k+kj]|2.
wj[l]=G(I^j[l],uj[l])|w^j[l]|exp(iangle(uj[l])),
|w^j[l]|=argmin|wj[l]|χ2(I^j[l]|wj[l]|2)2/|wj[l]|2+1γ1(|uj[l]||wj[l]|)2.
wj[l]=G(I^j[l],uj[l])=uj[l]|uj[l]|I^j[l].
θa,o=Thτa,o(Φmod(o)),θφ,o=Thτφ,o(Φφ,oangle(o)),
b^=Thτ(b)={Th2τhard(b)=b·1(|b|2τ),iflp=l0,Thτsoft(b)=sign(b)·max(|b|τ,0),iflp=l1,
ot+1[k]=vot[k]/γo+1γ1j=1jψ^jt[kkj](pt[kkj])*1/γo+1γ1j=1j|pt[kkj]|2;
pt+1[k]=j=1jψ^jt[k](ot+1(k+kj))*j=1j|ot+1[k+kj]|2;
ψjt+1[k]=ψjt[k]+ΠF{2ψ˜jt+1[k]ψjt+1[k]}ψ˜jt+1[k],
oi+1[k]=oi[k]+α(ψ˜q(i)i[kkq(i)]ψq(i)i[kkq(i)])(pi[kkq(i)])*|pi[kkq(i)]|max2,
pi+1[k]=pi[k]+β(ψ˜q(i)i[k]ψq(i)i[k])(os(i)[k+kq(i)])*|os(i)[k+kq(i)]|max2.
oht=othot,pht=pthpt,hot=(ot)Ho/ot22,hpt=(pt)Hp/pt22,
E0,object(t)=ooht22o22,E0,probe(t)=ppht22p22.
L1/o*=1γo(ovo)+1γ1j=1JujHo*(ujwj)=0.
1γo(ovo)+1γ1j=1J(A·diag(p)Gjoo)Ho*(ujwj)=1γo(ovo)+1γ1j=1J(A·diag(p)Gjo)H(A·diag(p)Gjoowj)=0.
[1γoIno×no+1γ1j=1J(Gjo)Tdiag(p*)AHA·diag(p)Gjo]o=1γovo+1γ1j=1J(Gjo)T·diag(pH)AHwj.
[1γoIno×no+1γ1j=1J(Gjo)Tdiag(|p|2)Gjo]o=1γovo+1γ1j=1J(Gjo)T·diag(pH)AHwj.
L1/p*=j=1JujHp*(ujwj)=0.
j=1Jdiag(GjooH)AHAdiag(Gjoo)p=j=1J(A·diag(Gjoo))Hwj.
j=1Jdiag(Gjo|o|2)p=j=1Jdiag(GjooH)AHwj.
L˜1=χ2j=1J(I^j|wj|2)·/|wj|2+1γ1j=1Jujwj2.
minwj[l]χ2(I^j[l]|wj[l]2)2/|wj[l]|2+1γ1|uj[l]wj[l]|2,
angle(w^j[l])=angle(uj[l]).
χ2(I^j[l]|wj[l]|2)2/|wj[l]|2+1γ1(|uj[l]||wj[l]|)2,
|w^j[l]|=argmin|wj[l]|χ2(I^j[l]|wj[l]|2)2/|wj[l]|2+1γ1uj[l]||wj[l]2.
w^j[l]=|w^j[l]|exp(i·angle(uj[l])),
wj[l]=G(I^j[l],uj[l]).
θi=argminθiτ·θip+12(θiBi)2.

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