Abstract

For iterative phase retrieval algorithms in near field x-ray propagation imaging experiments with a single distance measurement, it is indispensable to have a strong constraint based on a priori information about the specimen; for example, information about the specimen’s support. Recently, Loock and Plonka proposed to use the a priori information that the exit wave is sparsely represented in a certain directional representation system, a so-called shearlet system. In this work, we extend this approach to complex-valued signals by applying the new shearlet constraint to amplitude and phase separately. Further, we demonstrate its applicability to experimental data.

© 2016 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Transport of intensity phase reconstruction to solve the twin image problem in holographic x-ray imaging

M. Krenkel, M. Bartels, and T. Salditt
Opt. Express 21(2) 2220-2235 (2013)

Regularized Newton methods for x-ray phase contrast and general imaging problems

Simon Maretzke, Matthias Bartels, Martin Krenkel, Tim Salditt, and Thorsten Hohage
Opt. Express 24(6) 6490-6506 (2016)

Gauging low-dose X-ray phase-contrast imaging at a single and large propagation distance

Ralf Hofmann, Alexander Schober, Steffen Hahn, Julian Moosmann, Jubin Kashef, Madeleine Hertel, Venera Weinhardt, Daniel Hänschke, Lukas Helfen, Iván A. Sánchez Salazar, Jean-Pierre Guigay, Xianghui Xiao, and Tilo Baumbach
Opt. Express 24(4) 4331-4348 (2016)

References

  • View by:
  • |
  • |
  • |

  1. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995).
    [Crossref]
  2. P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. 29(1), 133–146 (1996).
    [Crossref]
  3. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature 384, 335–338 (1996).
    [Crossref]
  4. K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
    [Crossref]
  5. A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68(7), 2774–2782 (1997).
    [Crossref]
  6. M. Bartels, M. Krenkel, J. Haber, R. N. Wilke, and T. Salditt, “X-ray holographic imaging of hydrated biological cells in solution,” Phys. Rev. Lett. 114, 048103 (2015).
    [Crossref]
  7. S. Maretzke, “A uniqueness result for propagation-based phase contrast imaging from a single measurement,” Inverse Probl. 31, 065003 (2015).
    [Crossref]
  8. J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics (John Wiley & Sons, 2011).
    [Crossref]
  9. R. A. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49(16), 3573–3583 (2004).
    [Crossref]
  10. D. M. Paganin, Coherent X-Ray Optics (Oxford University, 2006).
    [Crossref]
  11. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73(11), 1434–1441 (1983).
    [Crossref]
  12. P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holoto-mography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75(19), 2912–2914 (1999).
    [Crossref]
  13. J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “Mixed transfer function and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett. 32(12), 1617–1619 (2007).
    [Crossref] [PubMed]
  14. H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23(5), 1179–1200 (2006).
    [Crossref]
  15. J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
    [Crossref]
  16. S. Marchesini, “Invited article: a unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 011301 (2007).
    [Crossref]
  17. V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A 20(1), 40–55 (2003).
    [Crossref]
  18. T. E. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220(1–3), 49–58 (2003).
    [Crossref]
  19. K. Giewekemeyer, S. P. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A 83, 023804 (2011).
    [Crossref]
  20. A.-L. Robisch, K. Kröger, A. Rack, and T. Salditt, “Near-field ptychography using lateral and longitudinal shifts,” New J. Phys. 17, 073033 (2015).
    [Crossref]
  21. V. Davidoiu, B. Sixou, M. Langer, and F. Peyrin, “Nonlinear approaches for the single-distance phase retrieval problem involving regularizations with sparsity constraints,” Appl. Opt. 52(17), 3977–3986 (2013).
    [Crossref] [PubMed]
  22. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35(2), 237–246 (1972).
  23. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3(1), 27–29 (1978).
    [Crossref] [PubMed]
  24. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21(15), 2758–2769 (1982).
    [Crossref] [PubMed]
  25. H. H. Bauschke, P. L. Combettes, and D. R. Luke, “Hybrid projection-reflection method for phase retrieval,” J. Opt. Soc. Am. A 20(6), 1025–1034 (2003).
    [Crossref]
  26. D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21(1), 37–50 (2005).
    [Crossref]
  27. S. Loock and G. Plonka, “Phase retrieval for Fresnel measurements using a shearlet sparsity constraint,” Inverse Probl. 30, 055005 (2014).
    [Crossref]
  28. D. Labate, W.-Q. Lim, G. Kutyniok, and G. Weiss, “Sparse multidimensional representation using shearlets,” Proc. SPIE 5914, 254–262 (2005).
  29. K. Guo, G. Kutyniok, and D. Labate, “Sparse multidimensional representations using anisotropic dilation and shear operators,” in Proceedings of the International Conference on the Interactions between Wavelets and Splines, G. Chen and M. Lai, eds. (Nashboro, 2006), pp. 189–201.
  30. G. Kutyniok and D. Labate, Shearlets: Multiscale Analysis for Multivariate Data (Birkhäuser, 2012).
    [Crossref]
  31. P. Kittipoom, G. Kutyniok, and W.-Q. Lim, “Construction of compactly supported shearlet frames,” Constr. Approx. 35, (1)21–72 (2012).
    [Crossref]
  32. D. L. Donoho, “Sparse components of images and optimal atomic decomposition,” Constr. Approx. 17(3), 353–382 (2001).
    [Crossref]
  33. K. Guo and D. Labate, “Optimally sparse multidimensional representation using shearlets,” SIAM J. Math. Anal. 39(1), 298–318 (2007).
    [Crossref]
  34. G. Kutyniok and W.-Q. Lim, “Compactly supported shearlets are optimally sparse,” J. Approx. Theory 163(11), 1564–1589 (2011).
    [Crossref]
  35. W.-Q. Lim, “Nonseparable shearlet transform,” IEEE Trans. Image Process. 22(5), 2056–2065 (2013).
    [Crossref] [PubMed]
  36. G. Kutyniok, W.-Q. Lim, and R. Reisenhofer, “Shearlab 3D: faithful digital shearlet transforms based on compactly supported shearlets,” ACM Trans. Math. Software 42(1) 100 (2015).
  37. D. L. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inform. Theory 41(3), 613–627 (1995).
    [Crossref]
  38. T. Salditt, M. Osterhoff, M. Krenkel, R. N. Wilke, M. Priebe, M. Bartels, S. Kalbfleisch, and M. Sprung, “Compound focusing mirror and x-ray waveguide optics for coherent imaging and nano-diffraction,” J. Synchrotron Radiat. 22(4), 867–878 (2015).
    [Crossref] [PubMed]

2015 (5)

M. Bartels, M. Krenkel, J. Haber, R. N. Wilke, and T. Salditt, “X-ray holographic imaging of hydrated biological cells in solution,” Phys. Rev. Lett. 114, 048103 (2015).
[Crossref]

S. Maretzke, “A uniqueness result for propagation-based phase contrast imaging from a single measurement,” Inverse Probl. 31, 065003 (2015).
[Crossref]

A.-L. Robisch, K. Kröger, A. Rack, and T. Salditt, “Near-field ptychography using lateral and longitudinal shifts,” New J. Phys. 17, 073033 (2015).
[Crossref]

G. Kutyniok, W.-Q. Lim, and R. Reisenhofer, “Shearlab 3D: faithful digital shearlet transforms based on compactly supported shearlets,” ACM Trans. Math. Software 42(1) 100 (2015).

T. Salditt, M. Osterhoff, M. Krenkel, R. N. Wilke, M. Priebe, M. Bartels, S. Kalbfleisch, and M. Sprung, “Compound focusing mirror and x-ray waveguide optics for coherent imaging and nano-diffraction,” J. Synchrotron Radiat. 22(4), 867–878 (2015).
[Crossref] [PubMed]

2014 (1)

S. Loock and G. Plonka, “Phase retrieval for Fresnel measurements using a shearlet sparsity constraint,” Inverse Probl. 30, 055005 (2014).
[Crossref]

2013 (2)

2012 (1)

P. Kittipoom, G. Kutyniok, and W.-Q. Lim, “Construction of compactly supported shearlet frames,” Constr. Approx. 35, (1)21–72 (2012).
[Crossref]

2011 (2)

G. Kutyniok and W.-Q. Lim, “Compactly supported shearlets are optimally sparse,” J. Approx. Theory 163(11), 1564–1589 (2011).
[Crossref]

K. Giewekemeyer, S. P. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A 83, 023804 (2011).
[Crossref]

2007 (3)

J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “Mixed transfer function and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett. 32(12), 1617–1619 (2007).
[Crossref] [PubMed]

S. Marchesini, “Invited article: a unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 011301 (2007).
[Crossref]

K. Guo and D. Labate, “Optimally sparse multidimensional representation using shearlets,” SIAM J. Math. Anal. 39(1), 298–318 (2007).
[Crossref]

2006 (1)

2005 (2)

D. Labate, W.-Q. Lim, G. Kutyniok, and G. Weiss, “Sparse multidimensional representation using shearlets,” Proc. SPIE 5914, 254–262 (2005).

D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21(1), 37–50 (2005).
[Crossref]

2004 (1)

R. A. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49(16), 3573–3583 (2004).
[Crossref]

2003 (3)

2001 (1)

D. L. Donoho, “Sparse components of images and optimal atomic decomposition,” Constr. Approx. 17(3), 353–382 (2001).
[Crossref]

1999 (2)

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

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holoto-mography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75(19), 2912–2914 (1999).
[Crossref]

1997 (1)

A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68(7), 2774–2782 (1997).
[Crossref]

1996 (3)

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. 29(1), 133–146 (1996).
[Crossref]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature 384, 335–338 (1996).
[Crossref]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[Crossref]

1995 (2)

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995).
[Crossref]

D. L. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inform. Theory 41(3), 613–627 (1995).
[Crossref]

1983 (1)

1982 (1)

1978 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35(2), 237–246 (1972).

Als-Nielsen, J.

J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics (John Wiley & Sons, 2011).
[Crossref]

Barnea, Z.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[Crossref]

Barrett, R.

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. 29(1), 133–146 (1996).
[Crossref]

Bartels, M.

M. Bartels, M. Krenkel, J. Haber, R. N. Wilke, and T. Salditt, “X-ray holographic imaging of hydrated biological cells in solution,” Phys. Rev. Lett. 114, 048103 (2015).
[Crossref]

T. Salditt, M. Osterhoff, M. Krenkel, R. N. Wilke, M. Priebe, M. Bartels, S. Kalbfleisch, and M. Sprung, “Compound focusing mirror and x-ray waveguide optics for coherent imaging and nano-diffraction,” J. Synchrotron Radiat. 22(4), 867–878 (2015).
[Crossref] [PubMed]

K. Giewekemeyer, S. P. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A 83, 023804 (2011).
[Crossref]

Barty, A.

Baruchel, J.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holoto-mography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75(19), 2912–2914 (1999).
[Crossref]

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. 29(1), 133–146 (1996).
[Crossref]

Bauschke, H. H.

Beetz, T.

Beta, C.

K. Giewekemeyer, S. P. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A 83, 023804 (2011).
[Crossref]

Boistel, R.

Chapman, H. N.

Charalambous, P.

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

Cloetens, P.

J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “Mixed transfer function and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett. 32(12), 1617–1619 (2007).
[Crossref] [PubMed]

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holoto-mography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75(19), 2912–2914 (1999).
[Crossref]

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. 29(1), 133–146 (1996).
[Crossref]

Combettes, P. L.

Cookson, D. F.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[Crossref]

Cui, C.

Davidoiu, V.

Donoho, D. L.

D. L. Donoho, “Sparse components of images and optimal atomic decomposition,” Constr. Approx. 17(3), 353–382 (2001).
[Crossref]

D. L. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inform. Theory 41(3), 613–627 (1995).
[Crossref]

Elser, V.

Fienup, J. R.

Gao, D.

A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68(7), 2774–2782 (1997).
[Crossref]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature 384, 335–338 (1996).
[Crossref]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35(2), 237–246 (1972).

Giewekemeyer, K.

K. Giewekemeyer, S. P. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A 83, 023804 (2011).
[Crossref]

Guigay, J. P.

J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “Mixed transfer function and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett. 32(12), 1617–1619 (2007).
[Crossref] [PubMed]

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holoto-mography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75(19), 2912–2914 (1999).
[Crossref]

Guigay, J.-P.

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. 29(1), 133–146 (1996).
[Crossref]

Guo, K.

K. Guo and D. Labate, “Optimally sparse multidimensional representation using shearlets,” SIAM J. Math. Anal. 39(1), 298–318 (2007).
[Crossref]

K. Guo, G. Kutyniok, and D. Labate, “Sparse multidimensional representations using anisotropic dilation and shear operators,” in Proceedings of the International Conference on the Interactions between Wavelets and Splines, G. Chen and M. Lai, eds. (Nashboro, 2006), pp. 189–201.

Gureyev, T. E.

T. E. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220(1–3), 49–58 (2003).
[Crossref]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature 384, 335–338 (1996).
[Crossref]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[Crossref]

Haber, J.

M. Bartels, M. Krenkel, J. Haber, R. N. Wilke, and T. Salditt, “X-ray holographic imaging of hydrated biological cells in solution,” Phys. Rev. Lett. 114, 048103 (2015).
[Crossref]

Hau-Riege, S. P.

He, H.

Howells, M. R.

Jacobsen, C.

Kalbfleisch, S.

T. Salditt, M. Osterhoff, M. Krenkel, R. N. Wilke, M. Priebe, M. Bartels, S. Kalbfleisch, and M. Sprung, “Compound focusing mirror and x-ray waveguide optics for coherent imaging and nano-diffraction,” J. Synchrotron Radiat. 22(4), 867–878 (2015).
[Crossref] [PubMed]

K. Giewekemeyer, S. P. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A 83, 023804 (2011).
[Crossref]

Kirz, J.

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

Kittipoom, P.

P. Kittipoom, G. Kutyniok, and W.-Q. Lim, “Construction of compactly supported shearlet frames,” Constr. Approx. 35, (1)21–72 (2012).
[Crossref]

Kohn, V.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995).
[Crossref]

Krenkel, M.

M. Bartels, M. Krenkel, J. Haber, R. N. Wilke, and T. Salditt, “X-ray holographic imaging of hydrated biological cells in solution,” Phys. Rev. Lett. 114, 048103 (2015).
[Crossref]

T. Salditt, M. Osterhoff, M. Krenkel, R. N. Wilke, M. Priebe, M. Bartels, S. Kalbfleisch, and M. Sprung, “Compound focusing mirror and x-ray waveguide optics for coherent imaging and nano-diffraction,” J. Synchrotron Radiat. 22(4), 867–878 (2015).
[Crossref] [PubMed]

Kröger, K.

A.-L. Robisch, K. Kröger, A. Rack, and T. Salditt, “Near-field ptychography using lateral and longitudinal shifts,” New J. Phys. 17, 073033 (2015).
[Crossref]

Krüger, S. P.

K. Giewekemeyer, S. P. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A 83, 023804 (2011).
[Crossref]

Kutyniok, G.

G. Kutyniok, W.-Q. Lim, and R. Reisenhofer, “Shearlab 3D: faithful digital shearlet transforms based on compactly supported shearlets,” ACM Trans. Math. Software 42(1) 100 (2015).

P. Kittipoom, G. Kutyniok, and W.-Q. Lim, “Construction of compactly supported shearlet frames,” Constr. Approx. 35, (1)21–72 (2012).
[Crossref]

G. Kutyniok and W.-Q. Lim, “Compactly supported shearlets are optimally sparse,” J. Approx. Theory 163(11), 1564–1589 (2011).
[Crossref]

D. Labate, W.-Q. Lim, G. Kutyniok, and G. Weiss, “Sparse multidimensional representation using shearlets,” Proc. SPIE 5914, 254–262 (2005).

K. Guo, G. Kutyniok, and D. Labate, “Sparse multidimensional representations using anisotropic dilation and shear operators,” in Proceedings of the International Conference on the Interactions between Wavelets and Splines, G. Chen and M. Lai, eds. (Nashboro, 2006), pp. 189–201.

G. Kutyniok and D. Labate, Shearlets: Multiscale Analysis for Multivariate Data (Birkhäuser, 2012).
[Crossref]

Kuznetsov, S.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995).
[Crossref]

Labate, D.

K. Guo and D. Labate, “Optimally sparse multidimensional representation using shearlets,” SIAM J. Math. Anal. 39(1), 298–318 (2007).
[Crossref]

D. Labate, W.-Q. Lim, G. Kutyniok, and G. Weiss, “Sparse multidimensional representation using shearlets,” Proc. SPIE 5914, 254–262 (2005).

G. Kutyniok and D. Labate, Shearlets: Multiscale Analysis for Multivariate Data (Birkhäuser, 2012).
[Crossref]

K. Guo, G. Kutyniok, and D. Labate, “Sparse multidimensional representations using anisotropic dilation and shear operators,” in Proceedings of the International Conference on the Interactions between Wavelets and Splines, G. Chen and M. Lai, eds. (Nashboro, 2006), pp. 189–201.

Langer, M.

Lewis, R. A.

R. A. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49(16), 3573–3583 (2004).
[Crossref]

Lim, W.-Q.

G. Kutyniok, W.-Q. Lim, and R. Reisenhofer, “Shearlab 3D: faithful digital shearlet transforms based on compactly supported shearlets,” ACM Trans. Math. Software 42(1) 100 (2015).

W.-Q. Lim, “Nonseparable shearlet transform,” IEEE Trans. Image Process. 22(5), 2056–2065 (2013).
[Crossref] [PubMed]

P. Kittipoom, G. Kutyniok, and W.-Q. Lim, “Construction of compactly supported shearlet frames,” Constr. Approx. 35, (1)21–72 (2012).
[Crossref]

G. Kutyniok and W.-Q. Lim, “Compactly supported shearlets are optimally sparse,” J. Approx. Theory 163(11), 1564–1589 (2011).
[Crossref]

D. Labate, W.-Q. Lim, G. Kutyniok, and G. Weiss, “Sparse multidimensional representation using shearlets,” Proc. SPIE 5914, 254–262 (2005).

Loock, S.

S. Loock and G. Plonka, “Phase retrieval for Fresnel measurements using a shearlet sparsity constraint,” Inverse Probl. 30, 055005 (2014).
[Crossref]

Ludwig, W.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holoto-mography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75(19), 2912–2914 (1999).
[Crossref]

Luke, D. R.

D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21(1), 37–50 (2005).
[Crossref]

H. H. Bauschke, P. L. Combettes, and D. R. Luke, “Hybrid projection-reflection method for phase retrieval,” J. Opt. Soc. Am. A 20(6), 1025–1034 (2003).
[Crossref]

Marchesini, S.

Maretzke, S.

S. Maretzke, “A uniqueness result for propagation-based phase contrast imaging from a single measurement,” Inverse Probl. 31, 065003 (2015).
[Crossref]

McMorrow, D.

J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics (John Wiley & Sons, 2011).
[Crossref]

Miao, J.

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

Noy, A.

Nugent, K. A.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[Crossref]

Osterhoff, M.

T. Salditt, M. Osterhoff, M. Krenkel, R. N. Wilke, M. Priebe, M. Bartels, S. Kalbfleisch, and M. Sprung, “Compound focusing mirror and x-ray waveguide optics for coherent imaging and nano-diffraction,” J. Synchrotron Radiat. 22(4), 867–878 (2015).
[Crossref] [PubMed]

Paganin, D.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[Crossref]

Paganin, D. M.

D. M. Paganin, Coherent X-Ray Optics (Oxford University, 2006).
[Crossref]

Peyrin, F.

Plonka, G.

S. Loock and G. Plonka, “Phase retrieval for Fresnel measurements using a shearlet sparsity constraint,” Inverse Probl. 30, 055005 (2014).
[Crossref]

Pogany, A.

A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68(7), 2774–2782 (1997).
[Crossref]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature 384, 335–338 (1996).
[Crossref]

Priebe, M.

T. Salditt, M. Osterhoff, M. Krenkel, R. N. Wilke, M. Priebe, M. Bartels, S. Kalbfleisch, and M. Sprung, “Compound focusing mirror and x-ray waveguide optics for coherent imaging and nano-diffraction,” J. Synchrotron Radiat. 22(4), 867–878 (2015).
[Crossref] [PubMed]

Rack, A.

A.-L. Robisch, K. Kröger, A. Rack, and T. Salditt, “Near-field ptychography using lateral and longitudinal shifts,” New J. Phys. 17, 073033 (2015).
[Crossref]

Reisenhofer, R.

G. Kutyniok, W.-Q. Lim, and R. Reisenhofer, “Shearlab 3D: faithful digital shearlet transforms based on compactly supported shearlets,” ACM Trans. Math. Software 42(1) 100 (2015).

Robisch, A.-L.

A.-L. Robisch, K. Kröger, A. Rack, and T. Salditt, “Near-field ptychography using lateral and longitudinal shifts,” New J. Phys. 17, 073033 (2015).
[Crossref]

Rosen, R.

Salditt, T.

A.-L. Robisch, K. Kröger, A. Rack, and T. Salditt, “Near-field ptychography using lateral and longitudinal shifts,” New J. Phys. 17, 073033 (2015).
[Crossref]

M. Bartels, M. Krenkel, J. Haber, R. N. Wilke, and T. Salditt, “X-ray holographic imaging of hydrated biological cells in solution,” Phys. Rev. Lett. 114, 048103 (2015).
[Crossref]

T. Salditt, M. Osterhoff, M. Krenkel, R. N. Wilke, M. Priebe, M. Bartels, S. Kalbfleisch, and M. Sprung, “Compound focusing mirror and x-ray waveguide optics for coherent imaging and nano-diffraction,” J. Synchrotron Radiat. 22(4), 867–878 (2015).
[Crossref] [PubMed]

K. Giewekemeyer, S. P. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A 83, 023804 (2011).
[Crossref]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35(2), 237–246 (1972).

Sayre, D.

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

Schelokov, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995).
[Crossref]

Schlenker, M.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holoto-mography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75(19), 2912–2914 (1999).
[Crossref]

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. 29(1), 133–146 (1996).
[Crossref]

Shapiro, D.

Sixou, B.

Snigirev, A.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995).
[Crossref]

Snigireva, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995).
[Crossref]

Spence, J. C. H.

Sprung, M.

T. Salditt, M. Osterhoff, M. Krenkel, R. N. Wilke, M. Priebe, M. Bartels, S. Kalbfleisch, and M. Sprung, “Compound focusing mirror and x-ray waveguide optics for coherent imaging and nano-diffraction,” J. Synchrotron Radiat. 22(4), 867–878 (2015).
[Crossref] [PubMed]

Stevenson, A. W.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature 384, 335–338 (1996).
[Crossref]

Teague, M. R.

Van Dyck, D.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holoto-mography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75(19), 2912–2914 (1999).
[Crossref]

Van Landuyt, J.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holoto-mography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75(19), 2912–2914 (1999).
[Crossref]

Weierstall, U.

Weiss, G.

D. Labate, W.-Q. Lim, G. Kutyniok, and G. Weiss, “Sparse multidimensional representation using shearlets,” Proc. SPIE 5914, 254–262 (2005).

Wilke, R. N.

T. Salditt, M. Osterhoff, M. Krenkel, R. N. Wilke, M. Priebe, M. Bartels, S. Kalbfleisch, and M. Sprung, “Compound focusing mirror and x-ray waveguide optics for coherent imaging and nano-diffraction,” J. Synchrotron Radiat. 22(4), 867–878 (2015).
[Crossref] [PubMed]

M. Bartels, M. Krenkel, J. Haber, R. N. Wilke, and T. Salditt, “X-ray holographic imaging of hydrated biological cells in solution,” Phys. Rev. Lett. 114, 048103 (2015).
[Crossref]

Wilkins, S. W.

A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68(7), 2774–2782 (1997).
[Crossref]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature 384, 335–338 (1996).
[Crossref]

ACM Trans. Math. Software (1)

G. Kutyniok, W.-Q. Lim, and R. Reisenhofer, “Shearlab 3D: faithful digital shearlet transforms based on compactly supported shearlets,” ACM Trans. Math. Software 42(1) 100 (2015).

Appl. Opt. (2)

Appl. Phys. Lett. (1)

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holoto-mography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75(19), 2912–2914 (1999).
[Crossref]

Constr. Approx. (2)

P. Kittipoom, G. Kutyniok, and W.-Q. Lim, “Construction of compactly supported shearlet frames,” Constr. Approx. 35, (1)21–72 (2012).
[Crossref]

D. L. Donoho, “Sparse components of images and optimal atomic decomposition,” Constr. Approx. 17(3), 353–382 (2001).
[Crossref]

IEEE Trans. Image Process. (1)

W.-Q. Lim, “Nonseparable shearlet transform,” IEEE Trans. Image Process. 22(5), 2056–2065 (2013).
[Crossref] [PubMed]

IEEE Trans. Inform. Theory (1)

D. L. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inform. Theory 41(3), 613–627 (1995).
[Crossref]

Inverse Probl. (3)

D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21(1), 37–50 (2005).
[Crossref]

S. Loock and G. Plonka, “Phase retrieval for Fresnel measurements using a shearlet sparsity constraint,” Inverse Probl. 30, 055005 (2014).
[Crossref]

S. Maretzke, “A uniqueness result for propagation-based phase contrast imaging from a single measurement,” Inverse Probl. 31, 065003 (2015).
[Crossref]

J. Approx. Theory (1)

G. Kutyniok and W.-Q. Lim, “Compactly supported shearlets are optimally sparse,” J. Approx. Theory 163(11), 1564–1589 (2011).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

J. Phys. D: Appl. Phys. (1)

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. 29(1), 133–146 (1996).
[Crossref]

J. Synchrotron Radiat. (1)

T. Salditt, M. Osterhoff, M. Krenkel, R. N. Wilke, M. Priebe, M. Bartels, S. Kalbfleisch, and M. Sprung, “Compound focusing mirror and x-ray waveguide optics for coherent imaging and nano-diffraction,” J. Synchrotron Radiat. 22(4), 867–878 (2015).
[Crossref] [PubMed]

Nature (2)

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature 384, 335–338 (1996).
[Crossref]

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

New J. Phys. (1)

A.-L. Robisch, K. Kröger, A. Rack, and T. Salditt, “Near-field ptychography using lateral and longitudinal shifts,” New J. Phys. 17, 073033 (2015).
[Crossref]

Opt. Commun. (1)

T. E. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220(1–3), 49–58 (2003).
[Crossref]

Opt. Lett. (2)

Optik (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35(2), 237–246 (1972).

Phys. Med. Biol. (1)

R. A. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49(16), 3573–3583 (2004).
[Crossref]

Phys. Rev. A (1)

K. Giewekemeyer, S. P. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A 83, 023804 (2011).
[Crossref]

Phys. Rev. Lett. (2)

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[Crossref]

M. Bartels, M. Krenkel, J. Haber, R. N. Wilke, and T. Salditt, “X-ray holographic imaging of hydrated biological cells in solution,” Phys. Rev. Lett. 114, 048103 (2015).
[Crossref]

Proc. SPIE (1)

D. Labate, W.-Q. Lim, G. Kutyniok, and G. Weiss, “Sparse multidimensional representation using shearlets,” Proc. SPIE 5914, 254–262 (2005).

Rev. Sci. Instrum. (3)

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995).
[Crossref]

A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68(7), 2774–2782 (1997).
[Crossref]

S. Marchesini, “Invited article: a unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 011301 (2007).
[Crossref]

SIAM J. Math. Anal. (1)

K. Guo and D. Labate, “Optimally sparse multidimensional representation using shearlets,” SIAM J. Math. Anal. 39(1), 298–318 (2007).
[Crossref]

Other (4)

K. Guo, G. Kutyniok, and D. Labate, “Sparse multidimensional representations using anisotropic dilation and shear operators,” in Proceedings of the International Conference on the Interactions between Wavelets and Splines, G. Chen and M. Lai, eds. (Nashboro, 2006), pp. 189–201.

G. Kutyniok and D. Labate, Shearlets: Multiscale Analysis for Multivariate Data (Birkhäuser, 2012).
[Crossref]

D. M. Paganin, Coherent X-Ray Optics (Oxford University, 2006).
[Crossref]

J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics (John Wiley & Sons, 2011).
[Crossref]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

(a) Setup of an x-ray propagation imaging experiment. The specimen is illuminated by a coherent x-ray probe. Traversing the object, the waves are modified according to the complex refractive index and thereafter the exit wave (object plane) propagates to a detector at a distance z behind the specimen (detector plane). The phase contrast arises from the free space propagation without further optical elements needed. (b) Illustration of the shearlet constraint, incorporated into the ER algorithm. The current approximation is alternately propagated between the object and the detector plane where it is forced to fulfill the shearlet constraint and the modulus constraint, respectively. Here, for the general case of a mixed object, where shearlet thresholding is performed on the phase and the amplitude of the approximation separately. SH and SH−1 denote the discrete shearlet transform and its inverse.

Fig. 2
Fig. 2

Simulated x-ray propagation imaging experiment. (a), (b) Phase and amplitude of the complex-valued exit wave field in the object plane (512×512 pixel). The dashed black frames indicate the support considered for the support & range constraint (341×256 pixel). (c) Simulated intensity measurement with Fresnel number F = 4 10−3 and artificial Poisson noise (50 photons per pixel). The coarse outlines of the fish ·and the elephant are still visible. (d), (e), (f) Exit waves reconstructed with the RAAR algorithm (after 100 iterations) and different object plane constraints. The phase is shown in the upper row and the amplitude in the lower row. The amplitudes obtained with the range constraint and the support & range constraint are displayed after only 3 iterations since the corresponding error increases with the number of iteration steps, see (g). (g) RMS error decay of the phase and amplitude from the three different reconstructions, calculated according to Eq. (9) and (10), while taking the region inside the box support into account.

Fig. 3
Fig. 3

Experimental data. (a) Reconstructed phases from experiment 1 obtained with 100 iterations of the RAAR algorithm and different object plane constraints. Reconstructed is only a 1061 × 1061 pixel central detail of the 2048 × 2048 pixel near field hologram. The inlays show the details of the reconstructions as indicated by the dashed white boxes. (b) PSDs of the three reconstructions in (a). The rings denote the location of spatial frequency components corresponding to a half-period resolution of 250 nm and 350 nm as indicated by the legend. (c) Histograms of the phase values from the three reconstructions in (a). For comparison we plotted the same range of phase values for all three reconstructions, however, some runaway values are not captured by this region. Note the different y-scales. The indicated maxima were determined by Gaussian fitting. (d) Reconstructed phases from experiment 2 obtained with 100 iterations of the RAAR algorithm and different object plane constraints. Depicted are the central 700 × 700 pixel details of the reconstructions. The larger magnification factor in experiment 2 compared to experiment 1 provides a higher resolution.

Tables (2)

Tables Icon

Table 1 Effects of different object plane constraints for mixed objects, pure phase objects and pure absorption objects. |ψ|, φ ( ψ ) N x × N Y denote the amplitude and phase of the discrete exit wave field. For the support & range constraint, it is known that the compact support of the exit wave field is contained in some region D ⊂ Ω.

Tables Icon

Table 2 Relevant setup parameters of the two near field diffraction experiments with the Siemens star (Section 4), including the effective near field parameters of the parallel beam geometry.

Equations (10)

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

n ( x , y , z ) = 1 δ ( x , y , z ) + i β ( x , y , z ) ,
ψ ( x , y , z ) = D z [ ψ ( x , y , 0 ) ] : = exp ( i k z ) 1 [ exp ( i z k x 2 + k y 2 2 k ) [ ψ ( x , y , 0 ) ] ] = exp ( i k z ) exp ( i k τ ) 1 [ exp ( i z k x 2 + k y 2 2 k ) [ exp ( i k τ 0 ( δ ( x , y , z ) i β ( x , y , z ) ) d z ) ] ] .
I ( x , y , z ) = | ψ ( x , y , z ) | 2 = | D z [ ψ ( x , y , 0 ) ] | 2 .
f f N L 2 2 C N 2 ( log N ) 3 , as N , C ,
P M ψ = D z g with g ( i , j ) : = { m ( i , j ) D z ψ ( i , j ) | D z ψ ( i , j ) | , | D z ψ ( i , j ) | 0 m ( i , j ) , | D z ψ ( i , j ) | = 0 for ( i , j ) Ω ,
( T θ c ) j : = { c j θ , c j > θ c j + θ , c j < θ 0 , | c j | < θ ,
P S θ a , θ p ψ : = S 1 T θ a S | ψ | exp ( i S 1 T θ p S φ ( ψ ) ) ,
ψ n + 1 = ( α 2 ( R S θ a , θ p R M + Id ) + ( 1 α ) P M ) ψ n ,
RMS phase n = [ 1 N x N y n x = 1 N x n y = 1 N y ( φ ( ψ ( n x , n y ) ) φ ( ψ n ( n x , n y ) ) ) 2 ] 1 / 2 ,
RMS amplitude n = [ 1 N x N y n x = 1 N x n y = 1 N y ( | ψ ( n x , n y ) | | ψ n ( n x , n y ) | ) 2 ] 1 / 2 .

Metrics