Abstract

A reconstruction algorithm is proposed to analyze the complex-valued electron-density distribution in an object with x-ray in-line holograms. The real and imaginary parts of the electron density correspond to local variations of x-ray phase shift and absorption, respectively. In the algorithm, the least-squares error of the holograms is iteratively minimized under sign constraints on the real and imaginary parts. The constraints, which are derived from the physical conditions in the interaction between x rays and materials, facilitate robust reconstruction. The reconstruction was applied to holograms of a biological specimen that caused both phase shift and absorption.

© 2006 Optical Society of America

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  1. 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]
  2. D. Gabor, "A new microscopic principle," Nature 161, 777-778 (1948).
    [CrossRef] [PubMed]
  3. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  4. M. H. Maleki and A. J. Devaney, "Noniterative reconstruction of complex-valued objects from two intensity measurements," Opt. Eng. 33, 3243-3253 (1994).
    [CrossRef]
  5. T. Xiao, H. Xu, Y. Zhang, J. Chen, and Z. Xu, "Digital image decoding for in-line x-ray holography using two holograms," J. Mod. Opt. 45, 343-353 (1998).
    [CrossRef]
  6. Y. Zhang and X. Zhang, "Reconstruction of a complex object from two in-line holograms," Opt. Express 11, 572-578 (2003).
    [CrossRef] [PubMed]
  7. P. Cloetens, W. Ludwig, D. Van Dyck, J. P. Guigay, M. Schlenker, and J. Baruchel, "Quantitative phase tomography by holographic reconstruction," in Developments in X-Ray Tomography II, U.Bonse ed., Proc. SPIE 3772, 279-290 (1999).
  8. W. Coene, G. Janssen, M. Op de Beeck, and D. Van Dyck, "Phase retrieval through focus variation for ultra-resolution in field-emission transmission electron microscopy," Phys. Rev. Lett. 69, 3743-3746 (1992).
    [CrossRef] [PubMed]
  9. Y. Zhang, G. Pedrini, W. Osten, and H. J. Tiziani, "Whole optical wave field reconstruction from double or multi in-line holograms by phase retrieval algorithm," Opt. Express 11, 3234-3241 (2003).
    [CrossRef] [PubMed]
  10. M. Op de Beeck, D. Van Dyck, and W. Coene, "Wave function reconstruction in HRTEM: the parabola method," Ultramicroscopy 64, 167-183 (1996).
    [CrossRef]
  11. B. R. Frieden, "Image enhancement and restoration," inPicture Processing and Digital Filtering, 2nd ed., T.S.Huang, ed. (Springer-Verlag, 1979), Chap. 5.
  12. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Chap. 13.
  13. P. Y. Popalambros and D. J. Wilde, Principles of Optimal Design: Modeling and Computation, 2nd ed. (Cambridge U. Press, 2000).
    [CrossRef]
  14. M. Howells, "Fundamental limits in x-ray holography," in X-Ray Microscopy II, D.Sayre, M.Howells, J.Kirz, and H.Rarback, eds. (Springer-Verlag, 1988), pp. 263-271.
  15. A. K. Katsaggelos, Digital Image Restoration (Springer-Verlag, 1991).
    [CrossRef]
  16. T. Ishikawa, K. Tamasaku, M. Yabashi, S. Goto, Y. Tanaka, H. Yamazaki, K. Takeshita, H. Kimura, H. Ohashi, T. Matsushita, and T. Ohata, "1-km beamline at SPring-8," in Advances in X-Ray Optics, A.K.Freund, T.Ishikawa, A.M.Khounsary, D.C.Mancini, A.G.Michette, and S.Oestreich, eds., Proc. SPIE 4145, 1-10 (2001).
  17. K. Kinoshita, T. Matsumura, Y. Inagaki, N. Hirai,M. Sugiyama, H. Kihara, N. Watanabe, and Y. Shimanuki, "Electronic zooming TV readout system for an x-ray microscope," in Soft X-Ray Microscopy, C.J.Jacobsen and J.E.Trebes, eds., Proc. SPIE 1741, 287-293 (1992).

2003 (2)

1998 (1)

T. Xiao, H. Xu, Y. Zhang, J. Chen, and Z. Xu, "Digital image decoding for in-line x-ray holography using two holograms," J. Mod. Opt. 45, 343-353 (1998).
[CrossRef]

1996 (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]

M. Op de Beeck, D. Van Dyck, and W. Coene, "Wave function reconstruction in HRTEM: the parabola method," Ultramicroscopy 64, 167-183 (1996).
[CrossRef]

1994 (1)

M. H. Maleki and A. J. Devaney, "Noniterative reconstruction of complex-valued objects from two intensity measurements," Opt. Eng. 33, 3243-3253 (1994).
[CrossRef]

1992 (1)

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

1948 (1)

D. Gabor, "A new microscopic principle," Nature 161, 777-778 (1948).
[CrossRef] [PubMed]

Baruchel, J.

P. Cloetens, W. Ludwig, D. Van Dyck, J. P. Guigay, M. Schlenker, and J. Baruchel, "Quantitative phase tomography by holographic reconstruction," in Developments in X-Ray Tomography II, U.Bonse ed., Proc. SPIE 3772, 279-290 (1999).

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Chap. 13.

Chen, J.

T. Xiao, H. Xu, Y. Zhang, J. Chen, and Z. Xu, "Digital image decoding for in-line x-ray holography using two holograms," J. Mod. Opt. 45, 343-353 (1998).
[CrossRef]

Cloetens, P.

P. Cloetens, W. Ludwig, D. Van Dyck, J. P. Guigay, M. Schlenker, and J. Baruchel, "Quantitative phase tomography by holographic reconstruction," in Developments in X-Ray Tomography II, U.Bonse ed., Proc. SPIE 3772, 279-290 (1999).

Coene, W.

M. Op de Beeck, D. Van Dyck, and W. Coene, "Wave function reconstruction in HRTEM: the parabola method," Ultramicroscopy 64, 167-183 (1996).
[CrossRef]

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

Devaney, A. J.

M. H. Maleki and A. J. Devaney, "Noniterative reconstruction of complex-valued objects from two intensity measurements," Opt. Eng. 33, 3243-3253 (1994).
[CrossRef]

Frieden, B. R.

B. R. Frieden, "Image enhancement and restoration," inPicture Processing and Digital Filtering, 2nd ed., T.S.Huang, ed. (Springer-Verlag, 1979), Chap. 5.

Gabor, D.

D. Gabor, "A new microscopic principle," Nature 161, 777-778 (1948).
[CrossRef] [PubMed]

Gao, D.

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]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Goto, S.

T. Ishikawa, K. Tamasaku, M. Yabashi, S. Goto, Y. Tanaka, H. Yamazaki, K. Takeshita, H. Kimura, H. Ohashi, T. Matsushita, and T. Ohata, "1-km beamline at SPring-8," in Advances in X-Ray Optics, A.K.Freund, T.Ishikawa, A.M.Khounsary, D.C.Mancini, A.G.Michette, and S.Oestreich, eds., Proc. SPIE 4145, 1-10 (2001).

Guigay, J. P.

P. Cloetens, W. Ludwig, D. Van Dyck, J. P. Guigay, M. Schlenker, and J. Baruchel, "Quantitative phase tomography by holographic reconstruction," in Developments in X-Ray Tomography II, U.Bonse ed., Proc. SPIE 3772, 279-290 (1999).

Gureyev, T. E.

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]

Hirai, N.

K. Kinoshita, T. Matsumura, Y. Inagaki, N. Hirai,M. Sugiyama, H. Kihara, N. Watanabe, and Y. Shimanuki, "Electronic zooming TV readout system for an x-ray microscope," in Soft X-Ray Microscopy, C.J.Jacobsen and J.E.Trebes, eds., Proc. SPIE 1741, 287-293 (1992).

Howells, M.

M. Howells, "Fundamental limits in x-ray holography," in X-Ray Microscopy II, D.Sayre, M.Howells, J.Kirz, and H.Rarback, eds. (Springer-Verlag, 1988), pp. 263-271.

Inagaki, Y.

K. Kinoshita, T. Matsumura, Y. Inagaki, N. Hirai,M. Sugiyama, H. Kihara, N. Watanabe, and Y. Shimanuki, "Electronic zooming TV readout system for an x-ray microscope," in Soft X-Ray Microscopy, C.J.Jacobsen and J.E.Trebes, eds., Proc. SPIE 1741, 287-293 (1992).

Ishikawa, T.

T. Ishikawa, K. Tamasaku, M. Yabashi, S. Goto, Y. Tanaka, H. Yamazaki, K. Takeshita, H. Kimura, H. Ohashi, T. Matsushita, and T. Ohata, "1-km beamline at SPring-8," in Advances in X-Ray Optics, A.K.Freund, T.Ishikawa, A.M.Khounsary, D.C.Mancini, A.G.Michette, and S.Oestreich, eds., Proc. SPIE 4145, 1-10 (2001).

Janssen, G.

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

Katsaggelos, A. K.

A. K. Katsaggelos, Digital Image Restoration (Springer-Verlag, 1991).
[CrossRef]

Kihara, H.

K. Kinoshita, T. Matsumura, Y. Inagaki, N. Hirai,M. Sugiyama, H. Kihara, N. Watanabe, and Y. Shimanuki, "Electronic zooming TV readout system for an x-ray microscope," in Soft X-Ray Microscopy, C.J.Jacobsen and J.E.Trebes, eds., Proc. SPIE 1741, 287-293 (1992).

Kimura, H.

T. Ishikawa, K. Tamasaku, M. Yabashi, S. Goto, Y. Tanaka, H. Yamazaki, K. Takeshita, H. Kimura, H. Ohashi, T. Matsushita, and T. Ohata, "1-km beamline at SPring-8," in Advances in X-Ray Optics, A.K.Freund, T.Ishikawa, A.M.Khounsary, D.C.Mancini, A.G.Michette, and S.Oestreich, eds., Proc. SPIE 4145, 1-10 (2001).

Kinoshita, K.

K. Kinoshita, T. Matsumura, Y. Inagaki, N. Hirai,M. Sugiyama, H. Kihara, N. Watanabe, and Y. Shimanuki, "Electronic zooming TV readout system for an x-ray microscope," in Soft X-Ray Microscopy, C.J.Jacobsen and J.E.Trebes, eds., Proc. SPIE 1741, 287-293 (1992).

Ludwig, W.

P. Cloetens, W. Ludwig, D. Van Dyck, J. P. Guigay, M. Schlenker, and J. Baruchel, "Quantitative phase tomography by holographic reconstruction," in Developments in X-Ray Tomography II, U.Bonse ed., Proc. SPIE 3772, 279-290 (1999).

Maleki, M. H.

M. H. Maleki and A. J. Devaney, "Noniterative reconstruction of complex-valued objects from two intensity measurements," Opt. Eng. 33, 3243-3253 (1994).
[CrossRef]

Matsumura, T.

K. Kinoshita, T. Matsumura, Y. Inagaki, N. Hirai,M. Sugiyama, H. Kihara, N. Watanabe, and Y. Shimanuki, "Electronic zooming TV readout system for an x-ray microscope," in Soft X-Ray Microscopy, C.J.Jacobsen and J.E.Trebes, eds., Proc. SPIE 1741, 287-293 (1992).

Matsushita, T.

T. Ishikawa, K. Tamasaku, M. Yabashi, S. Goto, Y. Tanaka, H. Yamazaki, K. Takeshita, H. Kimura, H. Ohashi, T. Matsushita, and T. Ohata, "1-km beamline at SPring-8," in Advances in X-Ray Optics, A.K.Freund, T.Ishikawa, A.M.Khounsary, D.C.Mancini, A.G.Michette, and S.Oestreich, eds., Proc. SPIE 4145, 1-10 (2001).

Ohashi, H.

T. Ishikawa, K. Tamasaku, M. Yabashi, S. Goto, Y. Tanaka, H. Yamazaki, K. Takeshita, H. Kimura, H. Ohashi, T. Matsushita, and T. Ohata, "1-km beamline at SPring-8," in Advances in X-Ray Optics, A.K.Freund, T.Ishikawa, A.M.Khounsary, D.C.Mancini, A.G.Michette, and S.Oestreich, eds., Proc. SPIE 4145, 1-10 (2001).

Ohata, T.

T. Ishikawa, K. Tamasaku, M. Yabashi, S. Goto, Y. Tanaka, H. Yamazaki, K. Takeshita, H. Kimura, H. Ohashi, T. Matsushita, and T. Ohata, "1-km beamline at SPring-8," in Advances in X-Ray Optics, A.K.Freund, T.Ishikawa, A.M.Khounsary, D.C.Mancini, A.G.Michette, and S.Oestreich, eds., Proc. SPIE 4145, 1-10 (2001).

Op de Beeck, M.

M. Op de Beeck, D. Van Dyck, and W. Coene, "Wave function reconstruction in HRTEM: the parabola method," Ultramicroscopy 64, 167-183 (1996).
[CrossRef]

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

Osten, W.

Pedrini, G.

Pogany, A.

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]

Popalambros, P. Y.

P. Y. Popalambros and D. J. Wilde, Principles of Optimal Design: Modeling and Computation, 2nd ed. (Cambridge U. Press, 2000).
[CrossRef]

Schlenker, M.

P. Cloetens, W. Ludwig, D. Van Dyck, J. P. Guigay, M. Schlenker, and J. Baruchel, "Quantitative phase tomography by holographic reconstruction," in Developments in X-Ray Tomography II, U.Bonse ed., Proc. SPIE 3772, 279-290 (1999).

Shimanuki, Y.

K. Kinoshita, T. Matsumura, Y. Inagaki, N. Hirai,M. Sugiyama, H. Kihara, N. Watanabe, and Y. Shimanuki, "Electronic zooming TV readout system for an x-ray microscope," in Soft X-Ray Microscopy, C.J.Jacobsen and J.E.Trebes, eds., Proc. SPIE 1741, 287-293 (1992).

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]

Sugiyama, M.

K. Kinoshita, T. Matsumura, Y. Inagaki, N. Hirai,M. Sugiyama, H. Kihara, N. Watanabe, and Y. Shimanuki, "Electronic zooming TV readout system for an x-ray microscope," in Soft X-Ray Microscopy, C.J.Jacobsen and J.E.Trebes, eds., Proc. SPIE 1741, 287-293 (1992).

Takeshita, K.

T. Ishikawa, K. Tamasaku, M. Yabashi, S. Goto, Y. Tanaka, H. Yamazaki, K. Takeshita, H. Kimura, H. Ohashi, T. Matsushita, and T. Ohata, "1-km beamline at SPring-8," in Advances in X-Ray Optics, A.K.Freund, T.Ishikawa, A.M.Khounsary, D.C.Mancini, A.G.Michette, and S.Oestreich, eds., Proc. SPIE 4145, 1-10 (2001).

Tamasaku, K.

T. Ishikawa, K. Tamasaku, M. Yabashi, S. Goto, Y. Tanaka, H. Yamazaki, K. Takeshita, H. Kimura, H. Ohashi, T. Matsushita, and T. Ohata, "1-km beamline at SPring-8," in Advances in X-Ray Optics, A.K.Freund, T.Ishikawa, A.M.Khounsary, D.C.Mancini, A.G.Michette, and S.Oestreich, eds., Proc. SPIE 4145, 1-10 (2001).

Tanaka, Y.

T. Ishikawa, K. Tamasaku, M. Yabashi, S. Goto, Y. Tanaka, H. Yamazaki, K. Takeshita, H. Kimura, H. Ohashi, T. Matsushita, and T. Ohata, "1-km beamline at SPring-8," in Advances in X-Ray Optics, A.K.Freund, T.Ishikawa, A.M.Khounsary, D.C.Mancini, A.G.Michette, and S.Oestreich, eds., Proc. SPIE 4145, 1-10 (2001).

Tiziani, H. J.

Van Dyck, D.

M. Op de Beeck, D. Van Dyck, and W. Coene, "Wave function reconstruction in HRTEM: the parabola method," Ultramicroscopy 64, 167-183 (1996).
[CrossRef]

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

P. Cloetens, W. Ludwig, D. Van Dyck, J. P. Guigay, M. Schlenker, and J. Baruchel, "Quantitative phase tomography by holographic reconstruction," in Developments in X-Ray Tomography II, U.Bonse ed., Proc. SPIE 3772, 279-290 (1999).

Watanabe, N.

K. Kinoshita, T. Matsumura, Y. Inagaki, N. Hirai,M. Sugiyama, H. Kihara, N. Watanabe, and Y. Shimanuki, "Electronic zooming TV readout system for an x-ray microscope," in Soft X-Ray Microscopy, C.J.Jacobsen and J.E.Trebes, eds., Proc. SPIE 1741, 287-293 (1992).

Wilde, D. J.

P. Y. Popalambros and D. J. Wilde, Principles of Optimal Design: Modeling and Computation, 2nd ed. (Cambridge U. Press, 2000).
[CrossRef]

Wilkins, S. 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]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Chap. 13.

Xiao, T.

T. Xiao, H. Xu, Y. Zhang, J. Chen, and Z. Xu, "Digital image decoding for in-line x-ray holography using two holograms," J. Mod. Opt. 45, 343-353 (1998).
[CrossRef]

Xu, H.

T. Xiao, H. Xu, Y. Zhang, J. Chen, and Z. Xu, "Digital image decoding for in-line x-ray holography using two holograms," J. Mod. Opt. 45, 343-353 (1998).
[CrossRef]

Xu, Z.

T. Xiao, H. Xu, Y. Zhang, J. Chen, and Z. Xu, "Digital image decoding for in-line x-ray holography using two holograms," J. Mod. Opt. 45, 343-353 (1998).
[CrossRef]

Yabashi, M.

T. Ishikawa, K. Tamasaku, M. Yabashi, S. Goto, Y. Tanaka, H. Yamazaki, K. Takeshita, H. Kimura, H. Ohashi, T. Matsushita, and T. Ohata, "1-km beamline at SPring-8," in Advances in X-Ray Optics, A.K.Freund, T.Ishikawa, A.M.Khounsary, D.C.Mancini, A.G.Michette, and S.Oestreich, eds., Proc. SPIE 4145, 1-10 (2001).

Yamazaki, H.

T. Ishikawa, K. Tamasaku, M. Yabashi, S. Goto, Y. Tanaka, H. Yamazaki, K. Takeshita, H. Kimura, H. Ohashi, T. Matsushita, and T. Ohata, "1-km beamline at SPring-8," in Advances in X-Ray Optics, A.K.Freund, T.Ishikawa, A.M.Khounsary, D.C.Mancini, A.G.Michette, and S.Oestreich, eds., Proc. SPIE 4145, 1-10 (2001).

Zhang, X.

Zhang, Y.

J. Mod. Opt. (1)

T. Xiao, H. Xu, Y. Zhang, J. Chen, and Z. Xu, "Digital image decoding for in-line x-ray holography using two holograms," J. Mod. Opt. 45, 343-353 (1998).
[CrossRef]

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]

D. Gabor, "A new microscopic principle," Nature 161, 777-778 (1948).
[CrossRef] [PubMed]

Opt. Eng. (1)

M. H. Maleki and A. J. Devaney, "Noniterative reconstruction of complex-valued objects from two intensity measurements," Opt. Eng. 33, 3243-3253 (1994).
[CrossRef]

Opt. Express (2)

Phys. Rev. Lett. (1)

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

Ultramicroscopy (1)

M. Op de Beeck, D. Van Dyck, and W. Coene, "Wave function reconstruction in HRTEM: the parabola method," Ultramicroscopy 64, 167-183 (1996).
[CrossRef]

Other (9)

B. R. Frieden, "Image enhancement and restoration," inPicture Processing and Digital Filtering, 2nd ed., T.S.Huang, ed. (Springer-Verlag, 1979), Chap. 5.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Chap. 13.

P. Y. Popalambros and D. J. Wilde, Principles of Optimal Design: Modeling and Computation, 2nd ed. (Cambridge U. Press, 2000).
[CrossRef]

M. Howells, "Fundamental limits in x-ray holography," in X-Ray Microscopy II, D.Sayre, M.Howells, J.Kirz, and H.Rarback, eds. (Springer-Verlag, 1988), pp. 263-271.

A. K. Katsaggelos, Digital Image Restoration (Springer-Verlag, 1991).
[CrossRef]

T. Ishikawa, K. Tamasaku, M. Yabashi, S. Goto, Y. Tanaka, H. Yamazaki, K. Takeshita, H. Kimura, H. Ohashi, T. Matsushita, and T. Ohata, "1-km beamline at SPring-8," in Advances in X-Ray Optics, A.K.Freund, T.Ishikawa, A.M.Khounsary, D.C.Mancini, A.G.Michette, and S.Oestreich, eds., Proc. SPIE 4145, 1-10 (2001).

K. Kinoshita, T. Matsumura, Y. Inagaki, N. Hirai,M. Sugiyama, H. Kihara, N. Watanabe, and Y. Shimanuki, "Electronic zooming TV readout system for an x-ray microscope," in Soft X-Ray Microscopy, C.J.Jacobsen and J.E.Trebes, eds., Proc. SPIE 1741, 287-293 (1992).

P. Cloetens, W. Ludwig, D. Van Dyck, J. P. Guigay, M. Schlenker, and J. Baruchel, "Quantitative phase tomography by holographic reconstruction," in Developments in X-Ray Tomography II, U.Bonse ed., Proc. SPIE 3772, 279-290 (1999).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

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Figures (6)

Fig. 1
Fig. 1

Experimental setup to obtain test holograms. Undulator is a synchrotron x-radiation source. DCM is a double-crystal monochromator with Si 111 reflection. The slit was opened to the square with a side length 0.1 mm .

Fig. 2
Fig. 2

Optical microscopy images of the specimen for test holograms: (a) whole image and (b) detailed image of the square area in (a). The two black pigment cells are indistinguishable from the optical image.

Fig. 3
Fig. 3

One experimental interference term.

Fig. 4
Fig. 4

Optimized functions (a) a ( x , y ) and (b) b ( x , y ) .

Fig. 5
Fig. 5

Composite image of the electron density of the pigment cells. The imaginary-part image (red gradation) is painted over the real-part image (gray scale).

Fig. 6
Fig. 6

Profiles of the electron density along the line A B in Fig. 4a. The solid and dotted curves were obtained with and without the nonnegative constraint for a ( x , y ) , respectively.

Equations (19)

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

u 1 ( x , y , L ) = r e ρ ( ξ , η , ζ ) exp ( 2 π i ζ λ ) exp ( 2 π i r λ ) r d ξ d η d ζ .
r = L ζ + ( x ξ ) 2 + ( y η ) 2 2 L + ζ [ ( x ξ ) 2 + ( y n ) 2 ] 2 L 2 + .
Max { ζ [ ( x ξ ) 2 + ( y η ) 2 ] } 2 L 2 λ .
u 1 ( x , y , L ) = r e L exp ( 2 π i L λ ) ρ z ( x , y ) exp [ i π λ L ( x 2 + y 2 ) ] ,
ρ z ( x , y ) = ρ ( x , y , z ) d z = a ( x , y ) i b ( x , y ) .
J ̂ ( x , y , L ) = u 0 ( x , y , L ) + u 1 ( x , y , L ) 2 u 0 ( x , y , L ) 2 ,
J ̂ ( x , y , L ) = a ( x , y ) C ( x , y , L ) + b ( x , y ) S ( x , y , L ) ,
C ( x , y , L ) = 2 r e L cos [ π λ L ( x 2 + y 2 ) ] ,
S ( x , y , L ) = 2 r e L sin [ π λ L ( x 2 + y 2 ) ] .
J ( x , y , L ) = I ( x , y , L ) I 0 ( x , y , L ) I 0 ( x , y , L ) .
e = n [ J ( x , y , L n ) J ̂ ( x , y , L n ) ] 2 d x d y ,
δ e ( k ) δ a ( k ) ( x , y ) = 2 [ J ( x , y , L n ) J ̂ ( k ) ( x , y , L n ) ] C ( x , y , L n ) ,
δ e ( k ) δ b ( k ) ( x , y ) = 2 [ J ( x , y , L n ) J ̂ ( k ) ( x , y , L n ) ] S ( x , y , L n ) .
a ( k + 1 ) ( x , y ) = a ( k ) ( x , y ) μ δ e ( k ) δ a ( k ) ( x , y ) ,
b ( k + 1 ) ( x , y ) = b ( k ) ( x , y ) μ δ e ( k ) δ b ( k ) ( x , y ) ,
μ = [ J ( x , y , L n ) J ̂ ( k ) ( x , y , L n ) ] f ( x , y , L n ) d x d y f 2 ( x , y , L n ) d x d y ,
f ( x , y , L n ) = δ e ( k ) δ a ( k ) ( x , y ) C ( x , y , L n ) + δ e ( k ) δ b ( k ) ( x , y ) * S ( x , y , L n ) .
ζ 8 Δ 2 λ .
Max { x 2 + y 2 } > λ L 2 Δ .

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