Abstract

For objects larger than the depth of focus of an imaging system, one must account for wavefield propagation effects within the object as is done in diffraction tomography, diffraction microscopy, and multislice ptychographic tomography. We show here that if the imaging method used reconstructs Na planes along each viewing direction, one can reduce the number of illumination directions required to fill Fourier space by a factor of 1/Na, relaxing the usual Crowther criterion for tomography. This provides a conceptual basis to explain two recent experiments where multiple axial planes were imaged per viewing direction, and tomographic images were obtained with good 3D spatial resolution, even though fewer illumination directions were used than one would have expected from the Crowther criterion.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Three-dimensional iterative multislice reconstruction for ptychographic X-ray computed tomography

Kei Shimomura, Makoto Hirose, Takaya Higashino, and Yukio Takahashi
Opt. Express 26(24) 31199-31208 (2018)

3D x-ray imaging of continuous objects beyond the depth of focus limit

M. A. Gilles, Y. S. G. Nashed, M. Du, C. Jacobsen, and S. M. Wild
Optica 5(9) 1078-1086 (2018)

References

  • View by:
  • |
  • |
  • |

  1. R. A. Crowther, D. J. DeRosier, and A. Klug, Proc. R. Soc. London A 317, 319 (1970).
    [Crossref]
  2. A. M. Maiden, M. J. Humphry, and J. M. Rodenburg, J. Opt. Soc. Am. A 29, 1606 (2012).
    [Crossref]
  3. P. Li and A. M. Maiden, Sci. Rep. 8, 2049 (2018).
    [Crossref]
  4. R. Hovden, P. Ercius, Y. Jiang, D. Wang, Y. Yu, H. D. Abruña, V. Elser, and D. A. Muller, Ultramicroscopy 140, 26 (2014).
    [Crossref]
  5. F. Charrière, A. Marian, F. Montfort, J. Kuehn, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, Opt. Lett. 31, 178 (2006).
    [Crossref]
  6. G. Popescu, Quantitative Phase Imaging of Cells and Tissues (McGraw-Hill, 2011).
  7. U. S. Kamilov, I. N. Papadopoulos, M. H. Shoreh, A. Goy, C. Vonesch, M. Unser, and D. Psaltis, Optica 2, 517 (2015).
    [Crossref]
  8. H. Nyquist, Trans. Am. Inst. Electr. Eng. 47, 617 (1928).
    [Crossref]
  9. C. E. Shannon, Proc. IRE 37, 10 (1949).
    [Crossref]
  10. P. Rattey and A. Lindgren, IEEE Trans. Acoust. Speech Signal Process. 29, 994 (1981).
    [Crossref]
  11. A. G. Lindgren and P. A. Rattey, Advances in Electronics and Electron Physics, C. Marton, ed. (Academic, 1981), Vol. 56, pp. 359–410.
  12. A. G. Lindgren and P. A. Rattey, Signal Process. 7, 371 (1984).
    [Crossref]
  13. W. N. Brouw, Radio Astronomy, B. Alder, S. Fernbach, and M. Rotenberg, eds., Methods in Computational Physics: Advances in Research and Applications (Academic, 1975), Vol. 14, pp. 131–175.
  14. J. D. O’Sullivan, IEEE Trans. Med. Imaging 4, 200 (1985).
    [Crossref]
  15. B. A. Dowd, G. H. Campbell, R. B. Marr, V. Nagarkar, S. Tipnis, L. Axe, and D. P. Siddons, Proc. SPIE 3772, 224 (1999).
    [Crossref]
  16. R. Gordon, R. Bender, and G. T. Herman, J. Theor. Biol. 29, 471 (1970).
    [Crossref]
  17. W. Hoppe, Acta Crystallogr. A 25, 495 (1969).
    [Crossref]
  18. R. Hegerl and W. Hoppe, Ber. Bunsenges. Phys. Chem. 74, 1148 (1970).
    [Crossref]
  19. R. W. Gerchberg and W. O. Saxton, Optik 35, 237 (1972).
  20. J. R. Fienup, Opt. Lett. 3, 27 (1978).
    [Crossref]
  21. H. M. L. Faulkner and J. Rodenburg, Phys. Rev. Lett. 93, 023903 (2004).
    [Crossref]
  22. J. Rodenburg, A. Hurst, A. Cullis, B. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
    [Crossref]
  23. M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, Nature 467, 436 (2010).
    [Crossref]
  24. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  25. J. Deng, D. J. Vine, S. Chen, Y. S. G. Nashed, T. Peterka, R. Ross, S. Vogt, and C. Jacobsen, Proc. SPIE 9592, 95920U (2015).
    [Crossref]
  26. T. M. Godden, R. Suman, M. J. Humphry, J. M. Rodenburg, and A. M. Maiden, Opt. Express 22, 12513 (2014).
    [Crossref]
  27. A. Suzuki, S. Furutaku, K. Shimomura, K. Yamauchi, Y. Kohmura, T. Ishikawa, and Y. Takahashi, Phys. Rev. Lett. 112, 053903 (2014).
    [Crossref]
  28. E. H. R. Tsai, I. Usov, A. Diaz, A. Menzel, and M. Guizar-Sicairos, Opt. Express 24, 29089 (2016).
    [Crossref]
  29. S. Gao, P. Wang, F. Zhang, G. T. Martinez, P. D. Nellist, X. Pan, and A. I. Kirkland, Nat. Commun. 8, 1 (2017).
    [Crossref]
  30. J. M. Cowley and A. F. Moodie, Proc. Phys. Soc. London B 70, 486 (1957).
    [Crossref]
  31. J. M. Cowley and A. F. Moodie, Acta Crystallogr. 10, 609 (1957).
    [Crossref]
  32. S. A. Sugimoto and Y. Ichioka, Appl. Opt. 24, 2076 (1985).
    [Crossref]
  33. Y. Y. Schechner, N. Kiryati, and R. Basri, Int. J. Comput. Vis. 39, 25 (2000).
    [Crossref]
  34. A. G. Valdecasas, D. Marshall, J. M. Becerra, and J. J. Terrero, Micron 32, 559 (2001).
    [Crossref]
  35. F. Piccinini, A. Tesei, W. Zoli, and A. Bevilacqua, Microsc. Res. Tech. 75, 1582 (2012).
    [Crossref]
  36. T. Ficker, Optik 144, 289 (2017).
    [Crossref]

2018 (1)

P. Li and A. M. Maiden, Sci. Rep. 8, 2049 (2018).
[Crossref]

2017 (2)

S. Gao, P. Wang, F. Zhang, G. T. Martinez, P. D. Nellist, X. Pan, and A. I. Kirkland, Nat. Commun. 8, 1 (2017).
[Crossref]

T. Ficker, Optik 144, 289 (2017).
[Crossref]

2016 (1)

2015 (2)

J. Deng, D. J. Vine, S. Chen, Y. S. G. Nashed, T. Peterka, R. Ross, S. Vogt, and C. Jacobsen, Proc. SPIE 9592, 95920U (2015).
[Crossref]

U. S. Kamilov, I. N. Papadopoulos, M. H. Shoreh, A. Goy, C. Vonesch, M. Unser, and D. Psaltis, Optica 2, 517 (2015).
[Crossref]

2014 (3)

R. Hovden, P. Ercius, Y. Jiang, D. Wang, Y. Yu, H. D. Abruña, V. Elser, and D. A. Muller, Ultramicroscopy 140, 26 (2014).
[Crossref]

T. M. Godden, R. Suman, M. J. Humphry, J. M. Rodenburg, and A. M. Maiden, Opt. Express 22, 12513 (2014).
[Crossref]

A. Suzuki, S. Furutaku, K. Shimomura, K. Yamauchi, Y. Kohmura, T. Ishikawa, and Y. Takahashi, Phys. Rev. Lett. 112, 053903 (2014).
[Crossref]

2012 (2)

F. Piccinini, A. Tesei, W. Zoli, and A. Bevilacqua, Microsc. Res. Tech. 75, 1582 (2012).
[Crossref]

A. M. Maiden, M. J. Humphry, and J. M. Rodenburg, J. Opt. Soc. Am. A 29, 1606 (2012).
[Crossref]

2010 (1)

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, Nature 467, 436 (2010).
[Crossref]

2007 (1)

J. Rodenburg, A. Hurst, A. Cullis, B. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[Crossref]

2006 (1)

2004 (1)

H. M. L. Faulkner and J. Rodenburg, Phys. Rev. Lett. 93, 023903 (2004).
[Crossref]

2001 (1)

A. G. Valdecasas, D. Marshall, J. M. Becerra, and J. J. Terrero, Micron 32, 559 (2001).
[Crossref]

2000 (1)

Y. Y. Schechner, N. Kiryati, and R. Basri, Int. J. Comput. Vis. 39, 25 (2000).
[Crossref]

1999 (1)

B. A. Dowd, G. H. Campbell, R. B. Marr, V. Nagarkar, S. Tipnis, L. Axe, and D. P. Siddons, Proc. SPIE 3772, 224 (1999).
[Crossref]

1985 (2)

J. D. O’Sullivan, IEEE Trans. Med. Imaging 4, 200 (1985).
[Crossref]

S. A. Sugimoto and Y. Ichioka, Appl. Opt. 24, 2076 (1985).
[Crossref]

1984 (1)

A. G. Lindgren and P. A. Rattey, Signal Process. 7, 371 (1984).
[Crossref]

1981 (1)

P. Rattey and A. Lindgren, IEEE Trans. Acoust. Speech Signal Process. 29, 994 (1981).
[Crossref]

1978 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, Optik 35, 237 (1972).

1970 (3)

R. Hegerl and W. Hoppe, Ber. Bunsenges. Phys. Chem. 74, 1148 (1970).
[Crossref]

R. Gordon, R. Bender, and G. T. Herman, J. Theor. Biol. 29, 471 (1970).
[Crossref]

R. A. Crowther, D. J. DeRosier, and A. Klug, Proc. R. Soc. London A 317, 319 (1970).
[Crossref]

1969 (1)

W. Hoppe, Acta Crystallogr. A 25, 495 (1969).
[Crossref]

1957 (2)

J. M. Cowley and A. F. Moodie, Proc. Phys. Soc. London B 70, 486 (1957).
[Crossref]

J. M. Cowley and A. F. Moodie, Acta Crystallogr. 10, 609 (1957).
[Crossref]

1949 (1)

C. E. Shannon, Proc. IRE 37, 10 (1949).
[Crossref]

1928 (1)

H. Nyquist, Trans. Am. Inst. Electr. Eng. 47, 617 (1928).
[Crossref]

Abruña, H. D.

R. Hovden, P. Ercius, Y. Jiang, D. Wang, Y. Yu, H. D. Abruña, V. Elser, and D. A. Muller, Ultramicroscopy 140, 26 (2014).
[Crossref]

Axe, L.

B. A. Dowd, G. H. Campbell, R. B. Marr, V. Nagarkar, S. Tipnis, L. Axe, and D. P. Siddons, Proc. SPIE 3772, 224 (1999).
[Crossref]

Basri, R.

Y. Y. Schechner, N. Kiryati, and R. Basri, Int. J. Comput. Vis. 39, 25 (2000).
[Crossref]

Becerra, J. M.

A. G. Valdecasas, D. Marshall, J. M. Becerra, and J. J. Terrero, Micron 32, 559 (2001).
[Crossref]

Bender, R.

R. Gordon, R. Bender, and G. T. Herman, J. Theor. Biol. 29, 471 (1970).
[Crossref]

Bevilacqua, A.

F. Piccinini, A. Tesei, W. Zoli, and A. Bevilacqua, Microsc. Res. Tech. 75, 1582 (2012).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Brouw, W. N.

W. N. Brouw, Radio Astronomy, B. Alder, S. Fernbach, and M. Rotenberg, eds., Methods in Computational Physics: Advances in Research and Applications (Academic, 1975), Vol. 14, pp. 131–175.

Bunk, O.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, Nature 467, 436 (2010).
[Crossref]

J. Rodenburg, A. Hurst, A. Cullis, B. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[Crossref]

Campbell, G. H.

B. A. Dowd, G. H. Campbell, R. B. Marr, V. Nagarkar, S. Tipnis, L. Axe, and D. P. Siddons, Proc. SPIE 3772, 224 (1999).
[Crossref]

Charrière, F.

Chen, S.

J. Deng, D. J. Vine, S. Chen, Y. S. G. Nashed, T. Peterka, R. Ross, S. Vogt, and C. Jacobsen, Proc. SPIE 9592, 95920U (2015).
[Crossref]

Colomb, T.

Cowley, J. M.

J. M. Cowley and A. F. Moodie, Proc. Phys. Soc. London B 70, 486 (1957).
[Crossref]

J. M. Cowley and A. F. Moodie, Acta Crystallogr. 10, 609 (1957).
[Crossref]

Crowther, R. A.

R. A. Crowther, D. J. DeRosier, and A. Klug, Proc. R. Soc. London A 317, 319 (1970).
[Crossref]

Cuche, E.

Cullis, A.

J. Rodenburg, A. Hurst, A. Cullis, B. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[Crossref]

David, C.

J. Rodenburg, A. Hurst, A. Cullis, B. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[Crossref]

Deng, J.

J. Deng, D. J. Vine, S. Chen, Y. S. G. Nashed, T. Peterka, R. Ross, S. Vogt, and C. Jacobsen, Proc. SPIE 9592, 95920U (2015).
[Crossref]

Depeursinge, C.

DeRosier, D. J.

R. A. Crowther, D. J. DeRosier, and A. Klug, Proc. R. Soc. London A 317, 319 (1970).
[Crossref]

Diaz, A.

Dierolf, M.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, Nature 467, 436 (2010).
[Crossref]

Dobson, B.

J. Rodenburg, A. Hurst, A. Cullis, B. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[Crossref]

Dowd, B. A.

B. A. Dowd, G. H. Campbell, R. B. Marr, V. Nagarkar, S. Tipnis, L. Axe, and D. P. Siddons, Proc. SPIE 3772, 224 (1999).
[Crossref]

Elser, V.

R. Hovden, P. Ercius, Y. Jiang, D. Wang, Y. Yu, H. D. Abruña, V. Elser, and D. A. Muller, Ultramicroscopy 140, 26 (2014).
[Crossref]

Ercius, P.

R. Hovden, P. Ercius, Y. Jiang, D. Wang, Y. Yu, H. D. Abruña, V. Elser, and D. A. Muller, Ultramicroscopy 140, 26 (2014).
[Crossref]

Faulkner, H. M. L.

H. M. L. Faulkner and J. Rodenburg, Phys. Rev. Lett. 93, 023903 (2004).
[Crossref]

Ficker, T.

T. Ficker, Optik 144, 289 (2017).
[Crossref]

Fienup, J. R.

Furutaku, S.

A. Suzuki, S. Furutaku, K. Shimomura, K. Yamauchi, Y. Kohmura, T. Ishikawa, and Y. Takahashi, Phys. Rev. Lett. 112, 053903 (2014).
[Crossref]

Gao, S.

S. Gao, P. Wang, F. Zhang, G. T. Martinez, P. D. Nellist, X. Pan, and A. I. Kirkland, Nat. Commun. 8, 1 (2017).
[Crossref]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, Optik 35, 237 (1972).

Godden, T. M.

Gordon, R.

R. Gordon, R. Bender, and G. T. Herman, J. Theor. Biol. 29, 471 (1970).
[Crossref]

Goy, A.

Guizar-Sicairos, M.

Hegerl, R.

R. Hegerl and W. Hoppe, Ber. Bunsenges. Phys. Chem. 74, 1148 (1970).
[Crossref]

Herman, G. T.

R. Gordon, R. Bender, and G. T. Herman, J. Theor. Biol. 29, 471 (1970).
[Crossref]

Hoppe, W.

R. Hegerl and W. Hoppe, Ber. Bunsenges. Phys. Chem. 74, 1148 (1970).
[Crossref]

W. Hoppe, Acta Crystallogr. A 25, 495 (1969).
[Crossref]

Hovden, R.

R. Hovden, P. Ercius, Y. Jiang, D. Wang, Y. Yu, H. D. Abruña, V. Elser, and D. A. Muller, Ultramicroscopy 140, 26 (2014).
[Crossref]

Humphry, M. J.

Hurst, A.

J. Rodenburg, A. Hurst, A. Cullis, B. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[Crossref]

Ichioka, Y.

Ishikawa, T.

A. Suzuki, S. Furutaku, K. Shimomura, K. Yamauchi, Y. Kohmura, T. Ishikawa, and Y. Takahashi, Phys. Rev. Lett. 112, 053903 (2014).
[Crossref]

Jacobsen, C.

J. Deng, D. J. Vine, S. Chen, Y. S. G. Nashed, T. Peterka, R. Ross, S. Vogt, and C. Jacobsen, Proc. SPIE 9592, 95920U (2015).
[Crossref]

Jefimovs, K.

J. Rodenburg, A. Hurst, A. Cullis, B. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[Crossref]

Jiang, Y.

R. Hovden, P. Ercius, Y. Jiang, D. Wang, Y. Yu, H. D. Abruña, V. Elser, and D. A. Muller, Ultramicroscopy 140, 26 (2014).
[Crossref]

Johnson, I.

J. Rodenburg, A. Hurst, A. Cullis, B. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[Crossref]

Kamilov, U. S.

Kewish, C. M.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, Nature 467, 436 (2010).
[Crossref]

Kirkland, A. I.

S. Gao, P. Wang, F. Zhang, G. T. Martinez, P. D. Nellist, X. Pan, and A. I. Kirkland, Nat. Commun. 8, 1 (2017).
[Crossref]

Kiryati, N.

Y. Y. Schechner, N. Kiryati, and R. Basri, Int. J. Comput. Vis. 39, 25 (2000).
[Crossref]

Klug, A.

R. A. Crowther, D. J. DeRosier, and A. Klug, Proc. R. Soc. London A 317, 319 (1970).
[Crossref]

Kohmura, Y.

A. Suzuki, S. Furutaku, K. Shimomura, K. Yamauchi, Y. Kohmura, T. Ishikawa, and Y. Takahashi, Phys. Rev. Lett. 112, 053903 (2014).
[Crossref]

Kuehn, J.

Li, P.

P. Li and A. M. Maiden, Sci. Rep. 8, 2049 (2018).
[Crossref]

Lindgren, A.

P. Rattey and A. Lindgren, IEEE Trans. Acoust. Speech Signal Process. 29, 994 (1981).
[Crossref]

Lindgren, A. G.

A. G. Lindgren and P. A. Rattey, Signal Process. 7, 371 (1984).
[Crossref]

A. G. Lindgren and P. A. Rattey, Advances in Electronics and Electron Physics, C. Marton, ed. (Academic, 1981), Vol. 56, pp. 359–410.

Maiden, A. M.

Marian, A.

Marquet, P.

Marr, R. B.

B. A. Dowd, G. H. Campbell, R. B. Marr, V. Nagarkar, S. Tipnis, L. Axe, and D. P. Siddons, Proc. SPIE 3772, 224 (1999).
[Crossref]

Marshall, D.

A. G. Valdecasas, D. Marshall, J. M. Becerra, and J. J. Terrero, Micron 32, 559 (2001).
[Crossref]

Martinez, G. T.

S. Gao, P. Wang, F. Zhang, G. T. Martinez, P. D. Nellist, X. Pan, and A. I. Kirkland, Nat. Commun. 8, 1 (2017).
[Crossref]

Menzel, A.

E. H. R. Tsai, I. Usov, A. Diaz, A. Menzel, and M. Guizar-Sicairos, Opt. Express 24, 29089 (2016).
[Crossref]

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, Nature 467, 436 (2010).
[Crossref]

Montfort, F.

Moodie, A. F.

J. M. Cowley and A. F. Moodie, Proc. Phys. Soc. London B 70, 486 (1957).
[Crossref]

J. M. Cowley and A. F. Moodie, Acta Crystallogr. 10, 609 (1957).
[Crossref]

Muller, D. A.

R. Hovden, P. Ercius, Y. Jiang, D. Wang, Y. Yu, H. D. Abruña, V. Elser, and D. A. Muller, Ultramicroscopy 140, 26 (2014).
[Crossref]

Nagarkar, V.

B. A. Dowd, G. H. Campbell, R. B. Marr, V. Nagarkar, S. Tipnis, L. Axe, and D. P. Siddons, Proc. SPIE 3772, 224 (1999).
[Crossref]

Nashed, Y. S. G.

J. Deng, D. J. Vine, S. Chen, Y. S. G. Nashed, T. Peterka, R. Ross, S. Vogt, and C. Jacobsen, Proc. SPIE 9592, 95920U (2015).
[Crossref]

Nellist, P. D.

S. Gao, P. Wang, F. Zhang, G. T. Martinez, P. D. Nellist, X. Pan, and A. I. Kirkland, Nat. Commun. 8, 1 (2017).
[Crossref]

Nyquist, H.

H. Nyquist, Trans. Am. Inst. Electr. Eng. 47, 617 (1928).
[Crossref]

O’Sullivan, J. D.

J. D. O’Sullivan, IEEE Trans. Med. Imaging 4, 200 (1985).
[Crossref]

Pan, X.

S. Gao, P. Wang, F. Zhang, G. T. Martinez, P. D. Nellist, X. Pan, and A. I. Kirkland, Nat. Commun. 8, 1 (2017).
[Crossref]

Papadopoulos, I. N.

Peterka, T.

J. Deng, D. J. Vine, S. Chen, Y. S. G. Nashed, T. Peterka, R. Ross, S. Vogt, and C. Jacobsen, Proc. SPIE 9592, 95920U (2015).
[Crossref]

Pfeiffer, F.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, Nature 467, 436 (2010).
[Crossref]

J. Rodenburg, A. Hurst, A. Cullis, B. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[Crossref]

Piccinini, F.

F. Piccinini, A. Tesei, W. Zoli, and A. Bevilacqua, Microsc. Res. Tech. 75, 1582 (2012).
[Crossref]

Popescu, G.

G. Popescu, Quantitative Phase Imaging of Cells and Tissues (McGraw-Hill, 2011).

Psaltis, D.

Rattey, P.

P. Rattey and A. Lindgren, IEEE Trans. Acoust. Speech Signal Process. 29, 994 (1981).
[Crossref]

Rattey, P. A.

A. G. Lindgren and P. A. Rattey, Signal Process. 7, 371 (1984).
[Crossref]

A. G. Lindgren and P. A. Rattey, Advances in Electronics and Electron Physics, C. Marton, ed. (Academic, 1981), Vol. 56, pp. 359–410.

Rodenburg, J.

J. Rodenburg, A. Hurst, A. Cullis, B. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[Crossref]

H. M. L. Faulkner and J. Rodenburg, Phys. Rev. Lett. 93, 023903 (2004).
[Crossref]

Rodenburg, J. M.

Ross, R.

J. Deng, D. J. Vine, S. Chen, Y. S. G. Nashed, T. Peterka, R. Ross, S. Vogt, and C. Jacobsen, Proc. SPIE 9592, 95920U (2015).
[Crossref]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, Optik 35, 237 (1972).

Schechner, Y. Y.

Y. Y. Schechner, N. Kiryati, and R. Basri, Int. J. Comput. Vis. 39, 25 (2000).
[Crossref]

Schneider, P.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, Nature 467, 436 (2010).
[Crossref]

Shannon, C. E.

C. E. Shannon, Proc. IRE 37, 10 (1949).
[Crossref]

Shimomura, K.

A. Suzuki, S. Furutaku, K. Shimomura, K. Yamauchi, Y. Kohmura, T. Ishikawa, and Y. Takahashi, Phys. Rev. Lett. 112, 053903 (2014).
[Crossref]

Shoreh, M. H.

Siddons, D. P.

B. A. Dowd, G. H. Campbell, R. B. Marr, V. Nagarkar, S. Tipnis, L. Axe, and D. P. Siddons, Proc. SPIE 3772, 224 (1999).
[Crossref]

Sugimoto, S. A.

Suman, R.

Suzuki, A.

A. Suzuki, S. Furutaku, K. Shimomura, K. Yamauchi, Y. Kohmura, T. Ishikawa, and Y. Takahashi, Phys. Rev. Lett. 112, 053903 (2014).
[Crossref]

Takahashi, Y.

A. Suzuki, S. Furutaku, K. Shimomura, K. Yamauchi, Y. Kohmura, T. Ishikawa, and Y. Takahashi, Phys. Rev. Lett. 112, 053903 (2014).
[Crossref]

Terrero, J. J.

A. G. Valdecasas, D. Marshall, J. M. Becerra, and J. J. Terrero, Micron 32, 559 (2001).
[Crossref]

Tesei, A.

F. Piccinini, A. Tesei, W. Zoli, and A. Bevilacqua, Microsc. Res. Tech. 75, 1582 (2012).
[Crossref]

Thibault, P.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, Nature 467, 436 (2010).
[Crossref]

Tipnis, S.

B. A. Dowd, G. H. Campbell, R. B. Marr, V. Nagarkar, S. Tipnis, L. Axe, and D. P. Siddons, Proc. SPIE 3772, 224 (1999).
[Crossref]

Tsai, E. H. R.

Unser, M.

Usov, I.

Valdecasas, A. G.

A. G. Valdecasas, D. Marshall, J. M. Becerra, and J. J. Terrero, Micron 32, 559 (2001).
[Crossref]

Vine, D. J.

J. Deng, D. J. Vine, S. Chen, Y. S. G. Nashed, T. Peterka, R. Ross, S. Vogt, and C. Jacobsen, Proc. SPIE 9592, 95920U (2015).
[Crossref]

Vogt, S.

J. Deng, D. J. Vine, S. Chen, Y. S. G. Nashed, T. Peterka, R. Ross, S. Vogt, and C. Jacobsen, Proc. SPIE 9592, 95920U (2015).
[Crossref]

Vonesch, C.

Wang, D.

R. Hovden, P. Ercius, Y. Jiang, D. Wang, Y. Yu, H. D. Abruña, V. Elser, and D. A. Muller, Ultramicroscopy 140, 26 (2014).
[Crossref]

Wang, P.

S. Gao, P. Wang, F. Zhang, G. T. Martinez, P. D. Nellist, X. Pan, and A. I. Kirkland, Nat. Commun. 8, 1 (2017).
[Crossref]

Wepf, R.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, Nature 467, 436 (2010).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Yamauchi, K.

A. Suzuki, S. Furutaku, K. Shimomura, K. Yamauchi, Y. Kohmura, T. Ishikawa, and Y. Takahashi, Phys. Rev. Lett. 112, 053903 (2014).
[Crossref]

Yu, Y.

R. Hovden, P. Ercius, Y. Jiang, D. Wang, Y. Yu, H. D. Abruña, V. Elser, and D. A. Muller, Ultramicroscopy 140, 26 (2014).
[Crossref]

Zhang, F.

S. Gao, P. Wang, F. Zhang, G. T. Martinez, P. D. Nellist, X. Pan, and A. I. Kirkland, Nat. Commun. 8, 1 (2017).
[Crossref]

Zoli, W.

F. Piccinini, A. Tesei, W. Zoli, and A. Bevilacqua, Microsc. Res. Tech. 75, 1582 (2012).
[Crossref]

Acta Crystallogr. (1)

J. M. Cowley and A. F. Moodie, Acta Crystallogr. 10, 609 (1957).
[Crossref]

Acta Crystallogr. A (1)

W. Hoppe, Acta Crystallogr. A 25, 495 (1969).
[Crossref]

Appl. Opt. (1)

Ber. Bunsenges. Phys. Chem. (1)

R. Hegerl and W. Hoppe, Ber. Bunsenges. Phys. Chem. 74, 1148 (1970).
[Crossref]

IEEE Trans. Acoust. Speech Signal Process. (1)

P. Rattey and A. Lindgren, IEEE Trans. Acoust. Speech Signal Process. 29, 994 (1981).
[Crossref]

IEEE Trans. Med. Imaging (1)

J. D. O’Sullivan, IEEE Trans. Med. Imaging 4, 200 (1985).
[Crossref]

Int. J. Comput. Vis. (1)

Y. Y. Schechner, N. Kiryati, and R. Basri, Int. J. Comput. Vis. 39, 25 (2000).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Theor. Biol. (1)

R. Gordon, R. Bender, and G. T. Herman, J. Theor. Biol. 29, 471 (1970).
[Crossref]

Micron (1)

A. G. Valdecasas, D. Marshall, J. M. Becerra, and J. J. Terrero, Micron 32, 559 (2001).
[Crossref]

Microsc. Res. Tech. (1)

F. Piccinini, A. Tesei, W. Zoli, and A. Bevilacqua, Microsc. Res. Tech. 75, 1582 (2012).
[Crossref]

Nat. Commun. (1)

S. Gao, P. Wang, F. Zhang, G. T. Martinez, P. D. Nellist, X. Pan, and A. I. Kirkland, Nat. Commun. 8, 1 (2017).
[Crossref]

Nature (1)

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, Nature 467, 436 (2010).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Optica (1)

Optik (2)

T. Ficker, Optik 144, 289 (2017).
[Crossref]

R. W. Gerchberg and W. O. Saxton, Optik 35, 237 (1972).

Phys. Rev. Lett. (3)

H. M. L. Faulkner and J. Rodenburg, Phys. Rev. Lett. 93, 023903 (2004).
[Crossref]

J. Rodenburg, A. Hurst, A. Cullis, B. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[Crossref]

A. Suzuki, S. Furutaku, K. Shimomura, K. Yamauchi, Y. Kohmura, T. Ishikawa, and Y. Takahashi, Phys. Rev. Lett. 112, 053903 (2014).
[Crossref]

Proc. IRE (1)

C. E. Shannon, Proc. IRE 37, 10 (1949).
[Crossref]

Proc. Phys. Soc. London B (1)

J. M. Cowley and A. F. Moodie, Proc. Phys. Soc. London B 70, 486 (1957).
[Crossref]

Proc. R. Soc. London A (1)

R. A. Crowther, D. J. DeRosier, and A. Klug, Proc. R. Soc. London A 317, 319 (1970).
[Crossref]

Proc. SPIE (2)

B. A. Dowd, G. H. Campbell, R. B. Marr, V. Nagarkar, S. Tipnis, L. Axe, and D. P. Siddons, Proc. SPIE 3772, 224 (1999).
[Crossref]

J. Deng, D. J. Vine, S. Chen, Y. S. G. Nashed, T. Peterka, R. Ross, S. Vogt, and C. Jacobsen, Proc. SPIE 9592, 95920U (2015).
[Crossref]

Sci. Rep. (1)

P. Li and A. M. Maiden, Sci. Rep. 8, 2049 (2018).
[Crossref]

Signal Process. (1)

A. G. Lindgren and P. A. Rattey, Signal Process. 7, 371 (1984).
[Crossref]

Trans. Am. Inst. Electr. Eng. (1)

H. Nyquist, Trans. Am. Inst. Electr. Eng. 47, 617 (1928).
[Crossref]

Ultramicroscopy (1)

R. Hovden, P. Ercius, Y. Jiang, D. Wang, Y. Yu, H. D. Abruña, V. Elser, and D. A. Muller, Ultramicroscopy 140, 26 (2014).
[Crossref]

Other (4)

G. Popescu, Quantitative Phase Imaging of Cells and Tissues (McGraw-Hill, 2011).

A. G. Lindgren and P. A. Rattey, Advances in Electronics and Electron Physics, C. Marton, ed. (Academic, 1981), Vol. 56, pp. 359–410.

W. N. Brouw, Radio Astronomy, B. Alder, S. Fernbach, and M. Rotenberg, eds., Methods in Computational Physics: Advances in Research and Applications (Academic, 1975), Vol. 14, pp. 131–175.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

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. Schematic representation of the Crowther criterion in conventional tomography. In conventional tomography, each slice y of the object along the direction y ^ of the rotation axis is mapped onto one row of a detector. One then obtains one-dimensional pure projection images of the object with N t transverse pixels of width Δ t in the transverse direction A, and a depth of precisely one pixel at zero spatial frequency in the axial direction (because there is no way to distinguish between different axial positions in a pure projection). For an angle θ = 0 ° , the Fourier transform of this image yields an array with N x = N t pixels in the transverse or u x direction and N z = 1 pixels in the axial or u z direction in transverse-axial Fourier space B. As the object is rotated, so is the information obtained in Fourier space, so the ( u x , u z ) Fourier space is filled, as shown in C. The Crowther criterion of Eq. (3) is effectively a statement that one must provide complete, gap-free coverage of all pixels around the periphery in transverse-axial Fourier space.
Fig. 2.
Fig. 2. Multislice ptychography [2] uses Fresnel propagation of the illumination probe to determine which of many axial slices an object is localized to. While propagating a single probe function between many axial planes (as well carrying out the standard ptychographic action of translating the probe to multiple transverse positions), at each axial plane the net wavefield is decomposed into contributions from the probe and from the optical modulation provided by any specimen features at that plane. This is accomplished through the fact that the probe is modified as it propagates plane-to-plane, so that prior experimental demonstrations [2,2629] have clearly shown the ability to separate even broad, low-spatial-frequency objects into their respective planes along the illumination direction. Multislice ptychography (as well as through-focus imaging methods that succeed to various degrees in separating objects into their respective planes along the illumination direction) therefore delivers axial slice images which can have objects of all transverse sizes (and therefore content at all transverse spatial frequencies) reconstructed at separate axial planes.
Fig. 3.
Fig. 3. Schematic representation of the Crowther criterion in multislice ptychographic tomography. Because multislice ptychography obtains separate reconstructed images for each axial slice (A; see also Fig. 2) and because these objects at each plane can be larger or small and, thus, contain information at all transverse spatial frequencies, the information that a multislice ptychographic image delivers in ( u x , u z ) or transverse-axial Fourier space is as shown in B. If one then carries out multislice ptychography from a variety of illumination angles, one fills in Fourier space, as shown in C. The equivalent of the Crowther criterion (Fig. 1C) for multislice ptychographic tomography is then determined by setting the angular separation Δ θ between illumination directions to a value so that one provies complete, gap-free coverage of all pixels around the periphery in transverse-axial Fourier space, leading to Eq. (6).

Equations (6)

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

N θ π D / d ,
Δ u t = u t N t / 2 = 2 N t 1 2 Δ t = 1 N t Δ t ,
N θ = π u t Δ u t = π 1 / ( 2 Δ t ) 1 / ( N t Δ t ) = π 2 N t ,
δ a = 4.88 ( Δ t ) 2 λ .
N a = 2 u a Δ u t = 2 / ( 2 Δ a ) 1 / ( N t Δ t ) = N a / ( N t Δ t ) 1 / ( N t Δ t ) = N a .
N θ , a N a Δ u t = π u t N θ , a N a 1 N t Δ t = π 1 2 Δ t N θ , a = 1 N a π 2 N t .

Metrics