Abstract

An algorithm is described for reconstructing a symmetric three-dimensional image from its Fourier intensity that is sampled below the Nyquist rate. The study is motivated by an image reconstruction problem in macromolecular x-ray crystallography. Application of the algorithm to simulated crystallographic problems shows that it converges to the correct solution, with no initial phase information, where algorithms currently used in crystallography fail. The algorithm is potentially useful in a variety of situations in macromolecular crystallography. The results presented also lend support to the possibility of ab initio phase retrieval in macromolecular crystallography when sufficient a priori information is available. Other applications in image reconstruction are possible.

© 1997 Optical Society of America

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  1. R. H. T. Bates, M. J. McDonnell, Image Restoration and Reconstruction (Clarendon, Oxford, 1989).
  2. R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990).
    [Crossref]
  3. N. E. Hurt, Phase Retrieval and Zero Crossings: Mathematical Methods in Image Reconstruction (Kluwer, Dordrecht, The Netherlands, 1989).
  4. H. Stark, Image Recovery: Theory and Application (Academic, London, 1987).
  5. M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 140–154 (1982).
    [Crossref]
  6. R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–526 (1987).
    [Crossref]
  7. R. P. Millane, “Redundancy in multidimensional deconvolution and phase retrieval,” in Digital Image Synthesis and Inverse Optics, A. F. Gmitro, P. S. Idell, I. J. LaHaie, eds., Proc. SPIE1351, 227–236 (1990).
    [Crossref]
  8. R. P. Millane, “Multidimensional phase problems,” J. Opt. Soc. Am. A 13, 725–734 (1996).
    [Crossref]
  9. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [Crossref] [PubMed]
  10. R. H. T. Bates, W. R. Fright, “Composite two-dimensional phase-restoration procedure,” J. Opt. Soc. Am. 73, 358–365 (1983).
    [Crossref]
  11. J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.
  12. M. Woolfson, F. Hai-Fu, Physical and Non-physical Methods of Solving Crystal Structures (Cambridge U. Press, New York, 1995).
  13. R. P. Millane, “Phase problems for periodic images: effects of support and symmetry,” J. Opt. Soc. Am. A 10, 1037–1045 (1993).
    [Crossref]
  14. J. Drenth, Principles of Protein X-ray Crystallography (Springer-Verlag, New York, 1994).
  15. R. H. T. Bates, “Uniqueness of solutions to two-dimensional Fourier phase problems for localized and positive images,” Comput. Vision Graphics Image Process. 25, 205–217 (1984).
  16. S. Onesti, A. D. Miller, P. Brick, “The crystal structure of the lysyl-tRNA synthetase (LysU) from Escherichia coli,” Structure 3, 163–176 (1995).
  17. J. Grimes, A. K. Basak, P. Roy, D. Stuart, “The crystal structure of bluetongue virus VP7,” Nature 373, 167–170 (1995).
    [Crossref] [PubMed]
  18. K. Braig, Z. Otwinowski, R. Hegde, D. Boisvert, A. Joachimiak, A. L. Horwich, P. B. Sigler, “The crystal structure of the bacterial chaperonin GroEL at 2.8 Å,” Nature 371, 578–586 (1994).
    [Crossref] [PubMed]
  19. A. A. Antson, J. Otridge, A. M. Brzozowski, E. J. Dodson, G. G. Dodson, K. S. Wilson, T. M. Smith, M. Yang, T. Kurecki, P. Gollnick, “The structure of trp RNA-binding attenuation protein,” Nature 374, 693–700 (1995).
    [Crossref] [PubMed]
  20. L. Liljas, “Structure of spherical virusus,” Int. J. Biol. Macromol. 13, 273–280 (1991).
  21. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  22. R. P. Millane, W. J. Stroud, “Phase retrieval for icosahedral particles,” in Signal Recovery and Synthesis, Vol. 11 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 20–22.
  23. R. P. Millane, W. J. Stroud, “Phase retrieval for imaging symmetric particles,” in Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems, R. L. Barbour, M. J. Carvlin, M. A. Fiddy, eds., Proc. SPIE2570, 276–281 (1995).
    [Crossref]
  24. R. P. Millane, W. J. Stroud, “Image reconstruction in x-ray crystallography using symmetry constraints,” in Proceedings of the Ninth Workshop on Image and Multidimensional Signal Processing (IEEE, Piscataway, N.J., 1996), pp. 154–155.
  25. R. P. Millane, W. J. Stroud, “Uniqueness and algorithms for macromolecular ab initio phasing,” Acta Crystallogr. Sect. A, Suppl. A52, C-66 (1996).
  26. M. J. Perez-Ilzarbe, “Phase retrieval from the power spectrum of a periodic object,” J. Opt. Soc. Am. A 9, 2138–2148 (1992).
    [Crossref]
  27. M. Kara, K. Kurki-Suonio, “Symmetrized multipole analysis of orientational distributions,” Acta Crystallogr. Sect. A 37, 201–210 (1981).
    [Crossref]
  28. C. V. Stauffacher, R. Usha, M. Harrington, T. Schmidt, M. V. Hosur, J. E. Johnson, “The structure of cowpea mosaic virus at 3.5 Å resolution,” in Crystallography in Molecular Biology, D. Moras, D. Suck, B. Strandberg, J. Drenth, T. Blundell, eds. (Plenum, New York, 1987), pp. 293–308.
  29. A. T. Brunger, x-plor Version 3.1 (Yale U. Press, New Haven, Conn., 1992).

1996 (2)

R. P. Millane, W. J. Stroud, “Uniqueness and algorithms for macromolecular ab initio phasing,” Acta Crystallogr. Sect. A, Suppl. A52, C-66 (1996).

R. P. Millane, “Multidimensional phase problems,” J. Opt. Soc. Am. A 13, 725–734 (1996).
[Crossref]

1995 (3)

S. Onesti, A. D. Miller, P. Brick, “The crystal structure of the lysyl-tRNA synthetase (LysU) from Escherichia coli,” Structure 3, 163–176 (1995).

J. Grimes, A. K. Basak, P. Roy, D. Stuart, “The crystal structure of bluetongue virus VP7,” Nature 373, 167–170 (1995).
[Crossref] [PubMed]

A. A. Antson, J. Otridge, A. M. Brzozowski, E. J. Dodson, G. G. Dodson, K. S. Wilson, T. M. Smith, M. Yang, T. Kurecki, P. Gollnick, “The structure of trp RNA-binding attenuation protein,” Nature 374, 693–700 (1995).
[Crossref] [PubMed]

1994 (1)

K. Braig, Z. Otwinowski, R. Hegde, D. Boisvert, A. Joachimiak, A. L. Horwich, P. B. Sigler, “The crystal structure of the bacterial chaperonin GroEL at 2.8 Å,” Nature 371, 578–586 (1994).
[Crossref] [PubMed]

1993 (1)

1992 (1)

1991 (1)

L. Liljas, “Structure of spherical virusus,” Int. J. Biol. Macromol. 13, 273–280 (1991).

1990 (1)

1987 (1)

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–526 (1987).
[Crossref]

1984 (1)

R. H. T. Bates, “Uniqueness of solutions to two-dimensional Fourier phase problems for localized and positive images,” Comput. Vision Graphics Image Process. 25, 205–217 (1984).

1983 (1)

1982 (2)

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

M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 140–154 (1982).
[Crossref]

1981 (1)

M. Kara, K. Kurki-Suonio, “Symmetrized multipole analysis of orientational distributions,” Acta Crystallogr. Sect. A 37, 201–210 (1981).
[Crossref]

1972 (1)

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

Antson, A. A.

A. A. Antson, J. Otridge, A. M. Brzozowski, E. J. Dodson, G. G. Dodson, K. S. Wilson, T. M. Smith, M. Yang, T. Kurecki, P. Gollnick, “The structure of trp RNA-binding attenuation protein,” Nature 374, 693–700 (1995).
[Crossref] [PubMed]

Basak, A. K.

J. Grimes, A. K. Basak, P. Roy, D. Stuart, “The crystal structure of bluetongue virus VP7,” Nature 373, 167–170 (1995).
[Crossref] [PubMed]

Bates, R. H. T.

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–526 (1987).
[Crossref]

R. H. T. Bates, “Uniqueness of solutions to two-dimensional Fourier phase problems for localized and positive images,” Comput. Vision Graphics Image Process. 25, 205–217 (1984).

R. H. T. Bates, W. R. Fright, “Composite two-dimensional phase-restoration procedure,” J. Opt. Soc. Am. 73, 358–365 (1983).
[Crossref]

R. H. T. Bates, M. J. McDonnell, Image Restoration and Reconstruction (Clarendon, Oxford, 1989).

Boisvert, D.

K. Braig, Z. Otwinowski, R. Hegde, D. Boisvert, A. Joachimiak, A. L. Horwich, P. B. Sigler, “The crystal structure of the bacterial chaperonin GroEL at 2.8 Å,” Nature 371, 578–586 (1994).
[Crossref] [PubMed]

Braig, K.

K. Braig, Z. Otwinowski, R. Hegde, D. Boisvert, A. Joachimiak, A. L. Horwich, P. B. Sigler, “The crystal structure of the bacterial chaperonin GroEL at 2.8 Å,” Nature 371, 578–586 (1994).
[Crossref] [PubMed]

Brick, P.

S. Onesti, A. D. Miller, P. Brick, “The crystal structure of the lysyl-tRNA synthetase (LysU) from Escherichia coli,” Structure 3, 163–176 (1995).

Brunger, A. T.

A. T. Brunger, x-plor Version 3.1 (Yale U. Press, New Haven, Conn., 1992).

Brzozowski, A. M.

A. A. Antson, J. Otridge, A. M. Brzozowski, E. J. Dodson, G. G. Dodson, K. S. Wilson, T. M. Smith, M. Yang, T. Kurecki, P. Gollnick, “The structure of trp RNA-binding attenuation protein,” Nature 374, 693–700 (1995).
[Crossref] [PubMed]

Dainty, J. C.

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

Dodson, E. J.

A. A. Antson, J. Otridge, A. M. Brzozowski, E. J. Dodson, G. G. Dodson, K. S. Wilson, T. M. Smith, M. Yang, T. Kurecki, P. Gollnick, “The structure of trp RNA-binding attenuation protein,” Nature 374, 693–700 (1995).
[Crossref] [PubMed]

Dodson, G. G.

A. A. Antson, J. Otridge, A. M. Brzozowski, E. J. Dodson, G. G. Dodson, K. S. Wilson, T. M. Smith, M. Yang, T. Kurecki, P. Gollnick, “The structure of trp RNA-binding attenuation protein,” Nature 374, 693–700 (1995).
[Crossref] [PubMed]

Drenth, J.

J. Drenth, Principles of Protein X-ray Crystallography (Springer-Verlag, New York, 1994).

Fienup, J. R.

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

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

Fright, W. R.

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–526 (1987).
[Crossref]

R. H. T. Bates, W. R. Fright, “Composite two-dimensional phase-restoration procedure,” J. Opt. Soc. Am. 73, 358–365 (1983).
[Crossref]

Gerchberg, R. W.

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

Gollnick, P.

A. A. Antson, J. Otridge, A. M. Brzozowski, E. J. Dodson, G. G. Dodson, K. S. Wilson, T. M. Smith, M. Yang, T. Kurecki, P. Gollnick, “The structure of trp RNA-binding attenuation protein,” Nature 374, 693–700 (1995).
[Crossref] [PubMed]

Grimes, J.

J. Grimes, A. K. Basak, P. Roy, D. Stuart, “The crystal structure of bluetongue virus VP7,” Nature 373, 167–170 (1995).
[Crossref] [PubMed]

Hai-Fu, F.

M. Woolfson, F. Hai-Fu, Physical and Non-physical Methods of Solving Crystal Structures (Cambridge U. Press, New York, 1995).

Harrington, M.

C. V. Stauffacher, R. Usha, M. Harrington, T. Schmidt, M. V. Hosur, J. E. Johnson, “The structure of cowpea mosaic virus at 3.5 Å resolution,” in Crystallography in Molecular Biology, D. Moras, D. Suck, B. Strandberg, J. Drenth, T. Blundell, eds. (Plenum, New York, 1987), pp. 293–308.

Hayes, M. H.

M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 140–154 (1982).
[Crossref]

Hegde, R.

K. Braig, Z. Otwinowski, R. Hegde, D. Boisvert, A. Joachimiak, A. L. Horwich, P. B. Sigler, “The crystal structure of the bacterial chaperonin GroEL at 2.8 Å,” Nature 371, 578–586 (1994).
[Crossref] [PubMed]

Horwich, A. L.

K. Braig, Z. Otwinowski, R. Hegde, D. Boisvert, A. Joachimiak, A. L. Horwich, P. B. Sigler, “The crystal structure of the bacterial chaperonin GroEL at 2.8 Å,” Nature 371, 578–586 (1994).
[Crossref] [PubMed]

Hosur, M. V.

C. V. Stauffacher, R. Usha, M. Harrington, T. Schmidt, M. V. Hosur, J. E. Johnson, “The structure of cowpea mosaic virus at 3.5 Å resolution,” in Crystallography in Molecular Biology, D. Moras, D. Suck, B. Strandberg, J. Drenth, T. Blundell, eds. (Plenum, New York, 1987), pp. 293–308.

Hurt, N. E.

N. E. Hurt, Phase Retrieval and Zero Crossings: Mathematical Methods in Image Reconstruction (Kluwer, Dordrecht, The Netherlands, 1989).

Joachimiak, A.

K. Braig, Z. Otwinowski, R. Hegde, D. Boisvert, A. Joachimiak, A. L. Horwich, P. B. Sigler, “The crystal structure of the bacterial chaperonin GroEL at 2.8 Å,” Nature 371, 578–586 (1994).
[Crossref] [PubMed]

Johnson, J. E.

C. V. Stauffacher, R. Usha, M. Harrington, T. Schmidt, M. V. Hosur, J. E. Johnson, “The structure of cowpea mosaic virus at 3.5 Å resolution,” in Crystallography in Molecular Biology, D. Moras, D. Suck, B. Strandberg, J. Drenth, T. Blundell, eds. (Plenum, New York, 1987), pp. 293–308.

Kara, M.

M. Kara, K. Kurki-Suonio, “Symmetrized multipole analysis of orientational distributions,” Acta Crystallogr. Sect. A 37, 201–210 (1981).
[Crossref]

Kurecki, T.

A. A. Antson, J. Otridge, A. M. Brzozowski, E. J. Dodson, G. G. Dodson, K. S. Wilson, T. M. Smith, M. Yang, T. Kurecki, P. Gollnick, “The structure of trp RNA-binding attenuation protein,” Nature 374, 693–700 (1995).
[Crossref] [PubMed]

Kurki-Suonio, K.

M. Kara, K. Kurki-Suonio, “Symmetrized multipole analysis of orientational distributions,” Acta Crystallogr. Sect. A 37, 201–210 (1981).
[Crossref]

Lane, R. G.

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–526 (1987).
[Crossref]

Liljas, L.

L. Liljas, “Structure of spherical virusus,” Int. J. Biol. Macromol. 13, 273–280 (1991).

McDonnell, M. J.

R. H. T. Bates, M. J. McDonnell, Image Restoration and Reconstruction (Clarendon, Oxford, 1989).

Millane, R. P.

R. P. Millane, “Multidimensional phase problems,” J. Opt. Soc. Am. A 13, 725–734 (1996).
[Crossref]

R. P. Millane, W. J. Stroud, “Uniqueness and algorithms for macromolecular ab initio phasing,” Acta Crystallogr. Sect. A, Suppl. A52, C-66 (1996).

R. P. Millane, “Phase problems for periodic images: effects of support and symmetry,” J. Opt. Soc. Am. A 10, 1037–1045 (1993).
[Crossref]

R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990).
[Crossref]

R. P. Millane, W. J. Stroud, “Image reconstruction in x-ray crystallography using symmetry constraints,” in Proceedings of the Ninth Workshop on Image and Multidimensional Signal Processing (IEEE, Piscataway, N.J., 1996), pp. 154–155.

R. P. Millane, “Redundancy in multidimensional deconvolution and phase retrieval,” in Digital Image Synthesis and Inverse Optics, A. F. Gmitro, P. S. Idell, I. J. LaHaie, eds., Proc. SPIE1351, 227–236 (1990).
[Crossref]

R. P. Millane, W. J. Stroud, “Phase retrieval for icosahedral particles,” in Signal Recovery and Synthesis, Vol. 11 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 20–22.

R. P. Millane, W. J. Stroud, “Phase retrieval for imaging symmetric particles,” in Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems, R. L. Barbour, M. J. Carvlin, M. A. Fiddy, eds., Proc. SPIE2570, 276–281 (1995).
[Crossref]

Miller, A. D.

S. Onesti, A. D. Miller, P. Brick, “The crystal structure of the lysyl-tRNA synthetase (LysU) from Escherichia coli,” Structure 3, 163–176 (1995).

Onesti, S.

S. Onesti, A. D. Miller, P. Brick, “The crystal structure of the lysyl-tRNA synthetase (LysU) from Escherichia coli,” Structure 3, 163–176 (1995).

Otridge, J.

A. A. Antson, J. Otridge, A. M. Brzozowski, E. J. Dodson, G. G. Dodson, K. S. Wilson, T. M. Smith, M. Yang, T. Kurecki, P. Gollnick, “The structure of trp RNA-binding attenuation protein,” Nature 374, 693–700 (1995).
[Crossref] [PubMed]

Otwinowski, Z.

K. Braig, Z. Otwinowski, R. Hegde, D. Boisvert, A. Joachimiak, A. L. Horwich, P. B. Sigler, “The crystal structure of the bacterial chaperonin GroEL at 2.8 Å,” Nature 371, 578–586 (1994).
[Crossref] [PubMed]

Perez-Ilzarbe, M. J.

Roy, P.

J. Grimes, A. K. Basak, P. Roy, D. Stuart, “The crystal structure of bluetongue virus VP7,” Nature 373, 167–170 (1995).
[Crossref] [PubMed]

Saxton, W. O.

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

Schmidt, T.

C. V. Stauffacher, R. Usha, M. Harrington, T. Schmidt, M. V. Hosur, J. E. Johnson, “The structure of cowpea mosaic virus at 3.5 Å resolution,” in Crystallography in Molecular Biology, D. Moras, D. Suck, B. Strandberg, J. Drenth, T. Blundell, eds. (Plenum, New York, 1987), pp. 293–308.

Sigler, P. B.

K. Braig, Z. Otwinowski, R. Hegde, D. Boisvert, A. Joachimiak, A. L. Horwich, P. B. Sigler, “The crystal structure of the bacterial chaperonin GroEL at 2.8 Å,” Nature 371, 578–586 (1994).
[Crossref] [PubMed]

Smith, T. M.

A. A. Antson, J. Otridge, A. M. Brzozowski, E. J. Dodson, G. G. Dodson, K. S. Wilson, T. M. Smith, M. Yang, T. Kurecki, P. Gollnick, “The structure of trp RNA-binding attenuation protein,” Nature 374, 693–700 (1995).
[Crossref] [PubMed]

Stark, H.

H. Stark, Image Recovery: Theory and Application (Academic, London, 1987).

Stauffacher, C. V.

C. V. Stauffacher, R. Usha, M. Harrington, T. Schmidt, M. V. Hosur, J. E. Johnson, “The structure of cowpea mosaic virus at 3.5 Å resolution,” in Crystallography in Molecular Biology, D. Moras, D. Suck, B. Strandberg, J. Drenth, T. Blundell, eds. (Plenum, New York, 1987), pp. 293–308.

Stroud, W. J.

R. P. Millane, W. J. Stroud, “Uniqueness and algorithms for macromolecular ab initio phasing,” Acta Crystallogr. Sect. A, Suppl. A52, C-66 (1996).

R. P. Millane, W. J. Stroud, “Image reconstruction in x-ray crystallography using symmetry constraints,” in Proceedings of the Ninth Workshop on Image and Multidimensional Signal Processing (IEEE, Piscataway, N.J., 1996), pp. 154–155.

R. P. Millane, W. J. Stroud, “Phase retrieval for imaging symmetric particles,” in Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems, R. L. Barbour, M. J. Carvlin, M. A. Fiddy, eds., Proc. SPIE2570, 276–281 (1995).
[Crossref]

R. P. Millane, W. J. Stroud, “Phase retrieval for icosahedral particles,” in Signal Recovery and Synthesis, Vol. 11 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 20–22.

Stuart, D.

J. Grimes, A. K. Basak, P. Roy, D. Stuart, “The crystal structure of bluetongue virus VP7,” Nature 373, 167–170 (1995).
[Crossref] [PubMed]

Usha, R.

C. V. Stauffacher, R. Usha, M. Harrington, T. Schmidt, M. V. Hosur, J. E. Johnson, “The structure of cowpea mosaic virus at 3.5 Å resolution,” in Crystallography in Molecular Biology, D. Moras, D. Suck, B. Strandberg, J. Drenth, T. Blundell, eds. (Plenum, New York, 1987), pp. 293–308.

Wilson, K. S.

A. A. Antson, J. Otridge, A. M. Brzozowski, E. J. Dodson, G. G. Dodson, K. S. Wilson, T. M. Smith, M. Yang, T. Kurecki, P. Gollnick, “The structure of trp RNA-binding attenuation protein,” Nature 374, 693–700 (1995).
[Crossref] [PubMed]

Woolfson, M.

M. Woolfson, F. Hai-Fu, Physical and Non-physical Methods of Solving Crystal Structures (Cambridge U. Press, New York, 1995).

Yang, M.

A. A. Antson, J. Otridge, A. M. Brzozowski, E. J. Dodson, G. G. Dodson, K. S. Wilson, T. M. Smith, M. Yang, T. Kurecki, P. Gollnick, “The structure of trp RNA-binding attenuation protein,” Nature 374, 693–700 (1995).
[Crossref] [PubMed]

Acta Crystallogr. Sect. A (1)

M. Kara, K. Kurki-Suonio, “Symmetrized multipole analysis of orientational distributions,” Acta Crystallogr. Sect. A 37, 201–210 (1981).
[Crossref]

Acta Crystallogr. Sect. A, Suppl. (1)

R. P. Millane, W. J. Stroud, “Uniqueness and algorithms for macromolecular ab initio phasing,” Acta Crystallogr. Sect. A, Suppl. A52, C-66 (1996).

Appl. Opt. (1)

Comput. Vision Graphics Image Process. (1)

R. H. T. Bates, “Uniqueness of solutions to two-dimensional Fourier phase problems for localized and positive images,” Comput. Vision Graphics Image Process. 25, 205–217 (1984).

IEEE Trans. Acoust. Speech Signal Process. (2)

M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 140–154 (1982).
[Crossref]

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–526 (1987).
[Crossref]

Int. J. Biol. Macromol. (1)

L. Liljas, “Structure of spherical virusus,” Int. J. Biol. Macromol. 13, 273–280 (1991).

J. Opt. Soc. Am. (1)

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

Nature (3)

J. Grimes, A. K. Basak, P. Roy, D. Stuart, “The crystal structure of bluetongue virus VP7,” Nature 373, 167–170 (1995).
[Crossref] [PubMed]

K. Braig, Z. Otwinowski, R. Hegde, D. Boisvert, A. Joachimiak, A. L. Horwich, P. B. Sigler, “The crystal structure of the bacterial chaperonin GroEL at 2.8 Å,” Nature 371, 578–586 (1994).
[Crossref] [PubMed]

A. A. Antson, J. Otridge, A. M. Brzozowski, E. J. Dodson, G. G. Dodson, K. S. Wilson, T. M. Smith, M. Yang, T. Kurecki, P. Gollnick, “The structure of trp RNA-binding attenuation protein,” Nature 374, 693–700 (1995).
[Crossref] [PubMed]

Optik (1)

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

Structure (1)

S. Onesti, A. D. Miller, P. Brick, “The crystal structure of the lysyl-tRNA synthetase (LysU) from Escherichia coli,” Structure 3, 163–176 (1995).

Other (12)

J. Drenth, Principles of Protein X-ray Crystallography (Springer-Verlag, New York, 1994).

R. H. T. Bates, M. J. McDonnell, Image Restoration and Reconstruction (Clarendon, Oxford, 1989).

N. E. Hurt, Phase Retrieval and Zero Crossings: Mathematical Methods in Image Reconstruction (Kluwer, Dordrecht, The Netherlands, 1989).

H. Stark, Image Recovery: Theory and Application (Academic, London, 1987).

R. P. Millane, “Redundancy in multidimensional deconvolution and phase retrieval,” in Digital Image Synthesis and Inverse Optics, A. F. Gmitro, P. S. Idell, I. J. LaHaie, eds., Proc. SPIE1351, 227–236 (1990).
[Crossref]

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

M. Woolfson, F. Hai-Fu, Physical and Non-physical Methods of Solving Crystal Structures (Cambridge U. Press, New York, 1995).

R. P. Millane, W. J. Stroud, “Phase retrieval for icosahedral particles,” in Signal Recovery and Synthesis, Vol. 11 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 20–22.

R. P. Millane, W. J. Stroud, “Phase retrieval for imaging symmetric particles,” in Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems, R. L. Barbour, M. J. Carvlin, M. A. Fiddy, eds., Proc. SPIE2570, 276–281 (1995).
[Crossref]

R. P. Millane, W. J. Stroud, “Image reconstruction in x-ray crystallography using symmetry constraints,” in Proceedings of the Ninth Workshop on Image and Multidimensional Signal Processing (IEEE, Piscataway, N.J., 1996), pp. 154–155.

C. V. Stauffacher, R. Usha, M. Harrington, T. Schmidt, M. V. Hosur, J. E. Johnson, “The structure of cowpea mosaic virus at 3.5 Å resolution,” in Crystallography in Molecular Biology, D. Moras, D. Suck, B. Strandberg, J. Drenth, T. Blundell, eds. (Plenum, New York, 1987), pp. 293–308.

A. T. Brunger, x-plor Version 3.1 (Yale U. Press, New Haven, Conn., 1992).

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Figures (8)

Fig. 1
Fig. 1

Flow chart of the hio algorithm adapted for the crystallographic application.

Fig. 2
Fig. 2

Relationship between the icosahedral symmetry axes (labeled 2, 3, and 5) and the cubic unit cell used in the simulations for 5-fold noncrystallographic symmetry.

Fig. 3
Fig. 3

Three error metrics versus iteration for reconstruction of the simple icosahedral image for five different random starting phases using (a) the er algorithm and (b) the hio algorithm.

Fig. 4
Fig. 4

Contour plots of sections through (a) the actual simple icosahedrally symmetric image and reconstructions with (b) the er algorithm and (c) the hio algorithm. The sections are through the center of the particle (left) and halfway between the center and the outer edge of the particle (right). Contour levels are at 0.2, 0.4, 0.6, and 0.8 of the maximum value of the image.

Fig. 5
Fig. 5

Surface (contoured at 0.2 of the maximum value of the image) of (a) the actual simple icosahedrally symmetric image and reconstructions with (b) the er algorithm and (c) the hio algorithm.

Fig. 6
Fig. 6

Three error metrics versus iteration for reconstruction of cpmv for five different starting random phases with (a) the er algorithm and (b) the hio algorithm.

Fig. 7
Fig. 7

Contour plots of sections through (a) the actual electron density of cpmv and reconstructions with (b) the er algorithm and (c) the hio algorithm. The sections are through the center of the virus particle (left) and halfway between the center and the outer edge of the virus particle (right). Contour levels are at 0.2, 0.4, 0.6, and 0.8 of the maximum value of the electron density.

Fig. 8
Fig. 8

Outer surface (contoured at 0.2 of the maximum value of the image) of (a) the actual cpmv particle and reconstructions with (b) the er algorithm and (c) the hio algorithm.

Equations (17)

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f(x, y, z)=e(x, y, z)l(x, y, z)=e(x, y, z)m,n,p=- δ(x-ma)×δ(y-nb)δ(z-pc),
F(u, v, w)=- f(x, y, z)×exp[i2π(ux+vy+wz)]dxdydz
F(u, v, w)=E(u, v, w)L(u, v, w)=E(u, v, w)Vh,k,l=- δ(u-h/a)×δ(v-k/b)δ(w-l/c),
Fh=V f(x) exp(i2πuh·x)dx,
f(x)=h Fh exp(-i2πuh·x).
e(x)=n=1N h(Rnx+Tn).
E(u)=n=1N H(Rnu)exp[-i2πu·(Rn-1Tn)],
e(x)=e(Rnx+Tn),n=1, 2, , N,
|E(u)|=|E(Rnu)|,
|E(Rnuh)|=|Fh|.
gn+1(x)=fn(x)gn(x)+β[cn(x)-fn(x)]if  |fn(x)-cn(x)|<if  |fn(x)-cn(x)|>  .
cn(x)=fn(x)W,xW,
cn(x)=1Mi=1Mfn(xi),xU,
gn+1(x)=cn(x).
Rn=h||Fhd|-|Ghn||h|Fhd|.
Φn=h|Fhd||ϕht-ϕhn|h|Fhd|.
en2=x[ft(x)-fn(x)]2x[ft(x)]2,

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