Abstract
Information theory provides a uniquely powerful apparatus for reconstructing an image from imperfect data in its dual space. An iterative procedure based on constrained entropy maximization is presented here for reconstruction of electron density from imperfect single-crystal X-ray diffraction data. In the ideal situation, continuous electron density, its periodicity given by the lattice, is the Fourier transform of an infinite set of structure factors whose squared moduli are observables. An ideal electron density is unattainable from experiment because of the common experimental inadequacies of incomplete and noisy data discussed by Gull and Daniell1. These inadequacies are partially overcome in the present procedure which yields true super-resolution and is economical even for protein crystal structures as illustrated.
This is a preview of subscription content, access via your institution
Access options
Subscribe to this journal
Receive 51 print issues and online access
$199.00 per year
only $3.90 per issue
Buy this article
- Purchase on Springer Link
- Instant access to full article PDF
Prices may be subject to local taxes which are calculated during checkout
Similar content being viewed by others
References
Gull, S. F. & Daniell, G. J. Nature 272, 686–690 (1978).
Benjamin, R. IEE Proc. F127, 341–353 (1980).
Oppenheim, A. V. & Schafer, R. W. Digital Signal Processing, Ch. 3 (Prentice-Hall, New Jersey, 1975).
Jaynes, E. T. IEEE Trans. SSC- 4, 227–241 (1968).
Ramachandran, G. N. & Srinivasan, R. Fourier Methods in Crystallography (Wiley–Interscience, New York, 1970).
Ables, J. G. Astr. Astrophys. Suppl. 15, 383–393 (1974).
Jaynes, E. T. Phys. Rev. 106, 620–630 (1957).
Collins, D. M. Acta crystallogr. A34, 533–541 (1978).
Watenpaugh, K. D., Sieker, L. C., Herriott, J. R. & Jensen, L. H. Acta crystallogr. B29, 943–956 (1973).
Hubbard, C. R., Quicksall, C. O. & Jacobson, R. A. The Fast Fourier Algorithm and the Programs ALFF, ALFFDP, ALFFPROJ, ALFFT, and FRIEDEL (Ames, Iowa, 1971).
Collins, D. M., Cotton, F. A., Hazen, E. E. Jr, Meyer, E. F. Jr & Morimoto, C. N. Science 190, 1047–1053 (1975).
Collins, D. M., Brice, M. D., la Cour, T. F. M. & Legg, M. J. Crystallographic Computing Techniques (eds Ahmed, F. R., Huml, K. & Sedláček, B.) 330–335 (Munksgaard, Copenhagen, 1976).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Collins, D. Electron density images from imperfect data by iterative entropy maximization. Nature 298, 49–51 (1982). https://doi.org/10.1038/298049a0
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1038/298049a0
This article is cited by
-
Anisotropic electrical conductivity changes in FeTiO3 structure transition under high pressure
Physics and Chemistry of Minerals (2024)
-
High-temperature synthesis and electronic bonding analysis of Ca-doped LaMnO3 rare-earth manganites
Rare Metals (2022)
-
Direct observation of one-dimensional disordered diffusion channel in a chain-like thermoelectric with ultralow thermal conductivity
Nature Communications (2021)
-
Charge distribution around Ba–O and Ti–O bonds in BaTi1−xZrxO3 through powder X-ray diffraction
Rare Metals (2021)
-
Influence of Eu3+-doped BaTiO3 phosphors on structural, optical and photoluminescence properties
Journal of Materials Science: Materials in Electronics (2021)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.