Introduction: Principles of Electron Tomography

  • Joachim Frank


Tomography is a method for reconstructing the interior of an object from its projections. The word tomography literally means the visualization of slices, and is applicable, in the strict sense of the word, only in the narrow context of the single-axis tilt geometry: for instance, in medical computerized axial tomography (CAT-scan imaging), the detector-source arrangement is tilted relative to the patient around a single axis (Fig. 1a). In electron microscopy, where the beam direction is fixed, the specimen holder is tilted around a single axis (Fig. 1b). However, the usage of this term has recently become more liberal, encompassing arbitrary geometries, provided that the specimen is actively tilted into multiple angles. In line with this relaxed convention, we will use the term electron tomography for any technique that employs the transmission electron microscope to collect projections of an object that is tilted in multiple directions and uses these projections to reconstruct the object in its entirety. Excluded from this definition are ‘single-particle’ techniques that make use of multiple occurrences of the object in different orientations, with or without the additional aid of symmetry (Fig. 1c). These techniques are covered elsewhere (non-symmetric: Frank, 1996, 2006; symmetric: Glaeser et al., 2007).


Fourier Space Single Axis Word Tomography Macromolecular Assembly Projection Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Amos, L.A., Henderson, R. and Unwin, P. N. T. (1982). Three-dimensional structure determination by electron microscopy of 2-dimensional crystals. Prog. Biophys. Mol. Biol. 39: 183–231.PubMedCrossRefGoogle Scholar
  2. Andrews, H. C. (1970). Computer Techniques in Image Processing. Academic Press, New York.Google Scholar
  3. Barnard, D. P., Turner, J.N., Frank, J. and McEwen, B.F. (1992). A 360° single-axis tilt stage for the high-voltage electron microscope. J. Microsc. 167:39–48.PubMedGoogle Scholar
  4. Bracewell, R. N. (1999). The Fourier Transform and Its Applications, 3rd edn. McGraw-Hill, New York.Google Scholar
  5. Bracewell, R. N. and Riddle, A. C. (1967). Inversion of fan-beam scans in radio astronomy. Astrophys. Soc. 150:427–434.CrossRefGoogle Scholar
  6. Chalcroft, J. P. and Davey, C. L. (1984). A simply constructed extreme-tilt holder for the Philips eucentric goniometer stage. J. Microsc. 134:41–48.Google Scholar
  7. Colsher, J. G. (1977). Iterative three-dimensional image reconstruction from tomographic projections. Comput. Graphics Image Process. 6:513–537.CrossRefGoogle Scholar
  8. Cormack, A. M. (1964). Representation of a function by its line integrals, with some radiological applications. I. J. Appl. Phys. 35:2908–2912.CrossRefGoogle Scholar
  9. Crowther, R. A., Amos, L. A., Finch, J. T. and Klug, A. (1970a). Three-dimensional reconstruction of spherical viruses by Fourier synthesis from electron micrographs. Nature 226: 421–425.PubMedCrossRefGoogle Scholar
  10. Crowther, R. A., DeRosier, D. J. and Klug, A. (1970b). The reconstruction of a three-dimensional structure from projections and its application to electron microscopy. Proc. R. Soc. B 317:319–340.CrossRefGoogle Scholar
  11. DeRosier, D. and Klug, A. (1968). Reconstruction of 3-dimensional structures from electron micrographs. Nature 217:130–134.CrossRefGoogle Scholar
  12. Frank, J. (1975). Averaging of low exposure electron micrographs of nonperiodic objects. Ultramicroscopy 1:159–162.PubMedCrossRefGoogle Scholar
  13. Frank, J. (1992). Introduction. In Electron Tomography (J. Frank, ed.) pp. 1–13. Plenum, New York.Google Scholar
  14. Frank, J. (1996). Three-dimensional Electron Microscopy of Macromolecules. Academic Press, San Diego.Google Scholar
  15. Frank, J. (2006). Three-dimensional Electron Microscopy of Macromolecules, 2nd edn. Oxford University Press, New York.Google Scholar
  16. Frank, J. and Radermacher, M. (1986). Three-dimensional reconstruction of nonperiodic macromolecular assemblies from electron micrographs. In Advanced Techniques in Electron Microscopy III (J. K. Koehler, ed.). Springer-Verlag, New York, pp. 1–72.Google Scholar
  17. Gilbert, P. F. C. (1972). The reconstruction of a three-dimensional structure from projections and its application to electron microscopy. II. Direct methods. Proc. R. Soc. B 182:89–117.CrossRefGoogle Scholar
  18. Glaeser, R. M., Downing, K. H., Chiu, W., Frank, J. and DeRosier, D. (2007). Electron Crystallography of Biological Macromolecules. Oxford University Press, New York.Google Scholar
  19. Hegerl, R. and Altbauer, A. (1982). The ‘EM’ program system. Ultramicroscopy 9:109–116.PubMedCrossRefGoogle Scholar
  20. Herman, G.T. (ed.) (1979). Image Reconstruction from Projections. Springer-Verlag, Berlin.Google Scholar
  21. Henderson, R. and Unwin, P. N. T. (1975). Three-dimensional model of purple membrane obtained by electron microscopy. Nature 257:28–32.PubMedCrossRefGoogle Scholar
  22. Herman, G. T. and Lewitt, R. M. (1979). Overview of image reconstruction from projections, in Image Reconstruction from Projections (G.T. Herman, ed.). Springer-Verlag, Berlin, pp. 1–7.Google Scholar
  23. Hoppe, W. (1972). Drei-dimensional abbildende Elektronenmikroskope. Z. Naturforsch. 27a: 919–929.Google Scholar
  24. Hoppe, W. (1981). Three-dimensional electron microscopy. Annu. Rev. Biophys. Bioeng. 10: 563–592.PubMedCrossRefGoogle Scholar
  25. Hoppe, W. (1983). Elektronenbeugung mit dem Transmissions-Elektronenmikroskop als phasenbestimmendem Diffraktometer-von der Ortsfrequenzfilterung zur dreidimensionalen Strukturanalyse an Ribosomen. Angew. Chem. 95:465–494.CrossRefGoogle Scholar
  26. Hoppe, W., Gassmann, J., Hunsmann, N., Schramm, H. J. and Sturm, M. (1974). Three-dimensional reconstruction of individual negatively stained fatty-acid synthetase molecules from tilt series in the electron microscope. Hoppe-Seyler’s Z. Physiol. Chem. 355:1483–1487.PubMedGoogle Scholar
  27. Klug, A. (1983). From macromolecules to biological assemblies (Nobel lecture). Angew. Chem. 22:565–636.CrossRefGoogle Scholar
  28. Koster, A. J., Chen, J.W., Sedat, J.W. and Agard, D. A. (1992). Automated microscopy for electron tomography. Ultramicroscopy 46:207–227.PubMedCrossRefGoogle Scholar
  29. Lanzavecchia, S., Cantele, F., Bellon, P. L., Zampighi, L., Kreman, M., Wright, E., and Zampighi, G. A. (2005). Conical tomography of freeze-fracture replicas: a method for the study of integral membrane proteins inserted in phospholipids bilayers. J. Struct. Biol. 149:87–98.PubMedCrossRefGoogle Scholar
  30. Lewitt, R. M. and Bates, R. H. T. (1978a). Image reconstruction from projections. I: General theoretical considerations. Optik (Stuttgart) 50:19–33.Google Scholar
  31. Lewitt, R. M. and Bates, R. H.T. (1978b). Image reconstruction from projections. III: Projection completion methods (theory). Optik (Stuttgart) 50:189–204.Google Scholar
  32. Lewitt, R. M., Bates, R. H. T. and Peters, T. M. (1978). Image reconstruction from projections. II: Modified back-projection methods. Optik (Stuttgart) 50:85–109.Google Scholar
  33. Radermacher, M. and Hoppe, W. (1980). Properties of 3D reconstruction from projections by conical tilting compared to single axis tilting. In Proceedings of the 7th European Congress on Electron Microscopy, Den Haag, Vol. I, pp. 132–133.Google Scholar
  34. Radermacher, M., Wagenknechet, T., Verschoor, A. and Frank, J. (1987a). Three-dimensional structure of the large ribosomal subunit from Escherichia coli. EMBO J. 6:1107–1114.PubMedGoogle Scholar
  35. Radermacher, M., Wagenknecht, T., Verschoor, A. and Frank, J. (1987b). Three-dimensional reconstruction from single-exposure random conical tilt series applied to the 50S ribosomal subunit of Escherichia coli. J. Microsc. 146:113–136.PubMedGoogle Scholar
  36. Radon, J. (1917). Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig. Math. Phys. Klasse 69:262–277.Google Scholar
  37. Saxton, W. O. and Frank, J. (1977). Motif detection in quantum noise-limited electron micrographs by cross-correlation. Ultramicroscopy 2:219–227.PubMedCrossRefGoogle Scholar
  38. Smith, P. R., Peters, T.M. and Bates, R.H.T. (1973). Image reconstruction from a finite number of projections. J. Phys. A 6:361–382.CrossRefGoogle Scholar
  39. Turner, J. N. (1981). Stages and stereo pair recording. Methods Cell Biol. 23:33–51.Google Scholar
  40. Typke, D., Hoppe, W., Sessier, W. and Burger, M. (1976). Conception of a 3-D imaging electron microscopy. In Proceedings of the 6th European Congress on Electron Microscopy (D. G. Brandon, ed.), Vol. 1, Tal International, Israel, pp. 334–335.Google Scholar
  41. Unwin, P. N. T. and Henderson, R. (1975). Molecular structure determination by electron microscopy of unstained crystalline specimens. J. Mol. Biol. 94:425–440.PubMedCrossRefGoogle Scholar
  42. Zampighi, G., Zampighi, L., Fain., N., Wright, E. M., Cantele, F. and Lanzavecchia, S. (2005). Conical tomography II: a method for the study of cellular organelles in thin sections. J. Struct. Biol. 151:263–274.PubMedCrossRefGoogle Scholar
  43. Zwick, M. and Zeitler, E. (1973). Image reconstruction from projections. Optik 38:550–565.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Joachim Frank
    • 1
  1. 1.Howard Hughes Medical Institute, Health Research, Inc. at the Wadsworth Center, Department of Biomedical SciencesState University of New York at AlbanyAlbanyUSA

Personalised recommendations