Abstract
The intuitive understanding of the process of three-dimensional reconstruction is based on a number of assumptions, which are easily made unconsciously; the most crucial is the belief that what is detected is some kind of projection through the structure. This “projection” need not necessarily be a (weighted) sum or integral through the structure of some physical property of the latter; in principle, a mono-tonically varying function would be acceptable, although solving the corresponding inverse problem might not be easy. In practice, however, the usual interpretation of “projection” is overwhelmingly adopted, and it was for this definition that Radon (1917) first proposed a solution. In the case of light, shone through a translucent structure of varying opacity, a three-dimensional transparency as it were, the validity of this projection assumption seems too obvious to need discussion. We know enough about the behavior of x-rays in matter to establish the conditions in which it is valid in radiography. In this chapter, we enquire whether it is valid in electron microscopy, where intuition might well lead us to suspect that it is not. Electron—specimen interactions are very different from those encountered in x-ray tomography, the specimens are themselves very different in nature, creating phase rather than amplitude contrast, and an optical system is needed to transform the information about the specimen that the electrons have acquired into a visible image. If the electrons encounter more than one structural feature in their passage through the specimen, the overall effect is far from easy to guess, whereas in the case of light shone through a transparent structure, it is precisely the variety of such overlaps or superpositions that we use to effect the reconstruction. If intuition were our only guide, we might easily doubt whether three-dimensional reconstruction from electron micrographs is possible: there is no useful projection approximation for the balls on a pin-table! Why then has it been so successful? To understand this, we must examine in detail the nature of the interactions between the electrons and the specimen and the characteristics of the image-forming process in the electron microscope. Does the information about the specimen imprinted on the electron beam as it emerges from the latter represent a projection through the structure? How faithfully is this information conveyed to the recorded image? These are the questions that we shall be exploring in the following sections.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arnal, F., Balladore, J. L., Soum, G., and Verdier, P. (1977). Calculation of the cross-sections of electron interaction with matter. Ultramicro.scopy 2:305–310.
Amos, L. A., Henderson, R., and Unwin, P. N. T. (1982). Three-dimensional structure determination by electron microscopy of two-dimensional crystals. Prog. Biophys. Mol. Biol. 39:183–231.
Born, M. and Wolf, E. (1980). Principles of Optics. Pergamon, New York.
Cohen, H. A., Schmid, M. F., and Chiu, W. (1984). Estimates of validity of projection approximation for three-dimensional reconstructions at high resolution. Ultramicroscopy 14:219–226.
Deans, S. R. (1983). The Radon Transform and Some of Its Applications. Wiley-Interscience, New York.
DeRosier, D. J. (1971). The reconstruction of three-dimensional images from electron micrographs. Contemp. Phys. 12:437–452.
DeRosier, D. J. and Klug, A. (1968). Reconstruction of three-dimensional structures from electron micrographs. Nature 217:130–134.
Frank, J. and Radermacher, M. (1986). Three-dimensional reconstruction of nonperiodic macromolecular assemblies from electron micrographs, in Advanced Techniques in Biological Electron
Microscopy III (J. K. Koehler, ed.), pp. 1–72. Springer, Berlin and New York. Glaeser, R. M. (1982). Electron microscopy, in Methods of Experimental Physics (G. Ehrenstein and H. Lecar, eds.), Vol. 20, pp. 391–444. Academic Press, New York.
Glaeser, R. M. (1985). Electron crystallography of biological macromolecules. Ann. Rev. Phys. Chem. 36:243–275.
Glaser, W. (1952). Grundlagen der Elektronenoptik. Springer, Vienna.
Grinton, G. R. and Cowley, J. M. (1971). Phase and amplitude contrast in electron micrographs of biological material. Optik 34:221–233.
Hawkes, P. W. (1980). Image processing based on the linear theory of image formation, in Computer Processing of Electron Microscope Images (P. W. Hawkes, ed.), pp. 1–33. Springer, Berlin and New York.
Hawkes, P. W. (1980b). Units and conventions in electron microscopy, for use in ultramicroscopy. Ultramicroscopy 5:67–70.
Hawkes, P. W. and Kasper, E. (forthcoming). Principles of Electron Optics Vol. 3. Academic Press, London.
Henderson, R. and Baldwin, J. M. (1986). Treatment of the gradient of defocus in images of tilted, thin crystals, in Proc. 44th Ann. Meetg. EMSA (G. W. Bailey, ed.), pp. 6–9. San Francisco Press, San Francisco.
Henderson, R., Baldwin, J. M., Downing, K. H., Lepault, J., and Zemlin, F. (1986). Structure of
Purple membrane from Halobacterium halobium Recording, measurement and evaluation of electron micrographs at 3.5 Å resolution. Ultramicroscopy 19:147–178.
Jap, B. K. and Glaeser, R. M. (1980). The scattering of high-energy electrons. II: Quantitative validity domains of the single-scattering approximations for organic crystals Acta CCrystallogr A 36:57–6
Radermacher, M. (1988). Three-dimensional reconstruction of single particles from random and non-random tilt series. J. Electron Microsc. Technique 9:359–394.
Radon, J. (1917). Über die Bestimmung von Funktionen durch ihre Integralwerte langs gewisser Mannigfaltigkeiten. Ber. Verh. K. Sächs. Ges. Wiss. Leipzig, Math- Phys. Kl. 69:262–277. For an English translation, see Deans (1983).
Reichelt, R. and Engel, A. (1984). Monte Carlo calculations of elastic and inelastic electron scattering in biological and plastic materials. Ultramicroscopy 13:279–294.
Reimer, L. (1989). Transmission Electron Microscopy, 2nd ed. Springer, Berlin and New York.
Reimer, L. and Sommer, K. H. (1968). Messungen und Berechnungen zum elektronenmikroskopischen Streukontrast füür 17 bis 1200 keV-Elektronen. Z. Naturforsch. 23a:1569–1582.
Robards, A. W. and Sleytr, U. B. (1985). Low Temperature Methods in Biological Electron Microscopy. Elsevier, New York.
Saxton, W. O. (1986). Focal series restoration in HREM, in Proc. Xlth Int. Cong. Electron Microscopy, suppl. to J. Electron Microsc. 35, Post-deadline paper 1.
Schiske, P. (1968). Zur Frage der Bildrekonstruktion durch Fokusreihen, in Electron Microscopy 1968 (D. S. Bocciarelli, ed.), Vol. I. pp. 147–148. Tipografia Poliglotta Vaticana, Rome
Schiske, P. (1973). Image processing using additional statistical information about the object, in Image Processing and Computer-Aided Design in Electron Optic.s (P. W. Hawkes, ed.), pp. 82–90. Academic Press, New York.
Schiske, P. (1982). A posteriori correction of object tilt for the CTEM. Ultramicro scoPy 9:17–26.
Spence, J. C. H. (1988). Experimental High-resolution Electron Microscopy, 2nd ed. Oxford University Press, New York.
Steinbrecht, R. A. and Zierold, K., eds. (1987). Cryotechniques in Biological Electron Microscopy. Springer, Berlin and New York.
Steven, A. C. (1981). Visualization of virus structure in three dimensions, in Methods in Cell Biology, Vol. 22, Three-Dimensional Ultrastructure in Biology (J. N. Turner, ed.), pp. 297–323. Academic Press, New York.
Turner, J. N., ed. (1981). Methods in Cell Biology, Vol. 22, Three-Dimensional Ultrastructure in Biology. Academic Press, New York.
Zeitler, E., ed. (1982). Cryomicroscopy and radiation damage. Ultramicro.scopy 10:1–178.
Zeitler, E., ed. (1984). Cryomicroscopy and radiation damage. II. Ultramicroscopy 14:161–316.
Zemlin, F. (1989). Dynamic focusing for recording images from tilted samples in small-spot scanning with a transmission electron microscope. J. Electron Microsc. Technique 11:251–257.
Frank, J. (1973). Computer processing of electron micrographs, in Advanced Techniques in Biological Electron Microscopy (J. K. Koehler, ed.), pp. 215–274. Springer, Berlin and New York. Frank, J. (1980). The role of correlation techniques in computer image processing, in Hawkes (1980), pp. 187–222.
Frank, J. (1981). Introduction and Three-dimensional reconstruction of single molecules, in Methods in Cell Biology, Vol. 22, Three-Dimensional Ultra.structure in Biology (J. N. Turner, ed.), pp. 119–213, 325–344. Academic Press, New York.
Frank, J. (1989). Image analysis of single macromolecules. Electron Microsc. Rev. 2:53–74
Hawkes, P. W., ed. (1980). Computer Processing of Electron Microscope Images. Springer, Berlin and New York.
Henderson, R. and Glaeser, R. M. (1985). Quantitative analysis of image contrast in electron micrographs of beam-sensitive crystals. Ultramicroscopy 16:139–150.
Hoppe, W. and Hegerl, R. (1980). Three-dimensional structure determination by electron microscopy (nonperiodic specimens), in Hawkes (1980), pp. 127–185.
Hoppe, W. and Typke, D. (1979). Three-dimensional reconstruction of aperiodic objects in electron microscopy, in Advances in Structure Research by Diffraction Methods (W. Hoppe and R. Mason, eds.), Vol. 7, pp. 137–190. Vieweg, Braunschweig.
Lewitt, R. M. (1983). Reconstruction algorithms: transform methods. Proc. IEEE 71:390–408.
Mellema, J. E. (1980). Computer reconstruction of regular biological objects, in Hawkes (1980), pp. 89–126.
Moody, M. F. (1990). Image analysis of electron micrographs, in Biophysical Electron Microscopy (P. W. Hawkes and U. Valdrè, eds.), Academic Press, New York.
Robinson, D. G., Ehlers, U., Herken, R., Herrmann, B., Mayer, F., and Schüürmann, F.-W. (1987). Methods of Preparation for Electron Microscopy, An Introduction for the Biomedical Sciences. Springer, Berlin and New York.
Sommerville, J. and Scheer, U., eds. (1987). Electron Microscopy in Molecular Biology, A Practical Approach. IRL Press, Oxford and Washington.
Stewart, M. (1988). Introduction to the computer image processing of electron micrographs of twodimensionally ordered biological structures. J. Electron Microsc. Technique 9:301–324.
Stewart, M. (1988). Computer image processing of electron micrographs of biological structures with helical symmetry. J. Electron Microsc. Technique 9:325–358.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1992 Springer Science+Business Media New York
About this chapter
Cite this chapter
Hawkes, P.W. (1992). The Electron Microscope as a Structure Projector. In: Frank, J. (eds) Electron Tomography. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-2163-8_2
Download citation
DOI: https://doi.org/10.1007/978-1-4757-2163-8_2
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4757-2165-2
Online ISBN: 978-1-4757-2163-8
eBook Packages: Springer Book Archive