The reconstruction of a three-dimensional structure from projections and its application to electron microscopy

@article{Crowther1970TheRO,
  title={The reconstruction of a three-dimensional structure from projections and its application to electron microscopy},
  author={R. Anthony Crowther and David J. DeRosier and Aaron Klug},
  journal={Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences},
  year={1970},
  volume={317},
  pages={319 - 340}
}
  • R. CrowtherD. DeRosierA. Klug
  • Published 23 June 1970
  • Mathematics
  • Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
A transmission electron micrograph is essentially a projection of the specimen in the direction of view. In order to reconstruct a three-dimensional image of the specimen, it is necessary to be able to combine data from a number of different views. A formal solution of this problem is given in terms of Fourier transforms. Its realization requires data reduction and interpolation. The final solution is given by a least squares approach, which also indicates how many views must be included to… 

Figures from this paper

The reconstruction of three-dimensional structure from projections and its application to electron microscopy II. Direct methods

  • P. Gilbert
  • Mathematics
    Proceedings of the Royal Society of London. Series B. Biological Sciences
  • 1972
This direct method is shown to be equivalent to the Fourier method and also to certain methods of reconstruction developed for the solution of analogous problems in other fields, and to be very suitable for reconstructing the density of objects with rotational or helical symmetry.

Fundamentals of three-dimensional reconstruction from projections.

Tomographic Reconstruction from Electron Micrographs

Reconstruction from a tilt series of electron micrographs is based on the assumption that each image represents a projection through the specimen, and that the 3D information can be recovered by

Algorithms for two-dimensional reconstruction.

It is pointed out that in reconstruction algorithms the use of all N points for each density calculation is unnecessary and generates a noise-like error in the reconstruction at all but a few points at which reconstruction of the density is made.

Three-dimensional reconstruction from a single oblique section of fish muscle M-band

An application of the method to determine the structure of the M-band of fish muscle agrees well with that produced by the more conventional approach involving tilted views, and can be achieved most conveniently by linear least squares filtering.

Three-Dimensional Reconstruction from Projections: A Review of Algorithms

The problem of reconstructing three-dimensional objects from a set of two-dimensional projected images has arisen and been solved independently in fields ranging from medicine and electron microscopy

Algorithms for reproducing objects from their X-rays

Cryo-electron-microscopy reconstruction of partially symmetric objects.

A procedure is described for reconstructing three-dimensional objects from two-dimensional projections based on the original Crowther, DeRosier, and Klug work on image reconstructions of icosahedral viruses and on the concept of noncrystallographic symmetry.

Three-Dimensional Reconstruction of Aperiodic Objects in Electron Microscopy

In X-ray crystallography it has become fully evident during the development of protein crystallography that the determination of molecular structures with a thickness of more than ca. 2 nm is not
...

References

SHOWING 1-6 OF 6 REFERENCES

Electron Microscopy of Unstained Biological Material: The Polytropic Montage

Preliminary results indicate that one may thus study unstained, unshadowed biological material at high resolution.

Three Dimensional Reconstructions of Spherical Viruses by Fourier Synthesis from Electron Micrographs

Methods are developed for computing three dimensional density maps from transmission electron micrographs using Fourier transforms, which gives clear reconstructions of their surface structures when applied to negatively stained spherical viruses.

A computer-linked cathode-ray tube microdensitometer for x-ray crystallography.

A versatile double-beam microdensitometer has been constructed in which the source of light is the spot of a high resolution cathode-ray tube and the optical density is read directly into the computer.

The detection of sub‐units within the crystallographic asymmetric unit

In this paper, Rossmann & Blow describe how they have detected the existence of partial, approximate symmetry from a knowledge of the intensities alone. The effect of noncrystallographic symmetry,

Diffraction by helical structures

T h e s y m m e t r y of hel ical s t r u c t u r e s a n d t he i r d i f f r ac t i on p a t t e r n s is d i scussed , a n d a l is t is g iven of t h e l ine g r o u p s for e n a n t i o m o r p