# 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} }

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…

## 939 Citations

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

- MathematicsProceedings 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.

### Computation of a three dimensional image of a periodic specimen from a single view of an oblique section.

- GeologyElectron microscopy reviews
- 1990

### Fundamentals of three-dimensional reconstruction from projections.

- MathematicsMethods in enzymology
- 2010

### Tomographic Reconstruction from Electron Micrographs

- Mathematics
- 2018

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.

- PhysicsPhysics in medicine and biology
- 1977

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

- PhysicsNature
- 1984

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

- PhysicsOther Conferences
- 1975

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…

### Cryo-electron-microscopy reconstruction of partially symmetric objects.

- MathematicsJournal of structural biology
- 1999

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

- Chemistry
- 1979

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…

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