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

TLDR

Transmission electron micrographs are projections of specimens, so reconstructing a 3‑D image requires combining data from multiple views. The reconstruction is formulated via Fourier transforms and solved by a least‑squares approach, with data reduction, interpolation, and a view‑number criterion, while a non‑Fourier direct method is also described but found less practical. The least‑squares solution specifies the required number of views for a desired resolution, and the alternative non‑Fourier method is shown to be less practical.

Abstract

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 give a valid reconstruction of a given particle to a given degree of resolu­tion. Interpolation procedures of varying power are given, to be employed according to the economy with which the available data must be used. An alternative procedure is described for direct reconstruction without the use of Fourier transforms, but it is shown to be in general less practicable than the Fourier approach.