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

Abstract

Under certain conditions it is possible to obtain the approximate projected density of a specimen from an electron micrograph, and by taking electron micrographs in different directions a number of different projections of a specimen can be determined. Several methods have been proposed for reconstructing the three-dimensional density distribution of an object from a series of projections whose planes are equally spaced in angle about an axis. One of these methods uses the Fourier transforms of the projections, while other methods operate directly on the projection data. The limitations of the direct method of ‘back-projection’ are discussed, and it is shown how a valid reconstruction may be achieved by back-projection from modified projected densities. 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. An expression is given for the resolution of the reconstruction obtainable from a finite number of projections. Because of its computational simplicity this method is very suitable for reconstructing the density of objects with rotational or helical symmetry.