Three-dimensional Reconstruction from Radiographs and Electron Micrographs: Application of Convolutions instead of Fourier Transforms

Abstract

A new technique is proposed for the mathematical process of reconstruction of a three-dimensional object from its transmission shadowgraphs; it uses convolutions with functions defined in the real space of the object, without using Fourier transforms. The The first step in the solution of the problem consists of replacing the three-dimensional (optical) density distribution by a set of two-dimensional density functions in a series of sections perpendicular to, say, the z-axis, as shown in Fig. 1. The object is placed in a parallel beam of radiation, incident normal to the z-axis, and shadowgraphs (in two dimensions) are obtained by rotating the object about the z-axis, which is also equivalent to rotating the imaging system, consisting of the source and the recorder, through different angles 0 (Fig. 1). As shown in the figure, each shadowgraph may be considered to be made up of linear strips, each strip corresponding to a different section perpendicular to the z-axis. For such a strip, corresponding to the section at z, the logarithm of the ratio of the intensity at a point P(z, 1) to the incident intensity is a measure of the integral of the (optical) density of the object along a line through the point parallel to the direction of incidence of the beam [2]. We shall call this function g(1; 0; z) * for the shadowgraph of the section at the setting 0. (2) for each section z. We thus obtain the data f(r, sp; z) using all the linear strips at z in the different shadowgraphs taken at various angles 0. The three-dimensional density distribution f(r, sp, z) in the object is then obtained in cylindrical polar coordinates by putting the above sections together. Thus, the problem reduces to the reconstruction of a series of twodimensional (planar) sections from a set of one-dimensional (linear) shadowgraphs. For this reason, we shall hereafter restrict our discussion essentially to one section of the object at right angles to the rotation axis.* The semicolons are used to indicate that g is measured as a function of 1 only, but depends also on the parameters 0, describing the angular setting, and z, defining the section perpendicular to the rotation axis. When the parameters are considered explicitly as variables, the semicolons are replaced by commas. 2236Abbreviations: F.T. = Fourier transform; FTP = polar coordinate Fourier transform method; FTC = Cartesian coordinate Fourier series method; CON = convolution method, newly proposed.

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