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

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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 object is rotated about an axis at right angles to the direction of a parallel beam of radiation, and sections of it normal to the axis are reconstructed from data obtained by scanning the corresponding linear strips in the shadowgraphs at different angular settings. Since the formulae in the convolution method involve only summations over one variable at a time, while a two-dimensional reconstruction with the Fourier transform technique requires double summations, the convolution method is much faster (typically by a factor of 30); the relative increase in speed is larger where greater resolution is required. Tests of the convolution method with computer-simulated shadowgraphs show that it is also more accurate than the Fourier transform method. It has good potentialities for application in electron microscopy and x-radiography. A new method of reconstructing helical structures by this technique is also suggested.

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