The inverse problem of radiography

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Abstract

Radiography is a common diagnostic tool in the field of medicine. The physical process by which a radiograph is formed imposes fundamental limits on the resolution of a radiograph, however. The problem of increasing the resolution of a radiograph can be formulated as the solution of a convolution-type integral equation, but solving this integral equation is extremely difficult if noise is present in the data.

A technique for solving integral equation problems was discovered several years ago, but it requires matrix inversion and is unwieldy for data sequences of hundreds or thousands of points. In this article it is shown how this existing technique can be adapted to solving convolution-type integral equations. By using the convolution properties of discrete Fourier transforms and the fast Fourier transform, it is possible to solve convolution-type integral equations even when the data sequences consist of hundreds or thousands of points.

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