In X-ray radiography, inversion of Abel's integral equation is an essential step in the reconstruction of the radial optical density profile of a cylindrically symmetric object from its radiograph. The original inversion formulae developed by Abel a century and a half ago are not well suited for use on measured data, since they require differentiation which greatly amplifies inherent random measurement errors. Recent developments in the field are presented, including a derivative-free analytic inversion formula and a spline-based numerical inversion method. These methods were applied to the interpretation of radiographs of industrial products. Several practical problems were encountered such as the Gibbs phenomenon near sharp density changes in the object. Countermeasures developed to eliminate these problems are discussed and their use is demonstrated in the reconstruction of the density profile of a cylindrical ceramic fuse housing from a single radiograph.
- Abel's integral equation
- X-ray radiography
- radial optical density profile