A Bayesian Approach to Nonlinear Inversion: Abel Inversion from X-Ray Attenuation Data

With a change of variables the Abel transform gives the projection of a 2-D distribution with circular symmetry. Thus the Abel inversion formula allows one to determine the radial dependence of such a distribution from its projection. However, this inversion formula is very sensitive to noise in the projection data. When the projection data are derived from radiographic measurements, further difficulties arise from the necessity to invert the exponential dependence of the measured x-radiation intensity upon material thickness. These difficulties are shown to be overcome by applying a maximum a posteriori (MAP) method, which was developed to accomodate nonlinear measurements, to this tomographic reconstruction problem. The MAP method yields a smooth solution in regions where the signal-to-noise ratio is low while maintaining good spatial resolution in regions where it is high.