in *Transport Theory, Invariant Imbedding, and Integral Equations, Lect. Notes in Pure and Appl. Math.* **115**, P. Nelson et al., eds., pp. 363-368 (Marcel Dekker, New York, 1989)

## A Bayesian approach to nonlinear inversion: Abel inversion from x-ray data

Kenneth M. Hanson

*Los Alamos National Laboratory*
### Abstract

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.

**Keywords:** Bayesian methods, tomographic reconstruction, Abel inversion, x-ray data

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Send e-mail to author at kmh@hansonhub.com