One-dimensional tomography: a comparison of Abel, onion-peeling, and filtered backprojection methods

It is shown that the Abel inversion, onion-peeling, and filtered backprojection methods can be intercompared without assumptions about the object being deconvolved. If the projection data are taken at equally spaced radial positions, the deconvolved field is given by weighted sums of the projections divided by the data spacing. The weighting factors are independent of the data spacing. All the methods are remarkably similar and have Abelian behavior: the field at a radial location is primarily determined by the weighted differences of a few projections around the radial position. Onion-peeling and an Abel inversion using two-point interpolation are similar. When the Shepp-Logan filtered backprojection method is reduced to one dimension, it is essentially identical to an Abel inversion using three-point interpolation. The weighting factors directly determine the relative noise performance: the three-point Abel inversion is the best, while onion peeling is the worst with approximately twice the noise. Based on ease of calculation, robustness, and noise, the three-point Abel inversion is recommended.