Noise-filtering properties of numerical methods for the inverse Abel transform

The noise-filtering properties of polynomial interpolation-based numerical methods for computing inverse Abel transforms are studied. It is shown that, when sample spacing allows the use of the modified Abel transform, the methods are all in the form of serial products (discrete convolutions). Impulse responses and power spectra are used to show that the amplification of a single perturbation is proportional to the square root of the sampling rate and that the amplification of the variance of normal zero-mean white noise is proportional to the sampling rate. It is also shown that increasing the degree of the interpolating polynomial leads to more smoothing in methods based on the inverse transform, but to less smoothing in methods based on an inversion of the forward transform. These results are supported by a numerical example