A fast algorithm for the inversion of Abel's transform

TL;DR

This paper introduces a rapid, accurate algorithm for inverting Abel's transform using Fourier and Legendre coefficients, suitable for noisy data, with proven stability and demonstrated effectiveness.

Contribution

The paper presents a novel algorithm that leverages Fourier transforms to efficiently compute the inverse Abel transform, improving speed and accuracy over existing methods.

Findings

The algorithm achieves fast computation of the inverse Abel transform.

It maintains stability and accuracy even with noisy measurements.

Numerical experiments confirm the effectiveness of the method.

Abstract

We present a new algorithm for the computation of the inverse Abel transform, a problem which emerges in many areas of physics and engineering. We prove that the Legendre coefficients of a given function coincide with the Fourier coefficients of a suitable periodic function associated with its Abel transform. This allows us to compute the Legendre coefficients of the inverse Abel transform in an easy, fast and accurate way by means of a single Fast Fourier Transform. The algorithm is thus appropriate also for the inversion of Abel integrals given in terms of samples representing noisy measurements. Rigorous stability estimates are proved and the accuracy of the algorithm is illustrated also by some numerical experiments.