Derivative-free inversion of Abel’s integral equation

Abstract

We present a new inversion formula for Abel’s integral equation which does not require derivatives of any of the functions involved. This is a particularly desirable feature when analyzing experimentally derived data, since differentiation enormously amplifies the random errors inherent in all measured data. The high quality of the results obtainable using the new formula is demonstrated by a typical numerical example for which it yields errors smaller by an order of magnitude than those obtained with the usual inversion formula.

Orientational order determination in liquid crystals by x-ray diffraction

“…However, error amplification for experimentally derived I is considerably reduced [16,17] when using g2. Note also that I(P) is measured in a diffraction experiment.…”

Material Density Distribution of a Radial Symmetric Product from a Single X-Ray Radiograph

“…(1) For the inversion of the Abel transform, an approach based on piecewise cubic spline fitting [9,10] was used. At points of discontinuity of the linear attenuation distribution, the reconstructed image exhibits deviations due to the Gibbs phenomena.…”

Numerical Inversion of the Abel Integral Equation using Homotopy Perturbation Method

“…In fact, two explicit analytic inversion formulae were given by Abel [1], but their direct application amplifies the experimental noise inherent in the radiance data significantly [4]. In 1982, a third, analytic but derivative free, inversion formula was obtained by Deutsch and Beniaminy [5] to avoid this problem. In addition, many numerical inversion methods [5 -14] have been developed with varying degree of success with the inherent limitations of all measured data.…”