Inversion of Abel’s integral equation for experimental data

Abstract

A stable high-accuracy method for calculating the inverse of Abel’s integral equation for experimentally derived data is presented. The method employs a piece-wise cubic spline function, least-squares fitted to the data, to represent the function as inverted. Three formulas, two of which are based on the well-known analytic inverses of Abel’s equation and a new one, which was recently developed by the authors and does not contain an explicit derivative, are given for calculating the inverse numerically. The results of numerical tests performed using these formulas on simulated data are presented and compared with the results obtained using other published methods. The comparative study indicates that our method, employing piece-wise least-squares cubic splines, accurately reproduces the inverse function, regardless of the inversion formula employed. It yields markedly better results than do all the other methods compared in this study, when highly error free, sparse, or very noisy data are inverted. The errors at the ends of the data interval (‘‘termination errors’’) are also shown to be smaller than those of the other methods compared in this study.

Methods for obtaining true particle size distributions from cross section measurements

“…Methods employing the inverse to the Abel integral equation generally attempt to neutralize this intensification, but never completely eradicate it [27].…”

“…Blödner used a method in which  D (x) was estimated using a sample problem, its inverse intensifies the errors associated with the apparent distribution [27].…”

Etude expérimentale de bulles de vapeur générées par laser

“…Les déplacements des franges sont mesurés sur un diamètre, perpendiculairement à la direction de la frange non perturbée. La variation de l'indice de réfraction est déduite de la différence de marche par une méthode d'inversion d'Abel [7].…”

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.…”