Abstract

The Abel transform and its inverse appear in a wide variety of problems in which it is necessary to reconstruct axisymetric functions from line-integral projections. We present a new family of algorithms, principally for Abel inversion, that are recursive and hence computationally efficient. The methods are based on a linear, space-variant, state-variable model of the Abel transform. The model is the basis for deterministic algorithms, applicable when data are noise free, and least-squares-estimation (Kalman filter) algorithms, which accommodate the noisy data case. Both one-pass (filtering) and two-pass (smoothing) estimators are considered. In computer simulations, the new algorithms compare favorably with previous methods for Abel inversion.

© 1985 Optical Society of America

Space-domain inversion of the incomplete Abel transform

Eric W. Hansen
J. Opt. Soc. Am. A 9(12) 2126-2137 (1992)

Modified Fourier-Hankel method based on analysis of errors in Abel inversion using Fourier transform techniques

Shuiliang Ma, Hongming Gao, and Lin Wu
Appl. Opt. 47(9) 1350-1357 (2008)

Abel inversion method using Chebyshev wavelets for tomographic reconstruction of an axisymmetric medium

Mahfoud Elfagrich, Abdessadek Ait Haj Said, and Kamal El Fagrich
Appl. Opt. 60(21) 6067-6075 (2021)

Improved methodology for performing the inverse Abel transform of flame images for color ratio pyrometry

Jochen A. H. Dreyer, Radomir I. Slavchov, Eric J. Rees, Jethro Akroyd, Maurin Salamanca, Sebastian Mosbach, and Markus Kraft
Appl. Opt. 58(10) 2662-2670 (2019)

Algorithms for reconstruction of partially known, band-limited Fourier-transform pairs from noisy data

Richard Barakat and Garry Newsam
J. Opt. Soc. Am. A 2(11) 2027-2039 (1985)