The reconstruction of the radial density distribution of a cylindrically symmetric object is a common task in different area of physics (e.g. plasma physics). Typically, optical measurements of objects like plasma columns or flames are integrated along the line of sight.
To obtain the underlying distribution from a measured projection, the inverse Abel transform has to be calculated.
This submission provides a Fourier-based algorithm which extracts the radial (2D) distribution from a one-dimensional projection measurement. The algorithm has been proposed and published by G. Pretzler (Z. Naturforsch. 46a, 639 (1991)). Compared to earlier approaches towards Abel inversion, this algorithm is relatively insensitive to noise and errors in the determination of the object's center (see G. Pretzler et al. , Z. Naturforsch. 47a, 955 (1994)).
The fundamental idea is to fit the whole measured profile to a set of cos-expansion-based integrals. (In the conventional approach, in contrast, the radial distribution is obtained by starting at the edges and incrementally iterating towards the center - making it more prone to errors.)
UPDATE: There have been some questions on why the Abel inversion result of generic test data seems to be off by a factor 2. I found a simple explanation: The algorithm assumes that the measurement of line-of-sight-integrated data starts on the center axis of your circular object.
In most experimental situations, however, the measurement takes place along the complete object, effectively doubling the optical path length. Therefore, if you want to analyze experimental data obtained by observing circular (cylindrical) objects, please divide the result F_REC by 2 to yield the right result.