Abel transform
In mathematics, the Abel transform, named for Niels Henrik Abel, is an integral transform often used in the analysis of spherically symmetric or axially symmetric functions. The Abel transform of a function f(r) is given by:
Assuming f(r) drops to zero more quickly than 1/r, the inverse Abel transform is given by
In image analysis, the forward Abel transform is used to project an optically thin, axially symmetric emission function onto a plane, and the reverse Abel transform is used to calculate the emission function given a projection (i.e. a scan or a photograph) of that emission function.
In absorption spectroscopy of cylindrical flames or plumes, the forward Abel transform is the integrated absorbance along a ray with closest distance y from the center of the flame, while the inverse Abel transform gives the local absorption coefficient at a distance r from the center. Abel transform is limited to applications with axially symmetric geometries. For more general asymmetrical cases, more general-oriented reconstruction algorithms such as Algebraic Reconstruction Technique (ART), Maximum Likelihood Expectation Maximization (MLEM), Filtered Back-Projection (FBP) algorithms should be employed.
In recent years, the inverse Abel transformation (and its variants) has become the cornerstone of data analysis in photofragment-ion imaging and photoelectron imaging. Among recent most notable extensions of inverse Abel transformation are the Onion Peeling and BAsis Set Expansion (BASEX) methods of photoelectron and photoion image analysis.
In two dimensions, the Abel transform F(y) can be interpreted as the projection of a circularly symmetric function f(r) along a set of parallel lines of sight which are a distance y from the origin. Referring to the figure on the right, the observer (I) will see
where f(r) is the circularly symmetric function represented by the gray color in the figure. It is assumed that the observer is actually at x = ∞ so that the limits of integration are ±∞ and all lines of sight are parallel to the x-axis. Realizing that the radius r is related to x and y via r2 = x2 + y2, it follows that
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