Abel transform

In mathematics, the Abel transform,^[1] 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

${\displaystyle F(y)=2\int _{y}^{\infty }{\frac {f(r)r}{\sqrt {r^{2}-y^{2}}}}\,dr.}$

Assuming that f(r) drops to zero more quickly than 1/r, the inverse Abel transform is given by

${\displaystyle f(r)=-{\frac {1}{\pi }}\int _{r}^{\infty }{\frac {dF}{dy}}\,{\frac {dy}{\sqrt {y^{2}-r^{2}}}}.}$

In image analysis, the forward Abel transform is used to project an optically thin, axially symmetric emission function onto a plane, and the inverse 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 transform (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 transform are the "onion peeling" and "basis set expansion" (BASEX) methods of photoelectron and photoion image analysis.