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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 is given by
Assuming that f 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 inverse Abel transform is used to calculate the emission function given a projection 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 , maximum likelihood expectation maximization , filtered back-projection algorithms should be employed.