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:

Related Topics:
Mathematics - Niels Henrik Abel - Integral transform

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:F(y)=2int_y^infty rac{f(r)r,dr}{sqrt{r^2-y^2}}.

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

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:

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f(r)=-rac{1}{pi}int_r^inftyrac{d F}{dy},rac{dy}{sqrt{y^2-r^2}}.

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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.

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