Abel transform

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

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

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

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.

Geometrical interpretation

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 at a distance y from the origin. Referring to the figure on the right, the observer (I) will see

${\displaystyle F(y)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f\left(\sqrt{x^2+y^2}\right)dx,}$

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 as r² = x² + y², it follows that

${\displaystyle dx=\frac{rdr}{\sqrt{r^2-y^2}}}$

for x > 0. Since f(r) is an even function in x, we may write

${\displaystyle F(y)=2{\int }_0^{\mathrm{\infty }}f\left(\sqrt{x^2+y^2}\right)dx=2{\int }_{|y|}^{\mathrm{\infty }}f(r)\frac{rdr}{\sqrt{r^2-y^2}},}$

which yields the Abel transform of f(r).

The Abel transform may be extended to higher dimensions. Of particular interest is the extension to three dimensions. If we have an axially symmetric function f(ρ, z), where ρ² = x² + y² is the cylindrical radius, then we may want to know the projection of that function onto a plane parallel to the z axis. Without loss of generality, we can take that plane to be the yz plane, so that

${\displaystyle F(y,z)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(\rho ,z)dx=2{\int }_y^{\mathrm{\infty }}\frac{f(\rho ,z)\rho d\rho }{\sqrt{{\rho }^2-y^2}},}$

which is just the Abel transform of f(ρ, z) in ρ and y.

A particular type of axial symmetry is spherical symmetry. In this case, we have a function f(r), where r² = x² + y² + z². The projection onto, say, the yz plane will then be circularly symmetric and expressible as F(s), where s² = y² + z². Carrying out the integration, we have

${\displaystyle F(s)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(r)dx=2{\int }_s^{\mathrm{\infty }}\frac{f(r)rdr}{\sqrt{r^2-s^2}},}$

which is again, the Abel transform of f(r) in r and s.