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 Assuming that 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 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.

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.

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.

Since f(r) is an even function in x, we may write which yields the Abel transform of f(r).

The Abel transform may be extended to higher 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.

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

to find Differentiating formally, Now substitute this into the inverse Abel transform formula: By Fubini's theorem, the last integral equals Consider the case where

is related to the spatial distribution of terminal, non-tethered monomers of the polymers.

The extended version of the Abel transform for discontinuous F is proven upon applying the Abel transform to shifted, continuous

has more than a single discontinuity, one has to introduce shifts for any of them to come up with a generalized version of the inverse Abel transform which contains n additional terms, each of them corresponding to one of the n discontinuities.

The Abel transform is one member of the FHA cycle of integral operators.

As f(r) is isotropic, its Radon transform is the same at different angles of the viewing axis.

Thus, the Abel transform is a function of the distance along the viewing axis only.

A geometrical interpretation of the Abel transform in two dimensions. An observer (I) looks along a line parallel to the x axis a distance y above the origin. What the observer sees is the projection (i.e. the integral) of the circularly symmetric function f ( r ) along the line of sight. The function f ( r ) is represented in gray in this figure. The observer is assumed to be located infinitely far from the origin so that the limits of integration are ±∞.