Abel Transform - Geometrical Interpretation

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

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 via r2 = x2 + y2, it follows that

The path of integration in r does not pass through zero, and since both f(r) and the above expression for dx are even functions, we may write:

Substituting the expression for dx in terms of r and rewriting the integration limits accordingly yields the Abel transform.

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

$F(y,z) =\int_{-\infty}^\infty f(\rho,z)\,dx =2\int_y^\infty \frac{f(\rho,z)\rho\,d\rho}{\sqrt{\rho^2-y^2}}$

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:

$F(s) =\int_{-\infty}^\infty f(r)\,dx =2\int_s^\infty \frac{f(r)r\,dr}{\sqrt{r^2-s^2}}$

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

Read more about this topic: Abel Transform