Abel Transform and Integral Geometry

In 1917 Radon published his famous paper [26] where he defined the operator which we call today the Radon transform. I already had a chance to write about this very significant paper [17]. Let us remind that Radon considers the problem of the reconstruction of a function of 2 variables f(x, y) through its integrals F(l) along all lines l. For this reconstruction he remarks that this operator f ↦F (the Radon transform) commutes with all Euclidean motions and therefore it is sufficient to find a way to reconstruct in one point (let say on (0, 0)) a function with radial symmetry in this point. If we have such a function ϕ(r), r ≥ 0, then its Radon transform will also be symmetric relative to rotations and depend only on the distance q of a line l from (0, 0) and we have a transform of functions of one variable ϕ(r) ↦ Φ(q). It is simple to compute this transform explicitly: $$ \varphi (q) = 2\int_0^\infty {\frac{{\phi (r)dr}} {{\sqrt {r^2 - q^2 } }}.} $$