# Inverse Radon transfrom with Riesz potentials

The Fourier inversion formula simply lead us the filtering with |r| in the frequency domain and its alternative formulation with the differential and Hilbert transform operators. Using the Riesz potentials we can give a family of solutions where the Fourier inversion formula is just one of the members.

The Riesz-potential can be defined of a function of n-dimensions as follows:

Here is a constant. Similarly to the Hilbert transform we can give a definition using the Fourier transform:

Here .
From this definition we can see

Let us write the Riesz-potential of f with the Fourier definition:

Let us change variables to an n-dimensional polar coordinate system with notations:

Let us fill in the result of the multidimensional Central Slice Theorem:

With operator notations we can write:

On both sides calculating the potentials:

and from that several inversion formulae can be obtained.

In the case of we get back the earlier shown inversion formula. It is worthwhile to highlight the case of now

which at least partially reinstates locality of the inversion formula.