# Filter Design for the Filtered Backprojection

The formula for filtered backprojection can be generalized in a more practical way than was shown with the Riesz potentials, now we will look at the convolution form. Let the Radontransform in n dimensions. Let us prove that

where the indices to the convolution sign * indicate the variables of the convolution.
The LHS convolution with the our usual notations:

Let us replace variable y with , here z is perpendicular to -ra. Then inserting:

Let us now choose the following V and v functions:

Inserting:

Let us look for such V functions that in a given band limit approximate the Dirac delta function so the original f function would be restored. It can be proven that

thus, if the Fourier transform of Vis a constant within the bandlimits, for v a family of filters can be designed. If we only allow for radial dependence of the filters we get our previous filtered backprojection formulas.