**Interpretation of the inverse Radon transform**

## Analysis of the Fourier inversion formula

Let us look at the terms of the inversion formula again.

#### The adjoint to the Radon-transform

For the outermost integral we have already introduced the notation of . To an operator the definition of the adjoint operator reads:

here <*f*,*g*> is the scalar product of *f* and *g* . The adjoint to the Radon transform is the backprojection operator:

Here we have changed variables from the original t and **y** to the rotated **x** variables in the integration.

#### The Hilbert transform hidden in the Radon inversion formula

In the next step let us look at the terms disregarding the backprojection operator:

In general it is true to a funcion *g* that

We have shown about the Hilbert Transform that

from this:

As the sgn function is present, the formula

behaves differently depending on whether n is even or odd.

Now we can also write the odd and even terms separately:

The lack of the Hilbert transform in the even dimensions have a fundamental impact:

- when the Hilbert transform is missing from the formula, for the reconstruction of a certain point of the distribution we need the Radon transform on hyperplanes going through only the small neighborhood of the point
- when the Hilbert transform is present, for the reconstruction at a point we need the whole sinogram

Note that at n=2 we obtain Radon's inversion formula. There are further analytical solutions to the inverse Radon problem, we will be dealing with that in the next section.