# Interpretation of the inverse Radon transform

#### Analysis of the Fourier inversion formula Let us look at the terms of the inversion formula again.

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.