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
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.