I am trying to reproduce this paper concerning the Optical realization of the Radon Transform, especially the simulation (section 3). Shortly, the experiment is just a "Fourier filtering" with a 4F system.
In the paper, they propose to use an optical element defined as $$T(r,\theta)=\frac{e^{2\pi i\theta r \beta }}{r}\text.$$
I tried this and it doesn't give the correct output. I think this is due to the fact that the output should be a circle representing the Radon transform in cartesian coordinates, meaning a polar symmetric element and the optical element obviously isn't, at least in the standard definition of polar coordinates (centered at 0).
Then I tried to change the optical element with respect to the argument mentioned above:
$$T(r,\theta)=\frac{e^{2\pi i r \beta }}{r}\text,$$
getting rid of the $\theta$ in the exponential so my element is now perfectly symmetric (as in Fig. 5 of the paper). The result:
It represents the Radon transform in cartesian coordinates of a Shepp-Logan phantom (240x240 pixels). $\beta$ controls the radius of the circle and is tuned to match the mathematical representation (reference). In image it looks like this:
As you see, the image is reconstructed but there's a problem concerning the amplitude. I do believe the problem can come from 2 parameters.
First, I should maybe use $\theta$ somewhere, but when I generate it using atan2 it doesn't give the correct result.
Second, change the way I define $r=\sqrt{x^2+y^2}$ in order to match the definition of polar representation used for the usual Radon transform (r being the diagonal of the picture taking information twice and centering it).
- Has anyone ever encountered a similar problem in image processing, amplitude missing?
- Or maybe is someone an expert in Radon transform (computed tomography or such) and knows how to generate the proper polar representation?
- Or does someone know what I should try next?
