Optical realization of the Radon Transform 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?
I actually ran into this same paper and had the same problem.

The paper linked below does a good job addressing the issue. Starting on page 13 the discuss the issue with this optical element.

https://mstamenk.github.io/assets/files/OpticalRadon.pdf