Well, it can be done without too much trouble (it needs a bit of coding).
The direct Radon transform is equivalent to the Hough Transform (you may use the module I wrote, and use it with a small enough step size for the angle and radius).
To calculate the inverse Radon transform, more work is needed. If you take the 1D Fourier Transform in the radius axis (for each line that defines a different angle), then the result is equal to a radial profile of the 2D Fourier Transform of the image. You just have to map the 1D FFT of the Radon (Hough) transform into a cartesian coordinate system, and finally, take the inverse FFT. This is quite fast, but some artifacts may arise from the mapping process.
Another way to solve this is to use a filtered back-projection. This is fast enough and conceptually easy but is prone to artifacts and noise amplification.
Anyhow, the inverse Radon transform is an ill-posed problem. Noise amplification and artifacts are to be expected. Iterative processes may yield much better results (for example using TGV as regularizator) but are significantly slower.
I would say that there are better ways to solve both of these problems than the Radon transform.
