Filtered back-projection

Summary

Filtered back-projection is the standard method to reconstruct a function in the plane from its Radon transform. It has two steps. First each projection is convolved with a filter whose Fourier transform is close to $\abs{\sigma}$. Then the filtered projections are fed into the dual Radon transform. With the exact ramp filter it is the Radon inversion formula in dimension $2$. Practical filters cut off high frequencies, and the method then reconstructs a smoothed version of the function. It was for a long time the main algorithm of computed tomography scanners.