The discrete Radon transform and its approximate inversion via linear programming Original Research Article

Let S be a finite subset of a lattice and let vs(L), the number of points of S ∩ L for each line L, denote the discrete Radon transform of S. The problem is to reconstruct S from a knowledge (possibly noisy) of the restriction of vs to a subset L of the set of all lines in any of a few given directions through the lattice. Reconstructing a density from its line integrals is a well-understood problem, but discreteness causes many difficulties and precludes use of continuous Radon inversion algorithms. Indeed it has been shown that when the directions are main directions of the lattice, the case for most applications, the problem is finite but is NP-hard, so that any reconstruction algorithm will surely have to consist of exponentially many steps in the size of S.