The discrete Radon transform and its approximate inversion via linear programming

Abstract

Let S be a finite subset of a lattice and let v s ( 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 v s 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. We address this problem by looking instead for a fuzzy set S with the given line sums, i.e. a function f( z) with 0 ⩽ f( z) ⩽ 1 for all points z in the lattice, for which Vf( L)  v s ( L). The set of all such f forms a convex set and those f = χs with each f( z) ϵ {0, 1} are extreme points. Finding a fuzzy set f with the given line sums is a linear programming problem and so there are efficient algorithms for finding f or proving that no such f (and hence no S) exists with the given line sums. If S is an additive set with respect to L , i.e. if we can write S = { z ∑ LϵLϵz g( L) > 0} for some functional g on L , we show that there is only one fuzzy set f with the given line sums. We prove here that if S is not additive then there are many fuzzy sets with the given line sums, although there still may be only one actual set. Linear programming methods that are based on interior point methods always produce solutions that lie in the center of the convex set of all solutions. As a result, if S is a set with given line sums and linear programming produces a solution that consists only of {0, 1} then this solution must be the original subset S, and S must be a set of uniqueness. Thus interior point LP's give a polynomial and practical way to obtain the assertion of uniqueness when strong uniqueness obtains. This problem arises in a practical situation, although it was earlier studied in the case of coordinate directions for its intrinsic interest. In the practical situation, S represents a piece of a real crystal, and the line sums in any fixed direction can be measured (possibly with uncertainties) using a transmission electron microscope. In order to study the behavior of the linear programming reconstruction algorithm and to explore the question of whether a “typical” set S with a “typical” L will be a case of uniqueness or near-uniqueness, we conducted simulation experiments. These indicate that the method is practical and reasonably efficient, at least on the examples we considered.