An algebraic solution to the 3-D discrete tomography problem (original) (raw)

Discrete tomography is the problem of reconstructing a binary image defined on a discrete lattice of points from its projections at only a few angles. It has applications in X-ray crystallography, in which the projections are the number of atoms in the crystal along a given line, and nondestructive testing. The 2-D version of this problem is fairly well understood, and several algorithms for solving it are known, most of which involve discrete mathematics or network theory. However, the 3-D problem is much harder to solve. This paper shows how the problem can be recast in a purely algebraic form. This results in: (1) new insight into the number of projection angles needed for an almost surely unique solution; (2) non-obvious dependencies in projection data; and (3) new algorithms for solving.