Relation to Discrete Tomography

While there were certainly several results prior to 1994 that can be classified as discrete tomography, a term introduced by Larry Shepp, the subject took off that year at a mini-symposium Shepp organized at DIMACS. The focus of discrete tomography is on the reconstruction of lattice sets, that is, finite subsets of the integer lattice Z\(^n\), from their discrete X-rays parallel to lattice directions (i.e., directions parallel to vectors with integer coordinates).

Lattice sets are natural models for atoms in a crystal, and the original motivation for discrete tomography came from a genuine application in high-resolution transmission electron microscopy (HRTEM). Indeed, the DIMACS meeting was inspired by work of a physicist, Peter Schwander, whose team developed a technique that effectively allows the discrete X-rays of a crystal to be measured in certain lattice directions parallel to vectors with small integer coordinates. The process is described in detail by Schwander [36]. The algorithms of computerized tomography are useless, since the high energies required to produce the discrete X-rays mean that no more than about three to five discrete X-rays of a crystal can be taken before it is damaged. Discrete tomography has been a very active field since its inception, due to attempts to find suitable algorithms and to the stimulus provided by applications to crystallography and other areas, such as data compression and data security. The books [20] and [21] edited by Herman and Kuba provide an overview of discrete tomography.

Shepp's original vision was that discrete tomography should be concerned with the reconstruction of functions defined on Z\(^n\) from their integrals along lattice lines, i.e., lines passing through at least two points of Z\(^n\). This definition allows a clear view of geometric and discrete tomography as overlapping but distinct subjects. Later, Herman and Kuba [21, p. 18] attempted to widen the scope. In their view, discrete tomography is concerned with determining an unknown function \(f\) with a discrete range from weighted sums over subsets of its domain (if the latter is discrete) or from weighted integrals over subspaces (sic) of its domain (if the latter is continuous). According to private communication from Herman and Kuba, this definition was concocted mainly to encompass as many applications as possible. Unfortunately, shifting the emphasis from a discrete domain to a discrete range allows many situations that have essentially nothing to do with discrete mathematics or methods. For example, the problem of determining a star body (or equivalently, its characteristic function) from its section function would become part of discrete tomography! It is more appropriate either to adopt Shepp's definition of discrete tomography, or to generalize it as Herman and Kuba do provided the domain of \(f\) is required to be discrete.