Impulses from Results of Inverse Problems to Discrete Tomography

A practical motivation of our inverse problem from discrete tomography comes from quality control in VLSI design. It demands a homogeneous crystaline layers consisting, e.g., of silicon. Homogenity “means non-roughness” of a considered atom cluster which is discrete and three dimensional. Scientists such as Gardner and Gritzmann ([5]) are interested in carrying over mathematical methods from high-resolution transmission electron microscopy, in order to obtain information about the atoms distribution. We give the following model description: Suppose a set of atoms located on a chip. We want to measure the distribution of the atoms (represented as balls in a lattice) by “shooting” parallel electronic beams and recording the reverse “X-rays” at hyperplanes. How many directions of beams do we need, how to choose them? We embed our interest in three dimensions into the general case of finite dimensions d. The special situation of “convex” sets is comparatively well-understood; so, four suitable directions are enough for clusters in Z–Z , d = 2 ([13]). The “nonconvex” situation, however, is much harder. Some authors represent a given or not given atom at some position by 0, 1 (or: blank, pixel, respectively) such that projection in reverse direction means addition of 1s. This case is of basic importance, because higher dimensional cases can be arranged in a rectangular planar way. Moreover, provided these sums being arranged as line or column sums of a binary matrix we are guided to allocation like combinatorial problems. However, the choice of k directions remains a delicate matter which can expoit any knowledge where 0s are lying. For a recent approximative algorithm in binary tomography and to literature on previous ones we refer to [8]. In the following, we look at this practical motivation of concluding from discrete measurements to the unknown discrete cluster as an inverse problem of “discrete → discrete” type.