Impulses from Results of Inverse Problems to Discrete Tomography

Abstract

Abstract Extended Abstract This talk presents new theoretical and constructive-practical interpretations of discrete tomography in various contexts of inverse problems . Impulses to further research in discrete tomography are given within the wide range of definition drawn by R. J. Gardner. Gritzmann ([6]), and inspired by the author's half year stay at Mathematical Institute, Cologne, and in Working Group of U. Faigle P. R. Schrader at ZAIK. 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 d , 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 Is. 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 Os 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. For our impulses, we mainly come from nonlinear (continuous) optimization, especially from generalized semi-infinite optimization, from optimal control of ordinary differential equations and time-minimal control, and from continuum mechanics. Studying structure, global stability and design of algorithms, we observe many inverse ( “continuous → continuous ”) features but also a number of discrete items. Some examples: combinatorial numbers (Morse indices) give information about local descent in continuous optimization and control, expoiting intrinsic combinatorial information of these problems (settled in the set of inequalities) leads to problem reduction, k -determinacy in singularity theory, and optimization of architecture and flight of airplanes being reduced to optimization of a connectedness graph based on cell decomposition. Moreover, proofs of characterization theorems on structural stability of continuous optimization and control problems reveal an inverse aspect of reconstruction ([12, 16, 17]). This gives rise to investigate our “discrete → discrete ” tomographical subject by “nonlinear → discrete (coding)” impulses, but also by “discrete → nonlinear ” ones. Herewith, we think of lift, unfolding or homotopy from discrete dimension 0 to “continuous” dimensions > 1. Practical examples come from graph drawing, stability of graphs or atom clusters (incorporating analytical “pre-history”, simulating perturbations, or studying affine linear perturbations: [1]), reverse engineering, LP relaxation in combinatorial optimization, and from generalizing exact designs by approximative designs in statistics. The negative answer of F. Twilt to a question of S. Smale on the reconstructability of (discrete) Newton graphs by (continuous) Newton flows demonstrates structural frontiers in the “discrete → nonlinear” field ([18]). The talk concludes by first results on three recent and future research lines in discrete tomography ([19]): (i) Wavelets detect roughness in layer structures on chips ([7]). (ii) Representing any supposed atoms cluster within the grid box as a single word over {0, 1}, the theory of linear codes helps by error detection and correction. Here, we look for Hamming codes or ask for the extent of cyclicity ([11]). (iii) Further invariance and equivariance information about the atoms distribution can be extracted by using optimal experimental design from statistics ([3, 4], cf. also [2]). This leads to dimensional reduction of our measurements. Finally, semidefinite and nonlinear integer programming problems have to be resolved in this probabilistic approach. For (i)- (iii) we present theorems, illustrations, numerical examples and extensions to data reconstruction, e.g., in economy, and to traffic problems studied, e.g., at ZAIK.