Uniqueness and Complexity in Discrete Tomography

We study the discrete inverse problem of reconstructing finite subsets of the n-dimensional integer lattice ℤn that are only accessible via their line sums (discrete X-rays) in a finite set of lattice directions. Special emphasis is placed on the question of when such sets are uniquely determined by the data and on the difficulty of the related algorithmic problems. Such questions are motivated by demands from the material sciences for the reconstruction of crystalline structures from images produced by quantitative high-resolution transmission electron microscopy.

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