, Volume 42, Issue 4, pp 740-758
Date: 17 Jul 2009

Uniqueness in Discrete Tomography of Delone Sets with Long-Range Order

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Abstract

We address the problem of determining finite subsets of Delone sets Λ⊂ℝ d with long-range order by X-rays in prescribed Λ-directions, i.e., directions parallel to nonzero interpoint vectors of Λ. Here, an X-ray in direction u of a finite set gives the number of points in the set on each line parallel to u. For our main result, we introduce the notion of algebraic Delone sets Λ⊂ℝ2 and derive a sufficient condition for the determination of the convex subsets of these sets by X-rays in four prescribed Λ-directions.

The author was supported by the German Research Council (Deutsche Forschungsgemeinschaft), within the CRC 701, and by EPSRC via Grant EP/D058465/1.