Discrete Tomography of Model Sets: Reconstruction and Uniqueness

13. On the reconstruction of static and dynamic discrete structures

“…There are, however, certain aspects which involve weights on the points of F . Hence we will introduce the basic notions for appropriate generalizations of characteristic functions of finite point sets, partly following [24].…”

“…There exist already books and articles, which give detailed accounts of various aspects of discrete tomography and its applications. We single out [20,24,25,27,35,36,51]. The present article differs from these surveys in various ways: the mathematical focus will be on recent developments (which have not been covered in previous surveys).…”

“…As a service to the reader and with a view towards a more complete picture we will restate some aspects which are basic for the present article but have been covered before. In order to limit the overlap to other surveys we will, however, neither elude on the tomographic reconstruction of quasicrystals (see [24]) or polyominoes (see [17]), nor on the polyatomic case (see [21,92]) or point X-rays (see [89]). Also, we will not study general k-dimensional X-rays (see [24,101]) but concentrate on the case k = 1.…”

“…In order to limit the overlap to other surveys we will, however, neither elude on the tomographic reconstruction of quasicrystals (see [24]) or polyominoes (see [17]), nor on the polyatomic case (see [21,92]) or point X-rays (see [89]). Also, we will not study general k-dimensional X-rays (see [24,101]) but concentrate on the case k = 1. This means that our exposition will be based on the X-ray transform rather than on the Radon transform (which is the case k = d − 1).…”

On Polyatomic Tomography over Abelian Groups: Some Remarks on Consistency, Tree Packings and Complexity

“…Since then, a rich theory has been developed which is centered around the task of determining (theoretically and algorithmically) discrete structures by information about their interaction with certain query sets. For surveys see [1,10,12], and other sources quoted there. More than half a century after Ryser's work discrete tomography is a well-established field with seminal results reaching into combinatorics, geometry, geometry of numbers, commutative algebra, optimization and other fields.…”