Abstract
A main problem in discrete tomography consists in looking for theoretical models which ensure uniqueness of reconstruction. To this, lattice sets of points, contained in a multidimensional grid \(\mathcal{A}=[m_1]\times[m_2]\times\dots \times[m_n]\) (where for p ∈ ℕ, [p] = {0,1,...,p − 1}), are investigated by means of X-rays in a given set S of lattice directions. Without introducing any noise effect, one aims in finding the minimal cardinality of S which guarantees solution to the uniqueness problem.
In a previous work the matter has been completely settled in dimension two, and later extended to higher dimension. It turns out that d + 1 represents the minimal number of directions one needs in ℤn (n ≥ d ≥ 3), under the requirement that such directions span a d-dimensional subspace of ℤn. Also, those sets of d + 1 directions have been explicitly characterized.
However, in view of applications, it might be quite difficult to decide whether the uniqueness problem has a solution, when X-rays are taken in a set of more than two lattice directions. In order to get computational simpler approaches, some prior knowledge is usually required on the object to be reconstructed. A powerful information is provided by additivity, since additive sets are reconstructible in polynomial time by using linear programming.
In this paper we compute the proportion of non-additive sets of uniqueness with respect to additive sets in a given grid \(\mathcal{A}\subset \mathbb{Z}^n\), in the important case when d coordinate directions are employed.
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Brunetti, S., Dulio, P., Peri, C. (2014). Non-additive Bounded Sets of Uniqueness in ℤn . In: Barcucci, E., Frosini, A., Rinaldi, S. (eds) Discrete Geometry for Computer Imagery. DGCI 2014. Lecture Notes in Computer Science, vol 8668. Springer, Cham. https://doi.org/10.1007/978-3-319-09955-2_19
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