We deal with the question of uniqueness, namely to decide when an unknown finite set of points in $\mathbb{Z}^2$ is uniquely determined by its $X$-rays corresponding to a given set $S$ of lattice directions. In \cite{Ha} L. Hajdu proved that for any fixed rectangle $\mathcal{A}$ in $\mathbb{Z}^2$ there exists a valid set $S$ of four lattice directions (at least when $\mathcal{A}$ is not too ``small''), depending only on the size of $\mathcal{A}$, such that any two subsets of $\mathcal{A}$ can be distinguished by means of their $X$-rays taken in the directions in $S$. The proof was given by explicitly constructing a suitable set $S$ in any possible case. We improve this result by showing that in fact, for any fixed rectangle $\mathcal{A}$ in $\mathbb{Z}^2$, whole families of suitable sets of four directions can be found, for which we provide a complete characterization. Moreover this characterization permits to easily solve some relevant related problems.

Peri, C., Brunetti, S., Dulio, P., Discrete tomography determination of bounded lattice sets from four X-rays, <<DISCRETE APPLIED MATHEMATICS>>, 2012; 2012 (N/A): N/A-N/A. [doi:10.1016/j.dam.2012.09.010] [http://hdl.handle.net/10807/36704]

Discrete tomography determination of bounded lattice sets from four X-rays

Peri, Carla;
2012

Abstract

We deal with the question of uniqueness, namely to decide when an unknown finite set of points in $\mathbb{Z}^2$ is uniquely determined by its $X$-rays corresponding to a given set $S$ of lattice directions. In \cite{Ha} L. Hajdu proved that for any fixed rectangle $\mathcal{A}$ in $\mathbb{Z}^2$ there exists a valid set $S$ of four lattice directions (at least when $\mathcal{A}$ is not too ``small''), depending only on the size of $\mathcal{A}$, such that any two subsets of $\mathcal{A}$ can be distinguished by means of their $X$-rays taken in the directions in $S$. The proof was given by explicitly constructing a suitable set $S$ in any possible case. We improve this result by showing that in fact, for any fixed rectangle $\mathcal{A}$ in $\mathbb{Z}^2$, whole families of suitable sets of four directions can be found, for which we provide a complete characterization. Moreover this characterization permits to easily solve some relevant related problems.
Inglese
Peri, C., Brunetti, S., Dulio, P., Discrete tomography determination of bounded lattice sets from four X-rays, <>, 2012; 2012 (N/A): N/A-N/A. [doi:10.1016/j.dam.2012.09.010] [http://hdl.handle.net/10807/36704]
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/10807/36704
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