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EnglishWe 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.
| Autori: | |
| Titolo: | Discrete tomography determination of bounded lattice sets from four X-rays |
| Digital Object Identifier (DOI): | http://dx.doi.org/10.1016/j.dam.2012.09.010 |
| Data di pubblicazione: | 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. |
| Lingua: | Inglese |
| Rivista: | |
| Citazione: | 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] |
| Appare nelle tipologie: | Articolo in rivista, Nota a sentenza |
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