In Discrete Tomography there is a wide literature concerning (weakly) bad configurations. These occur in dealing with several questions concerning the important issues of uniqueness and additivity. Discrete lattice sets which are additive with respect to a given set $S$ of lattice directions are uniquely determined by $X$-rays in the direction of $S$. These sets are characterized by the absence of weakly bad configurations for $S$. On the other side, if a set has a bad configuration with respect to $S$, then it is not uniquely determined by the $X$-rays in the directions of $S$, and consequently it is also non-additive. Between these two opposite situations there are also the non-additive sets of uniqueness, which deserve interest in Discrete Tomography, since their unique reconstruction cannot be derived via the additivity property. In this paper we wish to investigate possible interplays among such notions in a given lattice grid $\mathcal{A}$, under $X$-rays taken in directions belonging to a set $S$ of four lattice directions.

Peri, C., Brunetti, S., Dulio, P., On the Non-Additive Sets of Uniqueness in a Finite Grid, in Discrete Geometry for Computer Imagery, 17th IAPR International Conference, DGCI 2013, Seville, Spain, March 20-22, 2013, Proceedings, (Siviglia, 20-22 March 2013), Springer Verlag, Berlino 2013:<<Lecture Notes in Computer Science, Vol. 7749>>, 288-299. [10.1007/978-3-642-37067-0-25] [http://hdl.handle.net/10807/41552]

### On the Non-Additive Sets of Uniqueness in a Finite Grid

#### Abstract

In Discrete Tomography there is a wide literature concerning (weakly) bad configurations. These occur in dealing with several questions concerning the important issues of uniqueness and additivity. Discrete lattice sets which are additive with respect to a given set $S$ of lattice directions are uniquely determined by $X$-rays in the direction of $S$. These sets are characterized by the absence of weakly bad configurations for $S$. On the other side, if a set has a bad configuration with respect to $S$, then it is not uniquely determined by the $X$-rays in the directions of $S$, and consequently it is also non-additive. Between these two opposite situations there are also the non-additive sets of uniqueness, which deserve interest in Discrete Tomography, since their unique reconstruction cannot be derived via the additivity property. In this paper we wish to investigate possible interplays among such notions in a given lattice grid $\mathcal{A}$, under $X$-rays taken in directions belonging to a set $S$ of four lattice directions.
##### Scheda breve Scheda completa Scheda completa (DC)
2013
Inglese
Discrete Geometry for Computer Imagery, 17th IAPR International Conference, DGCI 2013, Seville, Spain, March 20-22, 2013, Proceedings
tHE 17th International Conference on DISCRETE GEOMETRY for COMPUTER IMAGERY (DGCI 2013)
Siviglia
20-mar-2013
22-mar-2013
978-3-642-37067-0
Peri, C., Brunetti, S., Dulio, P., On the Non-Additive Sets of Uniqueness in a Finite Grid, in Discrete Geometry for Computer Imagery, 17th IAPR International Conference, DGCI 2013, Seville, Spain, March 20-22, 2013, Proceedings, (Siviglia, 20-22 March 2013), Springer Verlag, Berlino 2013:<<Lecture Notes in Computer Science, Vol. 7749>>, 288-299. [10.1007/978-3-642-37067-0-25] [http://hdl.handle.net/10807/41552]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10807/41552
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