The faithful reconstruction of an unknown object from projections, i.e., from measurements of its density function along a finite set of discrete directions, is a challenging task. Some theoretical results prevent, in general, both to perform the reconstruction sufficiently fast, and, even worse, to be sure to obtain, as output, the unknown starting object. In order to reduce the number of possible solutions, one tries to exploit some a priori knowledge. In the present paper we assume to know the size of a lattice grid $\mathcal{A}$ containing the object to be reconstructed. Instead of looking for uniqueness in the whole grid $\mathcal{A}$, we want to address the problem from a \textit{local} point of view. More precisely, given a limited number of directions, we aim in showing, first of all, which is the subregion of $\mathcal{A}$ where pixels are uniquely reconstructible, and then in finding where the reconstruction can be performed quickly (in linear time). In previous works we have characterized the shape of the region of uniqueness (ROU) for any pair of directions. In this paper we show that the results can be extended to special sets of three directions, by splitting them in three different pairs. Moreover, we show that such a procedure cannot be employed for general triples of directions. We also provide applications concerning the obtained characterization of the ROU, and further experiments which underline some regularities in the shape of the ROU corresponding to sets of three not-yet-considered directions.

Dulio, P., Frosini, A., Pagani, S. M. C., Geometrical Characterization of the Uniqueness Regions Under Special Sets of Three Directions in Discrete Tomography, Comunicazione, in Discrete Geometry for Computer Imagery (LNCS 9647), (Nantes, 18-20 April 2016), Nicolas Normand, Jeanpierre Guédon, Florent Autrusseau, Heidelberg 2016: 105-116. 10.1007/978-3-319-32360-2_8 [http://hdl.handle.net/10807/79300]

### Geometrical Characterization of the Uniqueness Regions Under Special Sets of Three Directions in Discrete Tomography

#### Abstract

The faithful reconstruction of an unknown object from projections, i.e., from measurements of its density function along a finite set of discrete directions, is a challenging task. Some theoretical results prevent, in general, both to perform the reconstruction sufficiently fast, and, even worse, to be sure to obtain, as output, the unknown starting object. In order to reduce the number of possible solutions, one tries to exploit some a priori knowledge. In the present paper we assume to know the size of a lattice grid $\mathcal{A}$ containing the object to be reconstructed. Instead of looking for uniqueness in the whole grid $\mathcal{A}$, we want to address the problem from a \textit{local} point of view. More precisely, given a limited number of directions, we aim in showing, first of all, which is the subregion of $\mathcal{A}$ where pixels are uniquely reconstructible, and then in finding where the reconstruction can be performed quickly (in linear time). In previous works we have characterized the shape of the region of uniqueness (ROU) for any pair of directions. In this paper we show that the results can be extended to special sets of three directions, by splitting them in three different pairs. Moreover, we show that such a procedure cannot be employed for general triples of directions. We also provide applications concerning the obtained characterization of the ROU, and further experiments which underline some regularities in the shape of the ROU corresponding to sets of three not-yet-considered directions.
##### Scheda breve Scheda completa Scheda completa (DC)
Inglese
Discrete Geometry for Computer Imagery (LNCS 9647)
DGCI 2016, 19th IAPR International Conference
Nantes
Comunicazione
18-apr-2016
20-apr-2016
978-3-319-32359-6
Nicolas Normand, Jeanpierre Guédon, Florent Autrusseau
Dulio, P., Frosini, A., Pagani, S. M. C., Geometrical Characterization of the Uniqueness Regions Under Special Sets of Three Directions in Discrete Tomography, Comunicazione, in Discrete Geometry for Computer Imagery (LNCS 9647), (Nantes, 18-20 April 2016), Nicolas Normand, Jeanpierre Guédon, Florent Autrusseau, Heidelberg 2016: 105-116. 10.1007/978-3-319-32360-2_8 [http://hdl.handle.net/10807/79300]
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/10807/79300
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