TY - GEN

T1 - Non-additive bounded sets of uniqueness in $Z^n$

AU - Peri, Carla

AU - Dulio, Paolo

AU - Brunetti, Sara

PY - 2014

Y1 - 2014

N2 - 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 $p\in\mathbb{N}$, $[p]=\{0,1,...,p-1\}$), 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 $\mathbb{Z}^n$ ($n\geq d\geq 3$), under the requirement that such directions span a $d$-dimensional subspace of $\mathbb{Z}^n$. 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.

AB - 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 $p\in\mathbb{N}$, $[p]=\{0,1,...,p-1\}$), 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 $\mathbb{Z}^n$ ($n\geq d\geq 3$), under the requirement that such directions span a $d$-dimensional subspace of $\mathbb{Z}^n$. 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.

KW - Additive set

KW - Bad configuration

KW - Discrete tomography

KW - X-ray

KW - Additive set

KW - Bad configuration

KW - Discrete tomography

KW - X-ray

UR - http://hdl.handle.net/10807/65147

U2 - 10.1007/978-3-319-09955-2_19

DO - 10.1007/978-3-319-09955-2_19

M3 - Conference contribution

SN - 978-3-319-09954-5

T3 - LECTURE NOTES IN COMPUTER SCIENCE

SP - 226

EP - 237

BT - E. Barcucci et al. (Eds) DGCI 2014, Lecture notes in computer science

T2 - DGCI 2014: the 18th international conference on Discrete Geometry for Computer Imagery

Y2 - 10 September 2014 through 12 September 2014

ER -