Non-additive Bounded Sets of Uniqueness in ℤ n

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

 ∈ 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 ℤ

n

(

n

 ≥ 

d

 ≥ 3), under the requirement that such directions span a

d

-dimensional subspace of 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.