On the Non-additive Sets of Uniqueness in a Finite Grid

Abstract

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 A, under X-rays taken in directions belonging to a set S of four lattice directions.

A rounding theorem for unique binary tomographic reconstruction

“…More generally, the notions of additivity and uniqueness are equivalent when two directions are employed, whereas, for three or more directions, additivity is more demanding than uniqueness (see [13,14] for details). However, uniqueness results can be achieved even without the additivity assumption (see [8,9,20] and the related bibliographies).…”

“…As a consequence, we are mainly interested in looking for conditions that can limit the number of allowed solutions, and possibly for uniqueness conditions. Sometimes uniqueness results can be achieved by introducing some geometric conditions, such as convexity ( [15]) or additivity ( [8,11,14]).…”

Ghosts in Discrete Tomography

“…Additivity is also of main importance because the reconstruction problem for additive set is polynomial by using (relaxation of integer) linear programming. A related intriguing problem is to find non-additive sets of uniqueness (see [27]). A complete characterization of bad configurations (weakly or not weakly) has been obtained in [25] with a new algebraic approach employed then in several papers (see for instance [27][28][29][30]).…”

“…A related intriguing problem is to find non-additive sets of uniqueness (see [27]). A complete characterization of bad configurations (weakly or not weakly) has been obtained in [25] with a new algebraic approach employed then in several papers (see for instance [27][28][29][30]). S-bad configurations, with the extra condition of convexity, are known as S-polygons, and reveal to be useful both in geometric tomography and in discrete tomography (see for instance [31,32], and [33] for an algorithmic approach), as well as very interesting also from a purely geometric point of view (see for instance [34][35][36]).…”

Probabilistic Reconstruction of hv-convex Polyominoes from Noisy Projection Data

“…In Tomography, a faithful reconstruction of an object from its projections is sought, and the existence of special sets of projections, which guarantee uniqueness in a given lattice grid (see for instance [11,12]), is investigated. In general, the discrete reconstruction problem is NP-hard if the number of projections is more than two ( [18]).…”