Carla Peri

Discrete tomography determination of bounded lattice sets from four X-rays

We deal with the question of uniqueness, namely to decide when an unknown finite set of points in Z^2 is uniquely determined by its X-rays corresponding to a given set S of lattice directions. In Hajdu (2005) [11] proved that for any fixed rectangle A in Z^2 there exists a non trivial set S of four lattice directions, depending only on the size of A, such that any two subsets of A can be distinguished by means of their X-rays taken in the directions in S. The proof was given by explicitly constructing a suitable set S in any possible case. We improve this result by showing that in fact whole families of suitable sets of four directions can be found, for which we provide a complete characterization. This permits us to easily solve some related problems and the computational aspects.

On the non-additive sets of uniqueness in a finite grid

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

Characterization of {-1, 0, +1} valued functions in discrete tomography under sets of four directions

\mathcal{A}

Discrete Point X-rays

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

On the minimal convex annulus of a planar convex body

In this paper we use the algebraic approach to Discrete Tomography introduced by Hajdu and Tijdeman to study functions f : Z2 → {-1, 0, +1} which have zero line sums along the lines corresponding to certain sets of four directions.

On Blaschke's extension of Bonnesen's inequality

A discrete point X-ray of a finite subset FM of Rn at a point p gives the number of points in F lying on each line passing through p. A systematic study of discrete point X-rays is initiated, with an emphasis on uniqueness results and subsets of the integer lattice.

Ghosts in Discrete Tomography

In this paper we introduce the notion of a minimal convex annulusK (C) of a convex bodyC, generalizing the concept of a minimal circular annulus. Then we prove the existence — as for the minimal circular annulus — of a Radon partition of the set of contact points of the boundaries ofK (C) andC. Subsequently, the uniqueness ofK (C) is shown. Finally, it is proven that, for typicalC, the boundary ofC has precisely two points in common with each component of the boundary ofK (C).

Minimal shells containing a convex surface in Minkowski space

W. Blaschke established a Bonnesen-style inequality for the relative inradius and circumradius of a planar convex bodyK with respect to another. We sharpen this inequality by considering the radii of the minimal convex annulus ofK.

Explicit determination of bounded non-additive sets of uniqueness for four X-rays

Switching components, also named as bad configurations, interchanges, and ghosts (according to different scenarios), play a key role in the study of ambiguous configurations, which often appear in Discrete Tomography and in several other areas of research. In this paper we give an upper bound for the minimal size bad configurations associated to a given set