Medians of Discrete Sets according to a Linear Distance

Abstract

In this paper, we present some results concerning the median points of a discrete set according to a distance defined by means of two directions p and q. We describe a local characterization of the median points and show how these points can be determined from the projections of the discrete set along directions p and q. We prove that the discrete sets having some connectivity properties have at most four median points according to a linear distance, and if there are four median points they form a parallelogram. Finally, we show that the 4-connected sets which are convex along the diagonal directions contain their median points along these directions.

Stability in Discrete Tomography: some positive results

“…where κ only depends on the directions p and q (see for example [7]). Moreover we denote by i, j pq the point M such that p(M) = i and q(M) = j.…”

Random generation of Q-convex sets

“…It must be noticed that i, j pq is not in Z 2 in general. More precisely there exists κ coprime with δ such that i, j pq ∈ Z 2 ⇐⇒ j ≡ κi mod δ (see for example [9]). Figure 1 illustrates the case p = x − y and q = 2x + y: with these directions we have δ = 3 and κ = 2.…”

An Algorithm for Reconstructing Special Lattice Sets from Their Approximate X-Rays

“…It is easy to prove that the point M intersection of p(M ) = i with q(M ) = j belongs to Z 2 if and only if j ≡ κi (mod δ), where: δ = |ad − bc|, κ = (cu + dv)sign(ad − bc) (mod δ) and au + bv = 1 (see Fig. 3(a) and [7]). Without any loss of generality, we can assume that a Q-convex set F around p and q whose X-rays are such that:…”