Digitization in Khalimsky spaces

We consider the digital plane of integer points equipped with the Khalimsky topology. We suggest a digitization of straight lines such that the digitized image is homeomorphic to the Khalimsky line and a digitized line segment is a Khalimsky arc. It is demonstrated that a Khalimsky arc is the digitization of a straight line segment if and only if it satisfies a generalized version of the chord property introduced by Rosenfeld. 1. Digitization of straight lines We are interested in representing lines in the digital plane, Z2. One widely used digitization is the one considered by Azriel Rosenfeld [Ro74]. Define the set C(0) ={x;x1 = 0 and − 1/2 < x2 ≤ 1/2} ∪ {x;−1/2 < x1 ≤ 1/2 and x2 = 0}. For each p ∈ Z2, let C(p) = C(0) + p. Note that C(p) is a cross with center at p, that C(p) ∩ C(q) is empty if p 6= q, and that ⋃ p∈Z2 C(p) is equal to the grid lines (R × Z) ∪ (Z × R). We now define the Rosenfeld digitization of A ⊂ R2: DR : P(R)→ P(Z), DR(A) = {p ∈ Z; C(p) ∩A 6= ∅}. Since the union of the crosses is the grid lines, a straight line or a sufficiently long line segment has non-empty digitization. We briefly discuss some terminology in this context: C(p) is called a cell with nucleus p. In general, C(p) may be any subset of a space X, and is defined for every p in some subspace Z of X. Using the definition above, the digitization of a subset of X is a subset of Z, completely determined by the cells. If (X, d) is a metric space and C(p) ⊂ {x ∈ X; ∀b ∈ Z, d(x, p) ≤ d(x, b)}, then we talk of a Voronoi digitization. Such a digitization may be thought of as a reasonable metric approximation. Note that DR is a Voronoi digitization; C(p) is contained in the set {x ∈ R2; ‖x− p‖∞ ≤ 1 2}.