Consistent Digital Rays

Abstract

Given a fixed origin o in the d-dimensional grid, we give a novel definition of digital rays dig(op) from o to each grid point p. Each digital ray dig(op) approximates the Euclidean line segment op between o and p. The set of all digital rays satisfies a set of axioms analogous to the Euclidean axioms. We measure the approximation quality by the maximum Hausdorff distance between a digital ray and its Euclidean counterpart and establish an asymptotically tight Θ(log n) bound in the n × n grid. The proof of the bound is based on discrepancy theory and a simple construction algorithm. Without a monotonicity property for digital rays the bound is improved to O(1). Digital rays enable us to define the family of digital star-shaped regions centered at o which we use to design efficient algorithms for image processing problems.

Consistent Digital Line Segments

“…Here, we phrase this monotonicity axiom differently, but still, the system of axioms (S1)-(S5) remains equivalent to the one given in [1].…”

“…For any pair of points p and q in the grid Z 2 we want to define the digital line segment S(p, q) connecting them, that is, {p, q} ⊆ S(p, q) ⊆ Z 2 . Chun et al in [1] put forward the following four axioms that arise naturally from properties of line segments in Euclidean geometry.…”

“…Another condition was introduced in [1] to enforce the monotonicity of the segments, ruling out pathological examples like the one above. Here, we phrase this monotonicity axiom differently, but still, the system of axioms (S1)-(S5) remains equivalent to the one given in [1].…”

“…The main question raised in [1] was if it is possible to define a CDS such that a Euclidean segment and its digitalization have a reasonably small Hausdorff distance. More precisely, the goal is to find a CDS satisfying the following condition.…”

“…In particular, we show how to derive such a system of digital segments from any total order on the integers. As a consequence, using a well-chosen total order, we manage to define a system of digital segments such that all digital segments are, in Hausdorff metric, optimally close to their corresponding Euclidean segments, thus giving an explicit construction that resolves the main question of [1]. …”