Consistent Digital Rays.

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 $\overline {op}$ 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. [ABSTRACT FROM AUTHOR]