CONSTANT TIME DIGITAL GEOMETRY ALGORITHMS ON THE SCAN MODEL OF PARALLEL COMPUTATION

Abstract

Assume that a black/white n×n image is stored one pixel per element of a vector of length n2. We consider determining some characteristics of such images using the scan model of parallel computation. The scan model was introduced by G.E. Blelloch and is a single instruction multiple data (SIMD) vector model of computation. The primitive operations of the model work on vectors (one dimensional arrays) of values, with three types of primitive operations: Elementwise arithmetic and logical operations, permutation operations, and scan operations, a type of prefix computation (a scan operation takes a binary associative operator ⊗ and a vector [a1,…, an] and returns the vector [a1, a1⊗a2,…, a1⊗a2⊗…⊗an]). We show that many important characteristics of binary images can be determined in constant time on the scan model of parallel computation. These include the convex hull construction, diameter, width, smallest enclosing box, perimeter, area, detecting digital convexity, parallel and point visibility (determining for each pixel of the image the portion that is visible, i.e., not obstructed by any black pixel, in a given direction from infinity or from a given point, respectively) of an image, smallest, largest and Hausdorff distances between two images, linear separability of two images, and the recognition of digital lines, rectangles and arcs.