An O[1] time algorithm for the 3D euclidean distance transform on the CRCW PRAM model

Abstract

We develop a parallel algorithm for the 2D Euclidean distance transform (2D/spl I.bar/EDT, for short) of a binary image of size N /spl times/ N in O(1) time using N/sup 2+/spl delta/+/spl epsi// CRCW processors and a parallel algorithm for the 3D Euclidean distance transform (3D/spl I.bar/EDT, for short) of a binary image of size N /spl times/ N /spl times/ N in O(1) time using N/sup 3+/spl delta/+/spl epsi// CRCW processors, where /spl delta/=1/, /spl epsi/=1/(2/sup c+1/-1), h, and are constants and positive integers. Our 2D/spl I.bar/EDT (3D/spl I.bar/EDT) parallel algorithm can be used to build up Voronoi diagram and Voronoi polygons (polyhedra) in a 2D (3D) binary image also. All of these parallel algorithms can be performed in O(1) time using N/sup 2+/spl delta/+/spl epsi// (N/sup 3+/spl delta/+/spl epsi//) CRCW processors. To the best of our knowledge, all results derived above are the best O(1) time algorithms known.