Optimal Algorithms pp 14-24

# Computing digitized voronoi diagrams on a systolic screen and applications to clustering

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## Abstract

A *systolic screen* of size M is a √M × √M mesh-of-processors where each processing element P_{ij} represents the pixel (i,j) of a *digitized plane* П of √M × √M pixels. In this paper we study the computation of the Voronoi diagram of a set of n planar objects represented by disjoint images contained in П. We present O(√M) time algorithms to compute the Voronoi diagram for a large class of object types (e.g., points, line segments, circles, ellipses, and polygons of constant size) and distance functions (e.g., all L_{p} metrices).

Since the Voronoi diagram is used in many geometric applications, the above result has numerous consequences for the design of efficient image processing algorithms on a systolic screen. We obtain, e.g., an O(√M) time systolic screen algorithm for "optical clustering"; i.e., identifying those groups of objects in a digitized picture that are "close" in the sense of human perception.

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