A systolic screen of size M is a √M × √M mesh-of-processors where each processing element Pij 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 Lp 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.