Parallel algorithms for arbitrary dimensional Euclidean distance transforms with applications on arrays with reconfigurable optical buses

In this paper, we present algorithms for computing the Euclidean distance transform (EDT) of a binary image on the array with reconfigurable optical buses (AROB). First, we develop a parallel algorithm termed as Algorithm Expander which can be implemented in O(1) time on an AROB with N/spl times/N/sup /spl delta// processors, where /spl delta/=1/k, k is a constant and a positive integer. Algorithm Expander is designed to compute a higher dimensional EDT based on the computed lower dimensional EDT. It functions as a general EDT expander for us to expand EDT from a lower dimension to a higher dimension. We then develop parallel algorithms for the two-dimensional (2-D)/spl I.bar/EDT of a binary image array of size N/spl times/N in O(1) time on an AROB with N/spl times/N/spl times/N/sup /spl delta// processors and for the three-dimensional (3-D)/spl I.bar/EDT of a binary image of size N/spl times/N/spl times/N in O(1) time on an AROB with N/spl times/N/spl times/N/spl times/N/sup /spl delta// processors. To the best of our knowledge, all results derived above are the best O(1) time algorithms known. We then extend it to compute the nD/spl I.bar/EDT of a binary image of size N/sup n/ in O(n) time on an AROB with N/sup n+/spl delta// processors. We also apply our parallel EDT algorithms to build Voronoi diagram and Voronoi polyhetra (polygons), to find all maximal empty spheres and the largest empty sphere, and to compute the medial axis transform. All of these applications can be solved in the same time complexity on an AROB with the same number of processors as needed for solving the EDT problems in the same dimensions.