Optimal Parallel Algorithms for the 3D Euclidean Distance Transform on the CRCW and EREW PRAM Models

In this paper, we develop parallel algorithms for the three-dimensional Euclidean distance transform (3D EDT) on both the CRCW and EREW PRAM computation models. Based on the dimensionality reduction technique and the solution for proximate points problem, we achieve the optimality of the 3D EDT computation. For an N × N × N binary image array, our parallel algorithms for 3D EDT computation are in O(log logN) time using N 3 loglogN CRCW processors or in O(logN) time using N 3 logN EREW processors. We then extend it to the n-dimensional space to compute the n-dimensional Euclidean distance transform (nD EDT) of a binary image array of size N. The n-dimensional Euclidean distance transform of a binary image of size N can be computed in O(nloglogN) time using N n loglogN CRCW processors or in O(nlogN) time using N n logN EREW processors. As for applications, our parallel 3D EDT algorithm can be used to build up Voronoi diagram and Voronoi polyhedra, to find all maximal empty spheres and the largest empty sphere, and to compute the distance-based medial axis transform in a 3-D binary image. All of these parallel algorithms can be performed in O(log logN) time using N3 loglogN CRCW processors or in O(logN) time using N 3 logN EREW processors. Our algorithms for the nD EDT can be also applied to extend those applications described above to an n-dimensional space. To the best of our knowledge, all results derived in this paper are the best that never found before.