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In a two- or three-dimensional image array, the computation of Euclidean distance transform (EDT) is an important task. With the increasing application of 3D voxel images, it is useful to consider the distance transform of a 3D digital image array. Because the EDT computation is a global operation, it is prohibitively time consuming when performing the EDT for image processing. In order to provide the efficient transform computations, parallelism is employed. We first derive several important geometry relations and properties among parallel planes. We then, develop a parallel algorithm for the three-dimensional Euclidean distance transform (3D-EDT) on the EREW PRAM computation model. The time complexity of our parallel algorithm is O(log2 N) for an N×N×N image array and this is currently the best known result. A generalized parallel algorithm for the 3D-EDT is also proposed. We implement the proposed algorithms sequentially, the performance of which exceeds the existing algorithms (proposed by Yamada, 1984). Finally, we develop the corresponding parallel programs on both the emulated EREW PRAM model computer and the IBM SP2 to verify the speed-up properties of the proposed algorithms.