A linear time algorithm for computing exact Euclidean distance transforms of binary images in arbitrary dimensions

Abstract—A sequential algorithm is presented for computing the exact Euclidean distance transform (DT) of a k-dimensional binary image in time linear in the total number of voxelsN. The algorithm, which is based on dimensionality reduction and partial Voronoi diagram construction, can be used for computing the DT for a wide class of distance functions, including the Lp and chamfer metrics. At each dimension level, the DT is computed by constructing the intersection of the Voronoi diagram whose sites are the feature voxels with each row of the image. This construction is performed efficiently by using the DT in the next lower dimension. The correctness and linear time complexity are demonstrated analytically and verified experimentally. The algorithm may be of practical value since it is relatively simple and easy to implement and it is relatively fast (not only does it run in O N time but the time constant is small). A simple modification of the algorithm computes the weighted Euclidean DT, which is useful for images with anisotropic voxel dimensions. A parallel version of the algorithm runs in O\u85N=p time with p processors. Index Terms—Euclidean distance transform, closest feature transform, Voronoi diagram.