Distance transform to seeds: computation and application

The distance transform (DT) is the mapping which gives for each point of an object, its distance to the nearest point in the complementary of the object. The distance transform to seeds (DTS) is a generalization of the DT. It maps to each point of the object, its distance to the nearest point in a selected set of seeds. The DT has been studied since the sixties. Many algorithms have been proposed in different approaches, but the first linear time algorithm to compute the DT with the Euclidean metric has only been proposed in 1996 by Hirata [5]. In skeletonization, the notion of medial axis (MA), the set of centers of maximal balls, is very important, since the MA has sufficient information for the reconstruction of the original shape (reversibility property). A maximal ball is a ball included in the object and not completely covered by any other ball also included in the object. The computation of the MA depends on the computation of the DT. In a previous work [8], we have also defined the exact Euclidean medial axis in higher resolution (HMA). Based on the DT, we have provided an algorithm for the HMA in 2D and 3D, but not for nD. In this paper we show how to use the separable distance transform algorithm to compute the distance transform to seeds, and we show how it is applied on the computation of skeletons in higher resolution, using the HMA.