New reduced discrete Euclidean nD medial axis with optimal algorithm

Abstract

Skeletons have been playing an important role in shape analysis since the introduction of the medial axis in the sixties. The original medial axis definition is in the continuous Euclidean space. It is a skeleton with the following characteristics: centered, thin, homotopic (it has the same topology as the object), and reversible (sufficient for the reconstruction of the object). The discrete version of the Euclidean medial axis (MA) is also reversible and centered, but no longer homotopic nor thin. The combination of the MA with homotopic thinning algorithms can result in a topology preserving, reversible skeleton. However, such combination may generate thicker skeletons, and the choice of the thinning algorithm is not obvious. A robust and well defined framework for fast homotopic thinning available in the literature is defined in the domain of abstract complexes. Since the abstract complexes are represented in a doubled resolution grid, some authors have extended the MA to a doubled resolution grid and defined the discrete Euclidean medial axis in higher resolution (HMA). The HMA can be combined with the thinning framework defined on abstract complexes. Other authors have given an alternative definition of medial axis, which is a subset of the MA, called reduced discrete medial axis (RDMA). The RDMA is reversible, thinner than the MA, and it can be computed in optimal time. In this paper, we extend the RDMA to the doubled resolution grid and we define the high-resolution RDMA (HRDMA). We provide an optimal algorithm to compute the HRDMA. The HRDMA can be combined with the thinning framework defined on abstract complexes. The skeleton obtained by such combination is provided with strong characteristics, not found in the literature.