New Higher-Resolution Discrete Euclidean Medial Axis in nD with Linear Time Parallel Algorithm

Abstract

The notion of skeleton plays a major role in shape analysis since the introduction of the medial axis. The continuous medial axis is a skeleton with the following characteristics: centered, thin, homotopic, and reversible (sufficient for the reconstruction of the original object). The discrete Euclidean medial axis (MA) is also reversible and centered, but no longer homotopic nor thin. To preserve topology and reversibility, the MA is usually combined with homotopic thinning algorithms. Since there is a robust and well defined framework for fast homotopic thinning defined in the domain of abstract complexes, some authors have extended the MA to a doubled resolution grid and defined the discrete Euclidean Medial Axis in Higher Resolution (HMA), which can be combined to the framework defined on abstract complexes. Other authors gave an alternative definition of medial axis, which is a reversible subset of the MA, and is called Reduced Discrete Medial Axis (RDMA). The RDMA is thinner than the MA and 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). The HRDMA is reversible and it can be computed in optimal time. The HRDMA can be combined with the algorithms in abstract complexes, so a reversible and homotopic Euclidean skeleton can be computed in optimal time.