Gaussian Kernel Based Medial Zone Computation

The medial zone of a 3-dimensional semi-analytic domain Ω, denoted as MZ(Ω), can be thought of as a “thick” version of the shape’s skeleton with important applications in design, motion planning and geometric reasoning. The utility of the medial zones stems from the fact that they theoretically are homeomorphic to and have the same dimension as the original domain, so in this sense they capture the topology of the domain. However, the original approach to compute them involved a discrete Laplace estimation commonly used in edge detection algorithms, which was used to extract the points belonging to the medial zones. Despite being fast, this computational approach produced unwanted topological artifacts and did not satisfy the convergence properties of the medial zones as formulated, which limited their effectiveness. In this work a new approach was proposed to compute points of the medial zones based on Gaussian Kernels located on the medial axis(MA) points whose widths are mapped to an approximate but differentiable function constructed over the domain. This computing paradigm produced a family of homeomorphic medial zones that have the same topology as the domain and converge to either theMA of the domain or to the domain itself. The resulting method was general in that it can be applied to domains of arbitrary complexity. The effectiveness of this approach was demonstrated by practical examples.