Medial zones: Formulation and applications

The popularity of medial axis in shape modeling and analysis comes from several of its well known fundamental properties. For example, medial axis captures the connectivity of the domain, has a lower dimension than the space itself, and is closely related to the distance function constructed over the same domain.We propose the new concept of a medial zone of an n  -dimensional semi-analytic domain ΩΩ that subsumes the medial axis MA(Ω)MA(Ω) of the same domain as a special case, and can be thought of as a ‘thick’ skeleton having the same dimension as that of ΩΩ. We show that by transforming the exact, non-differentiable, distance function of domain ΩΩ into approximate but differentiable distance functions, and by controlling the geodesic distance to the crests of the approximate distance functions of domain ΩΩ, one obtains families of medial zones of ΩΩ that are homeomorphic to the domain and are supersets of MA(Ω)MA(Ω). We present a set of natural properties for the medial zones MZ(Ω)MZ(Ω) of ΩΩ and discuss practical approaches to compute both medial axes and medial zones for 3-dimensional semi-analytic sets with rigid or evolving boundaries. Due to the fact that the medial zones fuse some of the critical geometric and topological properties of both the domain itself and of its medial axis, re-formulating problems in terms of medial zones affords the ‘best of both worlds’ in applications such as geometric reasoning, robotic and autonomous navigation, and design automation.

► We define the new concept of a medial zone of an n-dimensional semi-analytic domain. ► Medial zones can be thought of as ‘thick’ skeletons having the same dimension as that of the domain. ► We show that the medial zone converges to either the domain itself or its medial axis. ► Medial zones are homeomorphic to the domain. ► We demonstrate the attractive properties of medial zones and explore two of their potential applications.