THB-splines multi-patch parameterization for multiply-connected planar domains via Template Segmentation.

Given a planar multiply-connected domain Ω, we provide a multi-patch parameterization exploiting a segmentation algorithm and the construction of templates. The segmentation is a fundamental step which allows to split the input domain into patches topologically equivalent to quadrilaterals. Templates are prototypes of segmentation that will be chosen according to the shape of the given domain. We will compute a bijective harmonic mapping h defined between the input Ω and the chosen template Ωˆ. The segmentation will be transferred from the prototype template to the input Ω by approximating the inverse mapping. Templates are equipped with a multi-patch structure and each patch Ωˆ(i) is parametrized individually by a geometry mapping Gi. The final multi-patch parameterization of Ω is achieved exploiting the combination of each geometry mapping Gi and the inverse of the mapping h obtained by projection into the THB-spline space.