Template Mapping Using Adaptive Splines and Optimization of the Parameterization

Abstract

We consider the construction of a spline map (a volumetric deformation) that transforms a template, which is given in the domain, into a target shape. More precisely, the domain is equipped with a set of surface patches (the template skeleton) and target patches for some of them (which are called the constraining patches) are specified. The constructed spline map approximately transforms the constraining patches into the associated target patches. Possible applications include isogeometric segmentation and parameterization of the computational domain. In particular, the approach should be useful when performing isogeometric segmentation and parameterization for a large class of computational domains possessing similar shapes. We present a solution approach, which is based on least-squares fitting. In order to deal with the influence of the parameterization, the well-established approaches of point and tangent distance minimization are employed for the iterative solution of the resulting nonlinear optimization problems. Additionally, we enrich the approach with spline space refinement. The efficiency and performance of the approach are investigated experimentally. We demonstrate that the optimization of the parameterization, which is used in the point or tangent distance minimization, is an essential step of the procedure. In addition, we use adaptive spline refinement in order to save computational resources. The proposed template mapping approach is also applied to a case of industrial interest, as well as to a volumetric example.