Solving the Hermite interpolation problem with scalar subdivision schemes

Subdivision schemes represent an efficient and simple class of methods to generate curves and surfaces by successive refinements of a set of points with an associated connectivity defining polygons or meshes in the respective cases. Many applications require the possibility of interpolating points and associated derivatives. The interpolation is guaranteed with the use of interpolatory scalar and interpolatory Hermite subdivision schemes in each respective case. Moving beyond those schemes, in this thesis we study the point interpolation and the Hermite interpolation problems with curves by using the class of scalar linear uniform subdivision schemes. The motivation for this research is the gaps evident in the literature regarding interpolation with certain approximating schemes. The gaps include the interpolation with dual subdivision schemes and the derivatives interpolation for any scalar scheme. We analyze both primal and dual cases taking into consideration odd and even symmetry of their masks. That analysis provides a characterization of the singularity for the interpolation operator represented with a block-circulant matrix. Our choice of interpolation parameters differs from the usual chosen ones at integer parameters, adds a degree of freedom, and offers the possibility of constructing a family of interpolating curves. In addition, when in the presence of a singular interpolation operator, we propose a filter for the least square solution based on the kernel of that operator. This strategy provides a solution which optimizes a given fairness functional. Under some considerations that choice is found with a quadratic optimization problem, avoiding the need of facing the optimization as other fitting solutions in the literature. With the strategies proposed our research resolves the outstanding problems. The results are used for the free-form design of curves, the exact offset computation, and an image segmentation algorithm based on subdivision curves.