Refinable bivariate quartic and quintic C2C2-splines for quadrilateral subdivisions

Refinable compactly supported bivariate C2C2 quartic and quintic spline function vectors on the four-directional mesh are introduced in this paper to generate matrix-valued templates for approximation and Hermite interpolatory surface subdivision schemes, respectively, for both the 2 and 1-to-4 split quadrilateral topological rules. These splines have their full local polynomial preservation orders. In addition, we extend our study to parametric approach and use the symmetric properties of our refinable quintic spline components as a guideline to reduce the number of free parameters in constructing second order C2C2 Hermite interpolatory quadrilateral subdivision schemes with precisely six components.