Quasi-interpolating bivariate dual [formula omitted]-subdivision using 1D stencils

Abstract

In this paper, we study bivariate dual 2-subdivision schemes using one-dimensional (1D) stencils and reproducing bivariate polynomials of high orders. We show that such dual 2-subdivision schemes always possess the quasi-interpolating property (i.e., they interpolate bivariate polynomials of high orders) and are intrinsically linked to both 1D primal interpolating subdivision schemes and 1D masks having linear-phase moments. Using only 1D stencils, such subdivision schemes can be straightforwardly implemented on any quadrilateral meshes and there is no need to design special subdivision rules near extraordinary or boundary vertices. In this paper, we concentrate on a particular quasi-interpolating dual 2-subdivision scheme using 4-point 1D stencils and interpolating all bivariate cubic polynomials. This dual 2-subdivision scheme has C2 smoothness near any ordinary vertex guaranteed by the smoothness of its underlying basis/refinable function, while it achieves C1 smoothness near an extraordinary vertex by our numerical calculation using the known technique of analyzing characteristic maps. As an illustration, we apply the proposed dual 2-subdivision scheme to several arbitrary polyhedra meshes to demonstrate the generated subdivision surfaces.