ANISOTROPIC SUBDIVISION WITH THE OPTIMAL REPRODUCTION PROPERTY

Abstract

This paper presents several families of anisotropic interpolatory subdivisions satisfying the optimal reproduction property with the dilation matrix $\text{diag}(3,2)$. Such schemes are obtained based on the discussion of the relationship between sum rules and polynomial generation for anisotropic schemes. The corresponding masks are symmetric about the origin and imply that, with a relaxed requirement on symmetry, different families of similar schemes with the optimal reproduction property can be obtained. We also show that, given the unimodular matrices $\mathrm{\Theta },{\mathrm{\Theta }}^{\prime }$, the interpolatory schemes with the dilation matrix $\mathrm{\Theta }\text{diag}(3,2){\mathrm{\Theta }}^{\prime }$ and the ones with the dilation matrix $\text{diag}(3,2)$ have the same reproduction property. The convergence of these new schemes is analyzed based on the convergence analysis in the isotropic case. Some numerical examples are given to illustrate the performance of these new schemes.