Quasi-stationary Subdivision Schemes in Arbitrary Dimensions

Stationary subdivision schemes have been extensively studied and have numerous applications in CAGD and wavelet analysis. To have high-order smoothness of the scheme, it is usually inevitable to enlarge the support of the mask that is used, which is a major difficulty with stationary subdivision schemes due to complicated implementation and dramatically increased special subdivision rules at extraordinary vertices. In this paper, we introduce the notion of a multivariate quasi-stationary subdivision scheme and fully characterize its convergence and smoothness. We will also discuss the general procedure of designing interpolatory masks with short support that yields smooth quasi-stationary subdivision schemes. Specifically, using the dyadic dilation of both triangular and quadrilateral meshes, for each smoothness exponent $m=1,2$, we obtain examples of $C^m$-convergent quasi-stationary $2I_2$-subdivision schemes with bivariate symmetric masks having at most $m$-ring stencils. Our examples demonstrate the advantage of quasi-stationary subdivision schemes, which can circumvent the difficulty above with stationary subdivision schemes.