Quasi-stationary Subdivision Schemes in Arbitrary Dimensions

TL;DR

This paper introduces multivariate quasi-stationary subdivision schemes that achieve high smoothness with short support, simplifying implementation and overcoming limitations of stationary schemes in arbitrary dimensions.

Contribution

It fully characterizes convergence and smoothness of quasi-stationary schemes and provides practical design methods for interpolatory masks with short support.

Findings

Constructed $C^m$-convergent schemes with short support

Demonstrated advantages over stationary subdivision schemes

Provided examples for dyadic meshes in 2D

Abstract

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.