Quasi-interpolation projectors for Subdivision Surfaces

TL;DR

The paper develops a general, local quasi-interpolation framework for subdivision surfaces by formulating subdivision-space projectors that reproduce the subdivision space via a local interpolation problem tied to the subdivision matrix $\bm{S}$ and limit-position matrix $\bm{L}$, yielding explicit projectors for Catmull–Clark, Loop, and Modified Loop schemes. By solving $\bm{A}\bm{c}=\bm{f}$ with $\bm{A}=\bm{L}\bm{S}$, the linear functionals are obtained from rows of $\bm{A}^{-1}$, enabling $Q(f)=\sum_i \lambda_i(f) B_i$ to reproduce $\mathcal{S}$. Numerical experiments show that Catmull–Clark and Loop quasi-interpolants achieve $3^{\text{rd}}$-order convergence in $L_2$ and $2^{\text{nd}}$-order in $L_\infty$ on regular meshes, while the Modified Loop quasi-interpolant attains optimal rates in both norms; boundary and extraordinary-point treatments are handled via local region strategies. The approach provides an efficient, explicit alternative to global solvers for isogeometric analysis on models with complex topology. It also highlights remaining theoretical gaps, notably proving $L_2$-order parity with the subdivision space and exploring higher-order extensions.

Abstract

Subdivision surfaces are considered as an extension of splines to accommodate models with complex topologies, making them useful for addressing PDEs on models with complex topologies in isogeometric analysis. This has generated a lot of interest in the field of subdivision space approximation. The quasi-interpolation offers a highly efficient approach for spline approximation, eliminating the necessity of solving large linear systems of equations. Nevertheless, the lack of analytical expressions at extraordinary points on subdivision surfaces makes traditional techniques for creating B-spline quasi-interpolants inappropriate for subdivision spaces. To address this obstacle, this paper innovatively reframes the evaluation issue associated with subdivision surfaces as a correlation between subdivision matrices and limit points, offering a thorough method for quasi-interpolation specifically designed for subdivision surfaces. This developed quasi-interpolant, termed the subdivision space projection operator, accurately reproduces the subdivision space. We provide explicit quasi-interpolation formulas for various typical subdivision schemes. Numerical experiments demonstrate that the quasi-interpolants for Catmull-Clark and Loop subdivision exhibit third-order approximation in the (L_2) norm and second-order in the (L_\infty) norm. Furthermore, the modified Loop subdivision quasi-interpolant achieves optimal approximation rates in both the (L_2) and (L_\infty) norms.

Figures (22)

• Figure 1: The subdivision rules for cubic B-splines, where the circles denote the new vertices.
• Figure 2: Initial and subdivided control points, where solid line represents local interval $\Omega_i$.
• Figure 3: Catmull–Clark subdivision rules and the limit position mask.
• Figure 4: The local domain $\Omega_i$ is shaded by gray. (a) The interpolation points around an extraordinary point are marked by black solid circles. (b) The associated vertices of the non-vanishing basis functions on the local domain $\Omega_i$ are marked by white circles. The mesh around $\Omega_i$ after one subdivision is shown by dashed grid.
• Figure 5: The coefficients of the linear functional $\lambda_i$ defined by \ref{['lamdaK']} corresponds to a vertex of valence $n$ for the quasi-interpolation of the Catmull–Clark subdivision.

Theorems & Definitions (1)

• Remark 4.1