Quasi-interpolation projectors for Subdivision Surfaces

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.