Approximate Subdivision Surface Evaluation in the Language of Linear Algebra

We present an interpretation of approximate subdivision surface evaluation in the language of linear algebra. Specifically, vertices in the refined mesh can be computed by left-multiplying the vector of control vertices by a sparse matrix we call the subdivision operator. This interpretation is rather general: it applies to any level of subdivision, it holds for many common subdivision schemes (including Catmull-Clark and Loop), it can be extended to support hierarchical edit operations, and it subsumes sharpness and feature-adaptive schemes. Furthermore, our interpretation encourages high-performance implementations built on numerical linear algebra libraries. It is most applicable to subdivision of static control meshes undergoing deformation, i.e. animation, in which case it allows users to trade-off time-to-first-frame and framerate. We implemented our strategy as an extension to Pixar’s production subdivision code and observed speedups of 2x to 14x using both multicore CPUs and GPUs. CR Categories: I.3.5 [Computer Graphics]: Picture/Image Generation—Display algorithms; I.3.7 [Computer Graphics]: Computational Geometry and Object Modelling—Curve, surface, solid, and object representation