Generalized Catmull-Clark Subdivision

The Catmull-Clark subdivision algorithm consists of an operator that can be decomposed into a refinement operator and a successively executed smoothing operator, where the refinement operator splits each face with m vertices into m quadrilateral subfaces and the smoothing operator replaces each internal vertex with an affine combination of its neighboring vertices and itself. Over regular meshes, this smoothing operator is identical to applying the (face-)midpoint operator twice, where each application of the midpoint operator maps a mesh to the dual mesh that connects the centers of adjacent faces. In this paper, we generalize the Catmull-Clark scheme by generalizing the smoothing operator on regular meshes and by combining several smoothing operations into one subdivision step. The generalized Catmull-Clark subdivision operators build an infinite class of quadrilateral subdivision schemes, which includes the Catmull-Clark scheme with restricted parameters and the midpoint schemes. We analyze the smoothness of the resulting subdivision surfaces at regular and at extraordinary points by estimating the norm of a second order difference scheme and by using established methods for analyzing midpoint subdivision. The surfaces are smooth for regular meshes and they are also smooth at extraordinary points for most generalized Catmull-Clark subdivision schemes.