Parametrization of General Catmull-Clark Subdivision Surfaces and its Applications

A new parametrization technique and its applications for general Catmull-Clark subdivision surfaces are presented. The new technique extends J. Stam’s work by redefining all the eigen basis functions in the parametric representation for general Catmull-Clark subdivision surfaces and giving each of them an explicit form. The entire eigenstructure of the subdivision matrix and its inverse are computed exactly and explicitly with no need to precompute anything. Therefore, the new representation can be used not only for evaluation purpose, but for analysis purpose as well. The new approach is based on an Ω-partition of the parameter space and a detoured subdivision path. This results in a block diagonal matrix with constant size diagonal blocks (7×7) for the corresponding subdivision process. Consequently, eigen decomposition of the matrix is always possible and is simpler and more efficient. Furthermore, since the number of eigen basis functions required in the new approach is only one half of the previous approach, the new parametrization is also more efficient for evaluation purpose. This is demonstrated by several applications of the new techniques.