Pseudo-splines provide a rich family of subdivision schemes with a wide range of choices that meet various demands for balancing the approximation power, the length of the support, and the regularity of the limit functions. Special cases of pseudo-splines include uniform odd-degree B-splines and the interpolatory 2n-point subdivision schemes, and the other pseudo-splines fill the gap between these two families. In this paper we show how the refinement step of a pseudo-spline subdivision scheme can be implemented efficiently using repeated local operations, which require only the data in the direct neighbourhood of each vertex, and how to generalize this concept to quadrilateral meshes with arbitrary topology. The resulting pseudo-spline surfaces can be arbitrarily smooth in regular mesh regions and C~1 at extraordinary vertices as our numerical analysis reveals.
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