Subdivision schemes with optimal bounded curvature near extraordinary vertices

We present a novel method to construct subdivision stencils near extraordinary vertices with limit surfaces having optimal bounded curvature at extraordinary positions. With the proposed method, subdivision stencils for newly inserted and updated vertices near extraordinary vertices are first constructed to ensure subdivision with G1 continuity and bounded curvature at extraordinary positions. The remaining degrees of freedom of the constructed subdivision stencils are further used to optimize the eigenbasis functions corresponding to the subsubdominant eigenvalues of the subdivision with respect to G2 continuity constraints. We demonstrate the method by replacing subdivision stencils near extraordinary vertices for Catmull-Clark subdivision and compare the results with the original Catmull-Clark subdivision and previous tuning schemes known with small curvature variation near extraordinary positions. The results show that the proposed method produces subdivision schemes with better or comparable curvature behavior around extraordinary vertices with comparatively simple subdivision stencils.