We propose an unified λ -subdivision scheme with a continuous family of tuned subdivisions for quadrilateral meshes. Main subdivision stencil parameters of the unified scheme are represented as spline functions of the subdominant eigenvalue λ of respective subdivision matrices and the λ value can be selected within a wide range to produce desired properties of refined meshes and limit surfaces with optimal curvature performance in extraordinary regions. Spline representations of stencil parameters are constructed based on discrete optimized stencil coefficients obtained by a general tuning framework that optimizes eigenvectors of subdivision matrices towards curvature continuity conditions. To further improve the quality of limit surfaces, a weighting function is devised to penalize sign changes of Gauss curvatures on respective second order characteristic maps. By selecting an appropriate λ , the resulting unified subdivision scheme produces anticipated properties towards different target applications, including nice properties of several other existing tuned subdivision schemes. Comparison results also validate the advantage of the proposed scheme with higher quality surfaces for subdivision at lower λ values, a challenging task for other related tuned subdivision schemes.

Abstract This paper presents subdivision schemes with subdivision stencils near an extraordinary vertex that are free from or with substantially reduced polar artifact in extraordinary regions while maintaining the best possible bounded curvature at extraordinary positions. The subdivision stencils are firstly constructed to meet tangent plane continuity with bounded curvature at extraordinary positions. They are further optimized towards curvature continuity at an extraordinary position with additional measures for removing or for minimizing the polar artifact in extraordinary regions. The polar artifact for subdivision stencils of lower valences is removed by applying an additional constraint to the subdominant eigenvalue to be the same as that of subdivision at regular vertices, while the polar artifact for subdivision stencils of higher valances is substantially reduced by introducing an additional thin‐plate energy function and a penalty function for maintaining the uniformity and regularity of the characteristic map. A new tuned subdivision scheme is introduced by replacing subdivision stencils of Catmull‐Clark subdivision with that from this paper for extraordinary vertices of valences up to nine. We also compare the refined meshes and limit surface quality of the resulting subdivision scheme with that of Catmull‐Clark subdivision and other tuned subdivision schemes. The results show that subdivision stencils from our method produce well behaved subdivision meshes with the least polar artifact while maintaining satisfactory limit surface quality.

Abstract 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.