Subdivision Schemes for Quadrilateral Meshes with the Least Polar Artifact in Extraordinary Regions

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.