Approximating Doo–Sabin limit surfaces using geometrically continuous Bézier patchesImage 999

We present an efficient, globally G 1 –continuous scheme for extracting Bézier patches from any mesh with valence-four vertices and polygonal faces, that is, the mesh topology that arises in Doo-Sabin subdivision. The Bézier points are given explicitly as local, weighted averages in the vicinity of each vertex of the mesh, yielding bi-quadratic patches in regular regions and bi-quartic or bi-quintic patches in the vicinity of irregular regions. In particular, we first derive simple bi-quadratic averaging masks that produce quadratic patches that join with C 1 continuity in regular regions. In the vicinity of irregular faces, we elevate the degree, then impose certain symmetric gluing data of degree two on the patch interfaces, and compute explicitly masks as solution of the G 1 constraints. The resulting scheme, named G 1 ADS, ensures machine precision adherence to the G 1 conditions, reproduces quadratic C 1 B-splines at the regular regions, and enjoys a minimal number of degree elevated patches in the vicinity of irregular regions. We evaluate the performance of G 1 ADS quantitatively and qualitatively on several challenging benchmarks, in terms of accuracy, curvature and isophote analysis, and we compare with the state-of-the-art method. Our results demonstrate that G 1 ADS is efficient, robust and more accurate, producing high quality surfaces that converge to the respective Doo-Sabin limit surface.