Curvature adaptive surface remeshing by sampling normal cycle. (June 2019) - Su, Kehua; Lei, Na; Chen, Wei; Cui, Li; Si, Hang; Chen, Shikui; Gu, Xianfeng
Abstract: Surface meshing plays a fundamental important role in Visualization and Computer Graphics, which produces discrete meshes to approximate a smooth surface. Many geometric processing tasks heavily depend on the qualities of the meshes, especially the convergence in terms of topology, position, Riemannian metric, differential operators and curvature measures. Normal cycle theory points out that in order to guarantee the convergence of curvature measures, the discrete meshes are required to approximate not only the smooth surface itself, but also the normal cycle of the surface. This theory inspires the development of the remeshing method based on conformal parameterization and planar Delaunay refinement, which uniformly samples the smooth surface, and produces Delaunay triangulations with bounded minimal corner angles. This method ensures the Hausdorff distances between the normal cycles of the resulting meshes and the smooth normal cycle converges to 0, the discrete Gaussian curvature and mean curvature measures of the resulting meshes converge to their counter parts on the smooth surface. In the current work, the conformal parameterization based remeshing algorithm is further improved to speed up the curvature convergence. Instead of uniformly sampling the surface itself, the novel algorithm samples the normal cycle of the surface. The algorithm pipeline is as follows: first, two parameterizations are constructed, one is the surface conformal parameterization based on dynamic Ricci flow, the other is the normal cycle area-preserving parameterization based on optimal mass transportation; second, the normal cycle parameterization is uniformly sampled; third, the Delaunay refinement mesh generation is carried out on the surface conformal parameterization. The produced meshes can be proven to converge to the smooth surface in terms of curvature measures. Experimental results demonstrate the efficiency and efficacy of proposed algorithm, the convergence speeds of the curvatures are prominently faster than those of conventional methods. Graphical abstract: Highlights: Theoretical foundation: discrete optimal mass transportation and surface Ricci flow. 3D to 2D: parametric area proportional to curvature and minimizing the distortion. The high quality 3D meshing is obtained by 2D Delaunay refinement. Results guaranteed to converge to the smooth surface in terms of curvature measures.