Improving the Computation Efficiency of Mean Value Coordinates for Quad Cages in 3D

To improve the computational efficiency of mean value coordinates for quad cages in 3D (QMVC), this paper presents an improved method. The improvement consists of two parts. The first part focuses on the parameterization of quad patches: as the relative distance between the sampling points and a quad patch decreases from far to near, the quad patch is successively treated as two triangles, a rational bilinear surface, and a standard bilinear surface. By treating the quad patches as two triangles, the calculation of quads’ vertex weights is significantly faster. The rational bilinear surface is introduced to allow for a smooth geometric transition from two triangles to the standard bilinear surface, with the position and rate of transition being freely controlled by predefined parameters. The second part involves an improvement in the numerical integration method: a more efficient strategy is employed for the subdivision of quad patches, further improving the computational efficiency. Experiments on four different models with the same parameters and on the same model with different parameters are conducted. The results demonstrate that by selecting appropriate parameters, the improved method can effectively balance the computational efficiency and the deformation quality. With the default parameters of the improved method, the nail box model achieves almost the same deformation effect as QMVC, but the computation time is reduced from 7 579 seconds to 3 808 seconds.