多层次优化的网格曲面离散样条曲线设计方法

Discrete Spline Curve Design on Surface Mesh via Cascaded Optimization

为同时满足鲁棒性、高效性以及广泛适用性等要求,提出一种基于多层次优化的网格域样条曲线设计方法.该方法放松了曲线严格位于曲面的约束,仅将曲线的离散控制点置于流形空间,并采用内点法的思想,运用基于块坐标下降法的全局优化方法进行数值求解,最后借助局部参数化将曲线段映射到网格曲面.在此基础上,文中提出由粗到细的多尺度层次求解策略,不仅能够更为准确地估算离散微分算子,提高求解精度,而且能够减少挪动采样点的计算量,提升求解效率.收敛性分析实验表明,多层次优化方法能够快速收敛,并在多尺度策略下获得更为光滑的结果.和现有的投影法和光顺法相比,该方法效率更高,且在可控性、普适性和鲁棒性上均表现出一定的优势.

To meet the requirements of robustness,high efficiency and wide applicability,this paper proposes a discrete spline curve design method on mesh domain based on global optimization.The method relaxes the strict manifold constraint,i.e.the curve is located on the surface by constraining discrete points of the curve.With the idea of interior point method,it adopts block coordinate descent method to solve the global optimization equation,and then maps the curve to the mesh surface through local parameterization.In addition,a multi-scales coarse-to-fine strategy is proposed to accelerate computation,which can not only estimate the discretization of the differential operator more accurately,but also decreases the computation of the sample point locations to improve the performance.Experiments on convergence analysis show that the cascaded optimization method converges quickly,obtaining smoother results with multi-scales strategy.In contrast to existing projection-based and smoothness-based methods,the proposed method has advantages in controllability,generality and robustness.