基于稀疏优化的可展曲面逼近方法  

Approximation method of developable surface based on sparse optimization

作　　者：龚伟华 金耀[1] GONG Weihua;JIN Yao(School of Information Science and Technology,Zhejiang Sci-Tech University,Hangzhou 310018,China)

机构地区：[1]浙江理工大学信息学院,杭州310018

出　　处：《浙江理工大学学报（自然科学版）》2022年第1期60-68,共9页Journal of Zhejiang Sci-Tech University(Natural Sciences)

基　　金：国家自然科学基金项目(61702458);绍兴市技术创新计划(揭榜挂帅)项目(2020B41006)。

摘　　要：可展曲面能无形变地映射至平面,在工业设计领域有着广泛应用。针对基于严格可展条件的重建方法可能存在逼近误差较大的问题,提出一种基于稀疏优化的网格曲面可展性逼近方法。该方法将"高斯曲率处处为零"的可展条件松弛为"高斯曲率几乎处处为零",运用L;范数定义曲面高斯曲率度量,并结合基于拉普拉斯坐标的逼近能量来控制曲面的形状误差。为求解该非线性非凸问题,采用泰勒公式将高斯曲率函数线性化,并使用交替方向乘子法对原问题若干子问题进行迭代计算。结果表明:该方法能够有效地控制高斯曲率分布,使高斯曲率场奇异点集中分布于个别顶点,并能较好地逼近原曲面;相比现有方法,各种模型上的逼近结果在可展性和逼近精度方面均有提升。Developable surfaces, which can be mapped to a plane without any distortion, are widely applied in the field of industrial design. In view of the problem of large approximation error induced by reconstruction methods based on strict developable condition, a developable approximation method for mesh surfaces based on sparse optimization is proposed. In this method, the developability condition of “Gaussian curvature is zero everywhere” is relaxed to “Gaussian curvature is almost zero everywhere”, and the Gaussian curvature measure of surface is defined by L;norm. The approximation energy based on Laplacian coordinates is used to control the shape error of surface. To solve such a nonlinear and non-convex problem, the Taylor formula is employed to linearize the Gaussian curvature function, and some subproblems of the original problem are calculated iteratively by Alternating Direction Method of Multipliers(ADMM). The experimental results show that this method can effectively control the distribution of Gaussian curvature, so that the singularities will concentrate on some individual vertices and well approximate the original surface. Compared with existing methods, the proposed method achieves an improved developable effect and approximation accuracy in a variety of models.

关 键 词：可展曲面 网格曲面 高斯曲率 稀疏优化 交替方向乘子法 

分 类 号：TP391.7[自动化与计算机技术—计算机应用技术] TB47[自动化与计算机技术—计算机科学与技术]