论文标题
点云的平均曲率运动
Mean curvature motion of point cloud varifolds
论文作者
论文摘要
本文研究了点云上的平均曲率运动的离散方案，并特别强调了奇异演变。为了定义varifold，应用了局部协方差分析来计算云中点的近似切线平面。平均曲率运动模型的核心成分是通过用小模板与内核通过卷积的第一个变化的正规化。如果考虑到足够小的模板和常规采样，则证明了与光滑表面的演化速度的一致性。此外，得出了隐式和半平时的离散化。隐式方案具有离散的屏障属性，以平滑，连续的演化已知，而半平整的屏障属性仍然可以确保我们所有数值实验中的近似值非常好，同时易于实现。结果表明，所提出的方法在噪声方面是鲁棒的，并恢复了平滑曲线的演变以及奇异性的形成，例如2D中的三重点或3D中的最小锥。
This paper investigates a discretization scheme for mean curvature motion on point cloud varifolds with particular emphasis on singular evolutions. To define the varifold a local covariance analysis is applied to compute an approximate tangent plane for the points in the cloud. The core ingredient of the mean curvature motion model is the regularization of the first variation of the varifold via convolution with kernels with small stencil. Consistency with the evolution velocity for a smooth surface is proven if a sufficiently small stencil and a regular sampling are taking into account. Furthermore, an implicit and a semiimplicit time discretization are derived. The implicit scheme comes with discrete barrier properties known for the smooth, continuous evolution, whereas the semiimplicit still ensures in all our numerical experiments very good approximation properties while being easy to implement. It is shown that the proposed method is robust with respect to noise and recovers the evolution of smooth curves as well as the formation of singularities such as triple points in 2D or minimal cones in 3D.
