论文标题
带有完整的Riemannian度量的一系列平面三角网格
A Manifold of Planar Triangular Meshes with Complete Riemannian Metric
论文作者
论文摘要
形状空间在各种应用中都是基本的，包括图像登记，变形，匹配，插值和形状优化。在这项工作中，我们考虑以给定连接性的三角形网格表示的二维形状。我们表明，通过这种网格表示可允许的配置的集合形成了平滑的歧管。对于平面三角网格的这种歧管，我们提出了一个地理上完整的Riemannian度量。它是该度量标准的一个显着特征，它可以保留网格连接性并防止网格沿地球曲线降解。我们详细介绍了其在哈密顿公式中的地理方程的符号数值积分器。数值实验表明，所提出的指标使细胞纵横比远离零，从而避免了沿任意长的大地测量曲线的网格降解。
Shape spaces are fundamental in a variety of applications including image registration, morphing, matching, interpolation, and shape optimization. In this work, we consider two-dimensional shapes represented by triangular meshes of a given connectivity. We show that the collection of admissible configurations representable by such meshes form a smooth manifold. For this manifold of planar triangular meshes we propose a geodesically complete Riemannian metric. It is a distinguishing feature of this metric that it preserves the mesh connectivity and prevents the mesh from degrading along geodesic curves. We detail a symplectic numerical integrator for the geodesic equation in its Hamiltonian formulation. Numerical experiments show that the proposed metric keeps the cell aspect ratios bounded away from zero and thus avoids mesh degradation along arbitrarily long geodesic curves.
