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.
翻译:形状空间是各种应用的基础, 包括图像登记、 变形、 匹配、 内插和形状优化。 在这项工作中, 我们考虑一个特定连接的三角网格代表的二维形状。 我们显示, 这种网格代表的可接受配置的收集形成一个光滑的方块。 对于这多种平面三角网格, 我们提出了一个具有大地测量学特征的完整里伊曼尼度量仪。 它保护网格连通性, 防止网格在大地测量曲线上退化。 我们在其汉密尔顿式配方中详细描述了大地测量方形的跨度数字集成器。 数字实验显示, 拟议的指标将细胞方块比从零处拉开, 从而避免了任意长的地标曲线的网状退化 。