Computing intrinsic distances on discrete surfaces is at the heart of many minimization problems in geometry processing and beyond. Solving these problems is extremely challenging as it demands the computation of on-surface distances along with their derivatives. We present a novel approach for intrinsic minimization of distance-based objectives defined on triangle meshes. Using a variational formulation of shortest-path geodesics, we compute first and second-order distance derivatives based on the implicit function theorem, thus opening the door to efficient Newton-type minimization solvers. We demonstrate our differentiable geodesic distance framework on a wide range of examples, including geodesic networks and membranes on surfaces of arbitrary genus, two-way coupling between hosting surface and embedded system, differentiable geodesic Voronoi diagrams, and efficient computation of Karcher means on complex shapes. Our analysis shows that second-order descent methods based on our differentiable geodesics outperform existing first-order and quasi-Newton methods by large margins.

翻译：在离散曲面上计算内蕴距离是几何处理及众多领域中最小化问题的核心。求解此类问题极具挑战性，因为它需要同时计算曲面上的距离及其导数。我们提出了一种针对三角网格上基于距离目标函数的内蕴最小化新方法。利用最短路径测地线的变分公式，我们基于隐函数定理计算距离的一阶和二阶导数，从而为高效的牛顿型最小化解算器打开大门。我们在广泛实例中展示了可微测地距离框架的适用性，包括任意亏格曲面上的测地网络与薄膜、宿主曲面与嵌入系统之间的双向耦合、可微测地Voronoi图，以及复杂形状上Karcher均值的高效计算。分析表明，基于我们可微测地线的二阶下降方法在性能上大幅超越现有的一阶和拟牛顿方法。