Many geometry processing techniques require the solution of partial differential equations (PDEs) on surfaces. Such surface PDEs often involve boundary conditions prescribed on the surface, at points or curves on its interior or along the geometric (exterior) boundary of an open surface. However, input surfaces can take many forms (e.g., meshes, parametric surfaces, point clouds, level sets, neural implicits). One must therefore generate a mesh to apply finite element-type techniques or derive specialized discretization procedures for each surface representation. We propose instead to address such problems through a novel extension of the closest point method (CPM) to handle interior boundary conditions specified at surface points or curves. CPM solves the surface PDE by solving a volumetric PDE defined over the Cartesian embedding space containing the surface; only a closest point function is required to represent the surface. As such, CPM supports surfaces that are open or closed, orientable or not, and of any codimension or even mixed-codimension. To enable support for interior boundary conditions, we develop a method to implicitly partition the embedding space across interior boundaries. CPM's finite difference and interpolation stencils are adapted to respect this partition while preserving second-order accuracy. Furthermore, an efficient sparse-grid implementation and numerical solver is developed that can scale to tens of millions of degrees of freedom, allowing PDEs to be solved on more complex surfaces. We demonstrate our method's convergence behaviour on selected model PDEs. Several geometry processing problems are explored: diffusion curves on surfaces, geodesic distance, tangent vector field design, and harmonic map construction. Our proposed approach thus offers a powerful and flexible new tool for a range of geometry processing tasks on general surface representations.

翻译：许多几何处理技术需要求解曲面上的偏微分方程。这类曲面偏微分方程通常涉及在曲面上的点、内部曲线或开放曲面的几何（外部）边界处给定的边界条件。然而，输入曲面可能具有多种形式（例如网格、参数曲面、点云、水平集、神经隐式表示）。因此，必须生成网格以应用有限元类技术，或为每种曲面表示推导专门的离散化方法。本文提出了一种新的最邻近点方法扩展，以处理在曲面上点或曲线处指定的内边界条件。该方法通过求解包含曲面的笛卡尔嵌入空间上的体偏微分方程来处理曲面偏微分方程；仅需一个最邻近点函数即可表示曲面。因此，该方法支持开放或封闭、可定向或不可定向的曲面，以及任意余维数甚至混合余维数的曲面。为支持内边界条件，我们开发了一种在内部边界处隐式划分嵌入空间的方法。最邻近点方法的有限差分和插值模板被调整以尊重这种划分，同时保留二阶精度。此外，我们开发了一种高效的稀疏网格实现和数值求解器，可扩展到数千万自由度，从而能够在更复杂的曲面上求解偏微分方程。我们在选定的模型偏微分方程上展示了该方法的收敛行为。探索了若干几何处理问题：曲面上的扩散曲线、测地距离、切向量场设计和调和映射构造。因此，所提出的方法为通用曲面表示上的几何处理任务提供了一种强大而灵活的新工具。