iLogMap：基于磁拉普拉斯算子的测地极坐标参数化
iLogMap: Geodesic Polar Coordinates Parameterization with the Magnetic Laplacian
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中文总结 AI 辅助
iLogMap通过将测地极坐标的角度分量转化为磁特征值问题，实现了曲面上的精确参数化，在保持角度精度的同时降低度量畸变，并应用于计算心脏病学中的心房和心室建模。
中文摘要 AI 辅助
测地极坐标（GPC）为曲面提供了一种内在参数化方法，但其精确估计仍具挑战性，尤其是在存在各向异性度量、高曲率和复杂拓扑的情况下。我们提出了iLogMap，一种在弯曲域中计算GPC的方法，该方法将对数映射的角度分量重新表述为测地距离周向方向场上的基态磁特征值问题。我们的方法可轻松扩展到各向异性度量张量和实体体积，从而在四面体网格中实现圆柱和球面参数化。在不同亏格的各种形状上的实验证实，与基于热的方法相比，iLogMap具有竞争力的角度精度和更低的度量畸变，在带边界的曲面和各向异性域上性能更优。我们展示了iLogMap在计算心脏病学应用中的实用性，其中我们使用它来初始化心房表面的螺旋相位并估计心室模型中的局部激活模式。
英文摘要
Geodesic polar coordinates (GPCs) provide an intrinsic parameterization over curved surfaces, but their accurate estimation remains challenging, particularly in the presence of anisotropic metrics, high curvature and complex topology. We introduce iLogMap, a method for computing GPCs in curved domains that recasts the angular component of the logarithmic map to a ground-state magnetic eigenproblem over the circumferential direction field of geodesic distance. Our method effortlessly extends to anisotropic metric tensors and solid volumes, enabling cylindrical and spherical parameterizations in tetrahedral meshes. Experiments on diverse shapes with varying genus confirm competitive angular accuracy and reduced metric distortion relative to heat-based methods, with improved performance on surfaces with boundary and domains with anisotropy. We demonstrate the utility of iLogMap in computational cardiology applications, where we use it to initialize spiral phases on atrial surfaces and estimate local activation patterns in ventricular models.
