iLogMap: Geodesic Polar Coordinates Parameterization with the Magnetic Laplacian

2026-09-09

AI Summary

iLogMap通过将对数映射的角分量重铸为磁本征问题，解决了在复杂曲面和各向异性度量下准确估计测地极坐标的问题。

Abstract

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.

Problem

• Geodesic Polar Coordinates
• Anisotropic Metrics
• High Curvature
• Complex Topology

Innovation

• Geodesic Polar Coordinates
• Magnetic Laplacian
• Anisotropic Metrics
• Tetrahedral Meshes