Accurate curved surface mapping method improves heart model simulations
iLogMap: Geodesic Polar Coordinates Parameterization with the Magnetic Laplacian
Computational Geometry
Summary
Mapping curved surfaces accurately is tricky, especially when shapes are complex or irregular. The paper introduces iLogMap, a new way to map surfaces by using a special mathematical approach based on magnetic fields to better capture angles around points. This method works well even when the surface properties vary in different directions or inside 3D shapes. The researchers show that iLogMap performs better than older methods, particularly for surfaces with edges or unusual geometry. They also demonstrate its usefulness by applying it to models of heart chambers to help study electrical activation patterns.
geodesic polar coordinateslogarithmic mapmagnetic Laplaciananisotropic metricsangular accuracycurved surfacestetrahedral meshesparameterizationcomputational cardiology
Authors
Tomás Banduc, Simone Pezzuto, Francisco Sahli Costabal
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.