A Fast Approach to Compute Smooth Geodesic Distance Fields Using for High-Quality Local Parameterization

Local parameterization has been widely used in digital geometry processing. Conventional algorithms typically rely on the computation of geodesic distance fields. However, for both exact geodesic algorithms and approximate algorithms, the marching direction of a geodesic path is highly sensitive to triangulation, resulting in an uneven angle distribution in the local parameterization result. To obtain high-quality local parameterization results that are as equidistant as possible, we propose a fast numerical approach to compute smooth geodesic distance fields. We represent the target geodesic distance field by a linear span of the basis vectors in the low-frequency subspace of the Laplace operator, which enables finding the solution by solving the heat equation. Furthermore, by extracting only the first k eigenvalues for preprocessing, we can use a simple eigen-decomposition operation to speed up the computation, achieving real-time user interaction on larger-scale 3D models. By comparing our method with the VTP algorithm and the Heat method in terms of single calculation speed and smoothness, we demonstrate its advantages. Additionally, in comparative experiments of local parameterization, our method shows improved isometry in local parameterization tasks.