A Fast Approach To Compute Smooth Geodesic Distance Fields Using For High-quality Local Parameterization

Local parameterization provides a reliable tool for shape analysis, and thus has been widely used in digital geometry processing. Conventional algorithms have to depend on the computation of geodesic distance fields. However, for whether exact geodesic algorithms or approximate ones, the marching direction of a geodesic path is highly sensitive to triangulation, causing an uneven angle distribution in the local parameterization result. To obtain high-quality local parameterization results and we observe smooth distance fields have fewer singularities. So we propose a fast numerical approach for computing 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 one to find the solution by solving the heat equation. Generally, the number of unknown variables is reduced to 10% or even a less percentage of the number of mesh vertices. We further use a simple eigen-decomposition operation, for defining the low-frequency subspace, to speed up the computation. Comparative studies validate the usefulness and effectiveness of our approach in local parameterization.