Modified Heat Method using GMRES for Geodesic Distance

Abstract

The computation of geodesic distance has applications in a wide variety of fields. Several attempts have been made in the last decade to compute geodesic on curved surfaces by formulating a distance function for the whole mesh including the heat method by Keenan crane. The proposed method first creates a vector unit field with a gradient equal to the true geodesic function and then integrates the unit vector field over the surface. Integrating the unit vector field over the surface requires solving a Poisson equation, which consists of two sparse linear systems of equations. The heat method proposed by Keenan crane uses a direct solver to solve the Poisson equation which increases the total memory and time consumption of the original heat method and makes it unsuitable for large meshes. The proposed method uses the generalized minimal residual method (GMRES) for solving the set of sparse linear systems of equations generated. The use of an iterative solver not only significantly reduces the memory consumption but also computes geodesic distance in less time than the heat method for large meshes which makes the proposed method preferable for large meshes. The result shows that the proposed method can efficiently compute the geodesic distance for bigger mesh in less time than the heat method and Biconjugate gradient stabilized method with significantly reduced memory usage for considered mesh data.