Grid discretization based method for anisotropic shortest path problem over continuous regions

The paper presents a method to find the shortest path between two points over a continuous region. The length of a path is the integral of the cost along the path. The cost can be anisotropic, meaning that it depends on both the position on the path and its direction. The method uses a simple rectangular grid to discretize the region, independent of the cost. Because the cost is anisotropic, rectilinear paths connecting adjacent grid points may not approximate the optimal path. To overcome this "limit-on-direction" problem, the method searches over shifted versions of the rectilinear paths. A Bellman-Ford style algorithm finds the best shifted path. Theoretical analysis and numerical experiments ensure efficiency of the algorithm.