Publications

Abstract. The eikonal equation is a subset of the more general static Hamilton–Jacobi equation. It describes a wave propagating through \(d\)-dimensional space with a specified spatially varying wave speed. Solutions to the eikonal equation have been used in numerous fields, including turbulent flow, geometric optics, and computer vision. Fast methods using the causality of this equation solve the equation in \(\mathcal O(N)\)and \(\mathcal O(N\log N)\)time. These methods require careful construction of upwind difference operators to approximate the gradient at mesh points. The gradients computed by these operators can be sensitive to skewed, unstructured meshes, resulting in a breakdown of the method without proper care. In this work, we present a unified method for generating first-order accurate solutions in two dimensions on structured meshes, unstructured meshes, or point clouds. We demonstrate that this method does not break down even on highly skewed unstructured meshes. This method should be extensible to higher-order discretizations and higher dimensions.