A variational formulation of the fast marching eikonal solver

I exploit the theoretical link between the eikonal equation and Fermat’s principle to derive a variational interpretation of the recently developed method for fast traveltime computations. This method, known as fast marching, possesses remarkable computational properties. Based originally on the eikonal equation, it can be derived equally well from Fermat’s principle. The new variational formulation has two important applications: First, the method can be extended naturally for traveltime computation on unstructured (triangulated) grids. Second, it can be generalized to handle other Hamilton-type equations through their correspondence with variational principles. Now we are in the rarefied atmosphere of theories of excessive beauty and we are nearing a high plateau on which geometry, optics, mechanics, and wave mechanics meet on a common ground. Only concentrated thinking, and a considerable amount of re-creation, will reveal the full beauty of our subject in which the last word has not been spoken yet.–Cornelius Lanczos, The variational principles of mechanics