# Ordered Line Integral Methods for Solving the Eikonal Equation

## TL;DR

This paper introduces new fast, accurate Dijkstra-like algorithms for solving the eikonal and factored eikonal equations on grids, improving efficiency and accuracy over existing methods through local variational minimization and pruning strategies.

## Contribution

The paper presents novel algorithms that significantly reduce computational effort while maintaining linear convergence and high accuracy for eikonal equations in 2D and 3D.

## Key findings

- Methods outperform standard fast marching in error-time tradeoff.
- Two algorithms effectively reduce FLOPs in 3D computations.
- Extensive simulations validate theoretical convergence and efficiency.

## Abstract

We present a family of fast and accurate Dijkstra-like solvers for the eikonal equation and factored eikonal equation which compute solutions on a regular grid by solving local variational minimization problems. Our methods converge linearly but compute significantly more accurate solutions than competing first order methods. In 3D, we present two different families of algorithms which significantly reduce the number of FLOPs needed to obtain an accurate solution to the eikonal equation. One method employs a fast search using local characteristic directions to prune unnecessary updates, and the other uses the theory of constrained optimization to achieve the same end. The proposed solvers are more efficient than the standard fast marching method in terms of the relationship between error and CPU time. We also modify our method for use with the additively factored eikonal equation, which can be solved locally around point sources to maintain linear convergence. We conduct extensive numerical simulations and provide theoretical justification for our approach. A library that implements the proposed solvers is available on GitHub.