An eikonal equation (from Greek εἰκών, image) is a non-linear first-order partial differential equation that is encountered in problems of wave propagation.

The classical eikonal equation in geometric optics is a differential equation of the form

denotes the gradient is the Euclidean norm. In the context of geometric optics is the refractive index of the medium.

More generally, an eikonal equation is an equation of the form

, then equation () becomes ().

Eikonal equations naturally arise in the WKB method and the study of Maxwell's equations. Eikonal equations provide a link between physical (wave) optics and geometric (ray) optics.

One fast computational algorithm to approximate the solution to the eikonal equation is the fast marching method.

History

The term "eikonal" was first used in the context of geometric optics by Heinrich Bruns. However, the actual equation appears earlier in the seminal work of William Rowan Hamilton on geometric optics.

Physical interpretation

Continuous shortest-path problems

. The solution to the eikonal equation

an exit-cost penalty.)

, the solution gives the signed distance.

corresponds to a time-optimal control problem using Bellman's optimality principle and a Taylor expansion. exists at all points, and more advanced techniques are necessary to prove this. This led to the development of viscosity solutions in the 1980s by Pierre-Louis Lions and Michael G. Crandall, and Lions won a Fields Medal for his contributions.

Electromagnetic potential

The physical meaning of the eikonal equation is related to the formula

is the electric field is the electric potential. There is a similar equation for velocity potential in fluid flow and temperature in heat transfer. The physical meaning of this equation in the electromagnetic example is that any charge in the region is pushed to move at right angles to the lines of constant potential, and along lines of force determined by the field of the E vector and the sign of the charge.

Ray optics and electromagnetism are related by the fact that the eikonal equation gives a second electromagnetic formula of the same form as the potential equation above where the line of constant potential has been replaced by a line of constant phase, and the force lines have been replaced by normal vectors coming out of the constant phase line at right angles. The magnitude of these normal vectors is given by the square root of the relative permittivity. The line of constant phase can be considered the edge of one of the advancing light waves (wavefront). The normal vectors are the rays the light is traveling down in ray optics.

Computational algorithms

Several fast and efficient algorithms to solve the eikonal equation have been developed since the 1990s. Many of these algorithms take advantage of algorithms developed much earlier for shortest path problems on graphs with nonnegative edge lengths. These algorithms take advantage of the causality provided by the physical interpretation and typically discretize the domain using a mesh or regular grid and calculate the solution at each discretized point. Eikonal solvers on triangulated surfaces were introduced by Kimmel and Sethian in 1998.

Sethian's fast marching method (FMM) into a regular grid and "marches" the solution from "known" values to the undiscovered regions, precisely mirroring the logic of Dijkstra's algorithm term comes from the use of a heap (typically binary). A number of modifications can be prescribed to FMM since it is classified as a label-setting method. In addition, FMM has been generalized to operate on general meshes that discretize the domain.

Label-correcting methods such as the Bellman–Ford algorithm can also be used to solve the discretized Eikonal equation also with numerous modifications allowed (e.g. "Small Labels First" or "Large Labels Last" ). Two-queue methods have also been developed that are essentially a version of the Bellman-Ford algorithm except two queues are used with a threshold used to determine which queue a gridpoint should be assigned to based on local information.

Sweeping algorithms such as the fast sweeping method (FSM) are highly efficient for solving Eikonal equations when the corresponding characteristic curves do not change direction very often. These algorithms are label-correcting but do not make use of a queue or heap, and instead prescribe different orderings for the gridpoints to be updated and iterate through these orderings until convergence. Some improvements were introduced such as "locking" gridpoints-dimensional subsets and then runs an individual FSM on each subset.-dimensional hyperplane until the entire domain is fully swept.

Hybrid methods have also been introduced that take advantage of FMM's efficiency with FSM's simplicity. For example, the Heap Cell Method (HCM) decomposes the domain into cells and performs FMM on the cell-domain, and each time a "cell" is updated FSM is performed on the local gridpoint-domain that lies within that cell. A parallelized version of HCM has also been developed.

Numerical approximation

in the x and y directions, respectively.

2D approximation on a Cartesian grid

. A first-order scheme to approximate the partial derivatives is

where

Due to the consistent, monotone, and causal properties of this discretization then

, this reduces to

.

:

n-D approximation on a Cartesian grid

is a standard unit basis vector. The approximation is then

Solving this quadratic equation yields:

).

then perform the one-dimensional update

and choose the smallest.

Mathematical description

An eikonal equation is one of the form

can be thought of as the initial condition We could also solve the equation on a subset of this plane, or on a curved surface, with obvious modifications.

The eikonal equation shows up in geometrical optics, which is a way of studying solutions of the wave equation. In geometric optics, the eikonal equation describes the phase fronts of waves. Under reasonable hypothesis on the "initial" data, the eikonal equation admits a local solution, but a global smooth solution (e.g. a solution for all time in the geometrical optics case) is not possible. The reason is that caustics may develop. In the geometrical optics case, this means that wavefronts cross.

, where H=H(x,p) and p=(p1,...,pn) is the variable that gets replaced by ∇u. Here x=(x1,...,xn)=(t,x).

on those curves) as

1. .

follows from standard ODE theorems (using the non-characteristic hypothesis). These curves fill out an open set along these curves.

we must have

and therefore

is a solution)

is curl free as it is the gradient of a function. As for the other term, we note

The result follows.

Applications

• A concrete application is the computation of radiowave attenuation in the atmosphere.

• Finding the shape from shading in computer vision.

• Geometric optics

• Continuous shortest path problems

• Image segmentation

• Study of the shape for a solid propellant rocket grain

Optics, oceanology and quantum mechanics

The analogy between the three fields of optics, oceanology and quantum mechanics can be summarized in the following table. The first row shows the full wave equations from optics, oceanology and QM, namely the Helmholtz equation and the Schrödinger equation respectively.

The second row is the first order WKB approximation, i.e. the slowly varying amplitude variation and the equations obtained are the Eikonal equation of optics and oceanology and the Hamilton-Jacobi equation.

See also

• Hamilton–Jacobi–Bellman equation

• Hamilton–Jacobi equation

• Fermat's principle

References

Further reading

External links

• The linearized eikonal equation

• English translation of "Das Eikonal" by Heinrich Bruns