Level-set method

The level set method is a numerical technique for tracking interfaces and shapes. The advantage of the level set method is that one can perform numerical computations involving curves and surfaces on a fixed Cartezian grid (this is called the Eulerian approach) without having to parametrize these objects. Also, the level set method makes it very easy to follow shapes which change topology, for example when a shape splits in two, develops holes, or the reverse of these operations. All this makes the level set method a great tool to model time-varying objects, like inflation of an airbag, or a drop of oil floating in water.

Let us illustrate the level set method in two dimensions. A closed curve $\Gamma$ in the plane is described with the level set method as the zero level set of a two-dimensional auxiliary function $\phi$,

$\Gamma =\{(x,y)|\phi (x,y)=0\}.$

The function $\phi$ is assumed to have positive values inside the region delimited by the curve $\Gamma$ and negative values outside. $\phi$ is called the level set function.

For a time-varying curve, we consider a time-varying level set function. If the curve moves in the normal direction with a speed of v, this movement can be represented by means of a so-called Hamilton-Jacobi equation for the level set function:

$\phi _{t}=v|\nabla \phi |.$

This is a partial differential equation, and can be solved numerically, for example by using finite differences on a Cartezian grid.

The level set method was developed in the 80's by the American mathematicians Stanley Osher and James Sethian and ever since it became very popular in many disciplines, such as computational fluid dynamics and computer graphics.