Level set method

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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 Cartesian grid without having to parameterize these objects (this is called the Eulerian approach). Also, the level set method makes it very easy to follow shapes that change topology, for example when a shape splits in two, develops holes, or the reverse of these operations. All these make the level set method a great tool for modeling time-varying objects, like inflation of an airbag, or a drop of oil floating in water.

In two dimensions, the level set method amounts to representing a closed curve $Γ$ in the plane as the zero level set of a two-dimensional auxiliary function $φ$ ,

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

and then manipulating $Γ$ implicitly, through the function $φ$ . This function is called a level set function. $φ$ is assumed to take positive values inside the region delimited by the curve $Γ$ and negative values outside.

The picture above illustrates several important ideas about the level set method. In the upper-left corner we see a shape, that is, a bounded region with a well-behaved boundary. Below it, the red surface is the graph of a level set function $φ$ determining this shape, and the flat blue region represents the $x - y$ plane. The boundary of the shape is then the zero level set of $φ$ , while the shape itself is the set of points in the plane for which $φ$ is positive or zero.

In the top row we see a shape changing topology by splitting in two. It would be quite hard to describe this transformation numerically by parameterizing the boundary of the shape and following its evolution. One would need an algorithm able to detect the moment the shape splits in two, and then construct parameterizations for the two newly obtained curves. On the other hand, if we look at the bottom row, we see that the level set function merely got translated downward. We see that it is much easier to work with a shape through its level set function than with the shape directly, when we need to watch out for all the possible deformations the shape might undergo.

If the zero level set moves in the normal direction to itself with a speed 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 Cartesian grid.

The level set method was developed in the 1980s by the American mathematicians Stanley Osher and James Sethian. It has become popular in many disciplines, such as image processing, computer graphics, computational geometry, optimization, and computational fluid dynamics.

A number of level set data structures have been developed to facilitate the use of the level set method in computer applications.

[edit] References

Osher, S. & Sethian, J. A. (1988) Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations. Journal of Computation Physics, vol. 79, pages 12–49.
Osher, Stanley J. & Fedkiw, Ronald P. (2002) Level Set Methods and Dynamic Implicit Surfaces. Springer-Verlag. ISBN 0-387-95482-1.

Sethian, James A. (1999) Level Set Methods and Fast Marching Methods : Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science (2nd ed.). Cambridge University Press. ISBN 0-521-64557-3.

[edit] External links

See Ronald Fedkiw's academic web page for many stunning pictures and animations showing how the level set method can be used to model real life phenomena, like fire, water, cloth, fracturing materials, etc.
Multivac is a C++ library for front tracking in 2D with level set methods.
James Sethian's web page on level set method.
Stanley Osher's homepage.