Mathematical Methods for Engineers II by Gilbert Strang (MIT OCW 18.086 March 3). William Gilbert Strang, usually known as simply Gilbert Strang, is a renowned American mathematician, with contributions to finite element theory, the calculus of variations, and wavelet analysis. He has made many contributions to mathematics education, including publishing six classic mathematics textbooks and one definitive monograph. Strang is a Professor of Mathematics at the Massachusetts Institute of Technology.
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,
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.
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:
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.