Level set method

Level set methods (LSM) are a conceptual framework for using level sets as a tool for numerical analysis of surfaces and shapes. The advantage of the level set model 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.

The figure on the right 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 ${\displaystyle \varphi }$ determining this shape, and the flat blue region represents the xy-plane. The boundary of the shape is then the zero level set of ${\displaystyle \varphi }$, while the shape itself is the set of points in the plane for which ${\displaystyle \varphi }$ is positive (interior of the shape) or zero (at the boundary).

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