Editing Level-set method (section)

== Overview ==
The figure on the right illustrates several ideas about LSM. In the upper left corner is a [[bounded region]] with a well-behaved boundary. Below it, the red surface is the graph of a level set function <math>\varphi</math> 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 <math>\varphi</math>, while the shape itself is the set of points in the plane for which <math>\varphi</math> is positive (interior of the shape) or zero (at the boundary).

In the top row, the shape's topology changes as it is split in two. It is challenging to describe this transformation numerically by [[Parametrization (geometry)|parameterizing]] the boundary of the shape and following its evolution. An algorithm can be used to detect the moment the shape splits in two and then construct parameterizations for the two newly obtained curves. On the bottom row, however, the plane at which the level set function is sampled is translated upwards, on which the shape's change in topology is described. It is less challenging to work with a shape through its level-set function rather than with itself directly, in which a method would need to consider all the possible deformations the shape might undergo.

Thus, in two dimensions, the level-set method amounts to representing a [[closed curve]] <math>\Gamma</math> (such as the shape boundary in our example) using an [[auxiliary function]] <math>\varphi</math>, called the level-set function. The curve <math>\Gamma</math> is represented as the zero-level set of <math>\varphi</math> by
:<math>\Gamma = \{(x, y) \mid \varphi(x, y) = 0 \},</math>
and the level-set method manipulates <math>\Gamma</math> ''implicitly'' through the function <math>\varphi</math>. This function <math>\varphi</math> is assumed to take positive values inside the region delimited by the curve <math>\Gamma</math> and negative values outside.<ref name="osher" /><ref name="sethian" />