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MillieGaming (talk | contribs) Cleaned up the lead section, consolidated the examples, and split the "Overview" section from the lead. In my opinion, rest of the article needs a fairly substantial rewrite to improve clarity and fix the tone. |
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The figure on the right illustrates several ideas about LSM. 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 <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 can be seen changing its topology by splitting in two. It would be difficult to describe this transformation numerically by [[Parametrization (geometry)|parameterizing]] the boundary of the shape and following its evolution. One would need an algorithm to be able 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 level set function accomplishes this change by translating downward. This is an example of when it can be easier to work with a shape through its level-set function than with the shape directly, where the method would need to consider and handle 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
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