Level-set method: Difference between revisions

Revision as of 08:51, 21 March 2024

Revision as of 16:38, 24 March 2024

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The figure on the right illustrates several ideas about LSM. In the upper-left corner there is a shape: 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 cantopology bechanges seenas changingit its topology byis splittingsplit 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 plane at which the level set function accomplishesis thissampled changeis bytranslated translatingdownwards, downward.on Thiswhich the shape's change in topology is andescribed. exampleThis ofshows whenthat it can be easier to work with a shape through its level-set function rather than with the shapeitself directly, wherein thewhich a 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