Level-set method: Difference between revisions

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[[File:Levelset-mean-curvature-spiral.ogv|thumb|Video of spiral being propagated by level sets ([[curvature flow]]) in 2D. Left image shows zero-level solution. Right image shows the level-set scalar field.]]

The '''Level-set methodsmethod''' ('''LSM''') constituteis a conceptual framework for using [[level set]]s as a tool for [[numerical analysis]] of [[Surface (topology)|surface]]s and [[shape]]s. Invented in 1988 byUnlike [[StanleyEuler Oshermethod|S.Eulerian Osher]] and [[James Sethian|J. A. Sethianmethods]], the key advantage of LSM is its ability tocan perform [[Numerical computation|numerical computations]] involving [[curve]]s and surfaces on a fixed [[Cartesian grid]] without having to [[Parametric surface|parameterize]] these objects (this is called the ''Eulerian approach'').<ref>{{Citation |last1 = Osher |first1 = S. |last2 = Sethian |first2 = J. A.| title = Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton–Jacobi formulations| journal = J. Comput. Phys.| volume = 79 |issue = 1 |year = 1988 |pages = 12&ndash;49 |url = http://math.berkeley.edu/~sethian/Papers/sethian.osher.88.pdf |doi=10.1016/0021-9991(88)90002-2|bibcode = 1988JCoPh..79...12O |hdl = 10338.dmlcz/144762 |citeseerx = 10.1.1.46.1266|s2cid = 205007680 }}</ref> Importantly, LSM makes it easier to followperform computations on shapes with sharp corners orand shapes that change [[topology]], for(such example,as whenby a shape splitssplitting in two, develops holes, or thedeveloping reverse of these operationsholes). These characteristics make LSM an effective method for modeling objects that vary in time-varying objects, likesuch inflation ofas an [[airbag]], inflating or a drop of oil floating in water.

[[Image:level set method.png|thumb|right|400px|An illustration of the level-set method]]

== Overview ==

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).

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If the field has a constant value subtracted from it in time, the zero level (which was the initial boundary) of the new fields will also be circular and will similarly collapse to a point. This is due to this being effectively the temporal integration of the [[Eikonal equation]] with a fixed front [[velocity]].

== Applications ==

In [[combustion]], this method is used to describe the instantaneous flame surface, known as the [[G equation]].

*In mathematical modeling of [[combustion]], this methodLSM is used to describe the instantaneous flame surface, known as the [[G equation]].

A number of *[[level set (data structures)|levelLevel-set data structures]] have been developed to facilitate the use of the level-set method in computer applications.

* [[Computational fluid dynamics]]

* [[Trajectory|Trajectory planning]]

* [[Mathematical optimization|Optimization]]

* [[Image processing]]

* [[Computational biophysics]]

* Discrete [[complex dynamics]]: (visualization of the [[b:Fractals/Iterations in the complex plane/Mandelbrot set|parameter plane]] and the [[b:Fractals/Iterations in the complex plane/Julia set|dynamic plane]])

==History==

The level-set method was developed in 1979 by Alain Dervieux,<ref>{{cite book |last1=Dervieux |first1=A. |last2=Thomasset |first2=F. |chapter=A finite element method for the simulation of a Rayleigh-Taylor instability |chapter-url= |title=Approximation Methods for Navier-Stokes Problems |publisher=Springer |series=Lecture Notes in Mathematics |volume=771 |date=1980 |isbn=978-3-540-38550-9 |pages=145–158 |doi=10.1007/BFb0086904}}</ref> and subsequently popularized by [[Stanley Osher]] and [[James Sethian]]. It has since become popular in many disciplines, such as [[image processing]], [[computer graphics]], [[computational geometry]], [[optimization (mathematics)|optimization]], [[computational fluid dynamics]], and [[computational biology]].

A number of [[level set (data structures)|level-set data structures]] have been developed to facilitate the use of the level-set method in computer applications.

==Applications==

* [[Computational fluid dynamics]]

* [[Combustion]]

* [[Trajectory|Trajectory planning]]

* [[Mathematical optimization|Optimization]]

* [[Image processing]]

* [[Computational biophysics]]

* Discrete [[complex dynamics]]: visualization of [[b:Fractals/Iterations in the complex plane/Mandelbrot set|parameter plane]] and [[b:Fractals/Iterations in the complex plane/Julia set|dynamic plane]]

==See also==

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==External links==

* See [[Ronald Fedkiw]]'s [http://graphics.stanford.edu/~fedkiw/ academic web page] for many stunning pictures and animations showing how the level-set method can be used to model real-life phenomena, like fire, water, cloth, fracturing materials, etc.

* [http://vivienmallet.net/fronts/ Multivac] is a C++ library for front tracking in 2D with level-set methods.

* [[James Sethian]]'s [http://math.berkeley.edu/~sethian/ web page] on level-set method.