Level-set method: Difference between revisions

Revision as of 01:31, 7 December 2023

Revision as of 07:42, 10 December 2023

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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.]]

'''Level-set methods''' ('''LSM''') are a conceptual framework for using [[level set]]s as a tool for [[numerical analysis]] of [[Surface (topology)|surface]]s and [[shape]]s. The advantage of the level-set model is that one can perform 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}}</ref> Also, the level-set method makes it very easyeasier to follow shapes that change [[topology]], for example, when a shape splits in two, develops holes, or the reverse of these operations. AllThese thesecharacteristics make the level-set method aan greateffective toolmethod for modeling time-varying objects, like inflation of an [[airbag]], or a drop of oil floating in water.

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

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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 <math>\varphi</math> determining this shape, and the flat blue region represents the ''xy'' 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 we see, the shape can be seen changing its topology by splitting in two. It would be quite harddifficult to describe this transformation numerically by parameterizing the boundary of the shape and following its evolution. One would need an algorithm able to detect the moment the shape splits in two, and then construct parameterizations for the two newly obtained curves. On the other hand, if we look at the bottom row, we see thathowever, the level set function merelyaccomplishes this change by translatedtranslating downward. This is an example of when it can be much easier to work with a shape through its level-set function than with the shape directly, where using the shape directlymethod 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. <math>\Gamma</math> is represented as the zero-level set of <math>\varphi</math> by

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==Example==

Consider a unit circle in <math display="inline">\mathbb{R}^2</math>, shrinking in on itself at a constant rate, i.e. each point on the boundary of the circle moves along its inwards pointing normal at some fixed speed. The circle will shrink and eventually collapse down to a point. If an initial distance field is constructed (i.e. a function whose value is the signed euclideanEuclidean distance to the boundary, positive interior, negative exterior) on the initial circle, the normalisednormalized gradient of this field will be the circle normal.

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.