LevelSet

A level set is the set of points where a scalar function takes a given value. In computational science, the level-set method embeds an interface as the zero level set of a higher dimensional function φ(x,t). The evolving interface Γ(t) is represented by φ(x,t) = 0, while the regions on either side correspond to φ > 0 and φ < 0.

If the interface moves with a velocity field V normal to the surface, the level-set function satisfies a partial differential equation of the form ∂φ/∂t + V · ∇φ = 0. A common specialization uses the normal speed F, giving ∂φ/∂t + F |∇φ| = 0. To preserve φ as a stable distance function, a reinitialization step is often applied to restore φ to a signed distance function to the zero level set.

History and significance: the level-set method was introduced by Osher and Sethian in 1988 as a robust framework for tracking moving interfaces that can undergo topological changes such as merging or breaking apart. Since then it has been developed for a wide range of problems in science and engineering.

Applications: level-set methods are used to track multiphase interfaces in fluids, to evolve shapes and contours in computer graphics, and to perform image segmentation in computer vision. They are also employed in materials science, solidification modeling, and shape optimization, where topological changes and complex geometries are common.

Variants and computational aspects: practical implementations use numerical schemes on grids, with upwind discretizations and reinitialization procedures. Accelerations such as narrow-band methods limit computations to regions near the interface. Level-set methods are often combined with other techniques, like volume-of-fluid, to enhance accuracy and efficiency.

Relation to implicit surfaces: the level-set function is an implicit surface representation that generalizes naturally to higher dimensions and complex topologies.