Distance Field Reinitialization for Localized Level Set Methods

Bibliography 83 Preface Since their introduction by Osher and Sethian [14], level set methods have been used to solve a wide range of problems, including, but not limited to • Photolithography Development, Etching and Deposition in Semi-conductor Manufacturing (see [2]), • Robotic Navigation with Constraints (see [9]). More examples of applications, further references, and a survey can be found in [17, 19]. Also, at the end of this introduction, some figures illustrate the wide area of usage for level set methods. The idea of the level set method is to embed the interface Γ, bounding a (possibly multiple connected) region Ω, as the zero level set of a continuous function Φ (x, t) in one higher dimension, i.e., Γ(t) = { x | Φ (x, t) = 0}. Φ is initialized to be negative inside Ω, positive outside Ω, and zero on Γ(t). Topological breaking and merging thus are well-defined and easily performed. Figure 1 shows such a level set function Φ (x, t). Adalsteinsson and Sethian point out in [1] that the drawback of the level set method, as devised in [14], stems from the expense. Since the interface is embedded as the level set of a higher dimensional function, the increase in computational labor required to solve the problem in a higher dimension can be considerable. This is due to the fact that the equations for the velocity field, which underlie physical laws of motion, have to be recomputed and solved throughout space as well. There are, though, some physical problems, e.g. multiphase incompressible fluid dynamics [20], where the extra dimension of the level set function adds only a small fraction of extra computing time. 1 2 PREFACE Some effort has been made in order to ease the computational labor required by the higher dimensional level set function. Adalsteinsson and Sethian [1] localized the method by defining a so-called tube around the interface that is to be tracked, where the computations take place. For points outside the tube the level set function will not be propagated through time. This " narrow band approach " lowers the computational labor per time step, because only parts of the domain have to be updated. Peng et al. introduced a different approach for localization of the level set method in [15]. They used only the values of the level set function and not the explicit location of points in the domain for …