Level Set Methods

Abstract

AbstractThe shape of a real‐world object can be represented by a parametric or a nonparametric contour in the image plane. The parametric contour representation is referred to as an explicit representation, and is defined in the Lagrangian coordinate system. In this coordinate system, two different objects have two different representations stemming from the different sets of control points that define the object contours. The control points constitute the finite elements, which is a common formalism used to represent shapes in the Lagrangian coordinates. In contrast to the parametric Lagrangian representation, the nonparametric representation defines the object contour implicitly in the Eulerian coordinates, which remains constant for two different objects. The level set method is a nonparametric representation defined in the Eulerian coordinate system and is used commonly in the computer vision community to represent the shape of an object in an image.The level set method has been introduced in the field of fluid dynamics by the seminal work of Osher and Sethian in 1988. After its introduction, it has been applied successfully in the fields of fluid mechanics, computational physics, computer graphics, and computer vision. In the level set representation, the value of each grid point (pixel) is set traditionally to the Euclidean distance between the grid point and the contour. Hence, moving the contour from one configuration to the other is achieved by changing the distance values in the grid points. During its motion, the contour can change its topology implicitly by splitting into two disjoint contours or by merging from two contours to one.The implicit nature of the representation becomes essential to handle the topology changes for the case when an initial configuration is required to solve a time‐dependent problem. Upon initialization, the level set converges to a solution by re‐evaluating the values at the grid points iteratively. This iterative procedure is referred to as the contour evolution.