Level Set Method In Medical Imaging: An overview

The level set method for capturing moving fronts was devised by Osher and Sethian in [3]. The goal is to compute and analyze the motion of an interface Γ that is moving according to its own geometries and/or the external physics of the underlying problem. The key idea behind this method is to implicitly represent the interface Γ as the zero level set of a smooth enough function φ, known as level set function. That is, Γ = {x/φ(x, t) = 0}. The level set function φ is initially expected to be a signed distance function with respect to the region Ω (eventually multi-connected) bounded by Γ. This means that φ is positive inside Ω, negative outside Ω, and vanishes on Γ. Topological changes of the moving interface Γ, such as breaking and merging are automatically handled during the course of evolution. In this paper, the theoretical presentation of the background necessary to formulate the level set problem will be presented. Then, the general concept of Hamilton-Jacobi (H.J.) equations will be intoduced, noting that both the time-dependent and the static perspectives of the level set method are instances of this general framework. The link between H.J. equations and the hyperbolic conservation laws (H.C.L.) will be established, and we will show how this link will serve in developing numerical schemes for solving the level set equation. Finally, we will list and analyze some applications of the level set method in the field of medical imaging with a focus on human brain applications.