报告摘要：

The level set method has been proved to be a versatile tool for tracing interfaces and partition domains. In this work we discuss variants of the PDE based level set method proposed earlier by Osher and Sethian. Traditionally interfaces are represented by the zero level set of continuous functions. We instead use piecewise constant level set (PCLS) functions, i.e. the level set function equals to a constant in each of the regions that we want to identify. Using the methods for interface problems, we need to minimize a smooth convex functional under a constraint. The level set functions are discontinuous at convergence, but the minimization functional is smooth and locally convex. The methods are truly variational, i.e. all the equations we need to solve are the Euler-Lagrangian equations from the minimization functionals. Thus, the fast Newton type of method can be easily used. The method works for 2D as well 3D problems. We show numerical results using the methods for segmentation of digital images. Application to inverse problems related to some elliptic equations and two-phase reservoir fluid models will be shown.

We shall present two variants of the piecewise constant level set methods (PCLSM). One of them is able to use just one level set function for identifying multiphase problems with arbitrary number of phases. Another variant, which we call the binary level set method, only requires the level set function equals 1 or -1. The geometrical quantities like the boundary length and area of the subdomain can be easily expressed as functions of the new level set functions. The constraint for the minimization problems can be handled by the augmented Lagrangian method or the MBO (Merriman, Bence and Osher) projection.

The problem for the convex minimization is in fact a convex integer programming problem. It would be interesting to see the possibility to use integer programming for this kind of problems.

报告时间： 2005年9月27日(周二) 下午3:30--4:30

报告地点：科技综合楼三层311报告厅