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Prof. Dr. Xue-Cheng Tai

Universität Bergen, Norwegen

Some variants of the level set methods and applications to image segmentation and other shape optimization problems

Abstract:

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 image segmentation 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. The method works for 2D as well 3D problems. We show numerical results using the methods for segmentation of digital images. We will also show some other applications related to inverse problems for elliptic equations and two-phase reservoir fluid models. Numerical experiments for interface problems involving motion by mean curvature will also be given. 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. Some fast methods for solving the constrained minimization problem are tested. We shall also especially explain the relationship between our method, the level set method of Osher and Sethian and the phase-field methods.
Datum: 20.10.2006
Zeit:14.15
Ort:FU Berlin, Institut für Mathematik II, Arnimallee 6, 14195 Berlin.
Raum:032 im Erdgeschoss

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