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 |
