ABSTRACT: Interface evolution problems are often solved elegantly by the level set method, which generally requires the time-consuming
reinitialization process. In order to avoid reinitialization, we reformulate the variational model as a constrained optimization
problem. Then we present an augmented Lagrangian method and a projection Lagrangian method to solve the constrained model
and propose two gradient-type algorithms. For the augmented Lagrangian method, we employ the Uzawa scheme to update the Lagrange
multiplier. For the projection Lagrangian method, we use the variable splitting technique and get an explicit expression for
the Lagrange multiplier. We apply the two approaches to the Chan-Vese model and obtain two efficient alternating iterative
algorithms based on the semi-implicit additive operator splitting scheme. Numerical results on various synthetic and real
images are provided to compare our methods with two others, which demonstrate effectiveness and efficiency of our algorithms.
KeywordsLevel set method–Reinitialization–Augmented Lagrangian method–Projection Lagrangian method–Chan-Vese model–Additive operator splitting
Journal of Mathematical Imaging and Vision 04/2012; 41(3):194-209. · 1.39 Impact Factor
Journal of Mathematical Imaging and Vision. 01/2011; 41:194-209.
Int. J. Comput. Math. 01/2011; 88:3026-3045.
J. Comput. Physics. 01/2010; 229:5062-5089.
ABSTRACT: We optimize eigenvalues in optimal shape design using binary level set methods. The interfaces of subregions are represented implicitly by the discontinuities of binary level set functions taking two values 1 or −1 at convergence. A binary constraint is added to the original model problems. We propose two variational algorithms to solve the constrained optimization problems. One is a hybrid type by coupling the Lagrange multiplier approach for the geometry constraint with the augmented Lagrangian method for the binary constraint. The other is devised using the Lagrange multiplier method for both constraints. The two iterative algorithms are both largely independent of the initial guess and can satisfy the geometry constraint very accurately during the iterations. Intensive numerical results are presented and compared with those obtained by level set methods, which demonstrate the effectiveness and robustness of our algorithms.