On an effective multigrid solver for solving a class of variational problems with application to image segmentation

摘要

In this paper we reformulate a class of non-linear variational models for global and selective image segmentation and obtain convergent multigrid solutions. In contrast, non-linear multigrid schemes do not converge for these problems with strong non-linearity and non-smoothness (jumps). Our new approach is to reformulate the non-linear models, using splitting techniques, to generate linear models in a higher dimension which are easier to solve and amenable to the linear multigrid framework. Although splitting techniques are well studied in isolation, direct application of a splitting idea is not sufficient and it is the combination of two splitting approaches and linear multigrid theory approaches which results in a highly effective multigrid algorithm. Numerical results demonstrate the fast convergence of the new multigrid methods.