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

ABSTRACT

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.

KEYWORDS:

• Iterative methods
• multigrid
• Euler-Lagrange equations
• segmentation models

AMS SUBJECT CLASSIFICATIONS:

• 65F10
• 65N55
• 35J20
• 49Q10

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Work supported by the Engineering and Physical Sciences Research Council (UK EPSRC) grant EP/N014499/1.