Abstract
Normalized cut or Ncut has been one of the most widely used models for image processing. A constraint can also be included in the framework of Ncut to represent a priori information for an effective image segmentation. This results in the so-called constrained Ncut problem. In this paper we present an observation that the constrained Ncut problem can be formulated as an indefinite system of equations under a mild condition on targeted segmentation results. We then show that the indefinite system can be effectively handled by the Augmented Lagrangian Uzawa iterative method together with a classical Algebraic Multigrid Method. Both mathematically and numerically, we demonstrate that the Augmented Lagrangian Uzawa method achieves a solution in one iteration. We show how the proposed method can be efficiently applied for the newly tested recursive two-way Ncut with constraints, i.e., a new constrained sequential segmentation as well. A number of numerical experiments are presented to confirm our theory and to show the superiority of the proposed method. In particular, numerical experiments show that the speed of our algorithm can be orders of magnitude faster than the previously proposed Projected Power Method, a significant improvement of conventional image segmentation algorithms.