Multigrid algorithm based on hybrid smoothers for variational and selective segmentation models

Automatic segmentation of an image to identify all meaningful parts is one of the most challenging as well as useful tasks in a number of application areas. This is widely studied. Selective segmentation, less studied, aims to use limited user-specified information to extract one or more interesting objects (instead of all objects). Constructing a fast solver remains a challenge for both classes of model. However, our primary concern is on selective segmentation. In this work, we develop an effective multigrid algorithm, based on a new non-standard smoother to deal with non-smooth coefficients, to solve the underlying partial differential equations of a class of variational segmentation models in the level-set formulation. For such models, non-smoothness (or jumps) is typical as segmentation is only possible if edges (jumps) are present. In comparison with previous multigrid methods which were shown to produce an acceptable mean smoothing rate for related models, the new algorithm can ensure a small and global smoothing rate that is a sufficient condition for convergence. Our rate analysis is by local Fourier analysis and, with it, we design the corresponding iterative solver, improving on an ineffective line smoother. Numerical tests show that the new algorithm outperforms multigrid methods based on competing smoothers.