  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 (PDEs) 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 {\it mean} smoothing rate for related models,
the new algorithm can ensure a small and {\it 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.
