  We propose an effective framework for multi-phase image segmentation and
semi-supervised data clustering by introducing a novel region force term into
the Potts model. Assume the probability that a pixel or a data point belongs to
each class is known a priori. We show that the corresponding indicator function
obeys the Bernoulli distribution and the new region force function can be
computed as the negative log-likelihood function under the Bernoulli
distribution. We solve the Potts model by the primal-dual hybrid gradient
method and the augmented Lagrangian method, which are based on two different
dual problems of the same primal problem. Empirical evaluations of the Potts
model with the new region force function on benchmark problems show that it is
competitive with existing variational methods in both image segmentation and
semi-supervised data clustering.
