An unconstrained multiphase thresholding approach for image segmentation

In this paper we provide a method to nd global minimizers
of certain non-convex 2-phase image segmentation problems. This is
achieved by formulating a convex minimization problem whose minimizers
are also minimizers of the initial non-convex segmentation problem,
similar to the approach proposed by Nikolova, Esedoglu and Chan. The
key dierence to the latter model is that the new model does not involve
any constraint in the convex formulation that needs to be respected when
minimizing the convex functional, neither explicitly nor by an articial
penalty term. This approach is related to recent results by Chambolle.
Eliminating the constraint considerably simplies the computational dif-
culties, and even a straightforward gradient descent scheme leads to a
reliable computation of the global minimizer. Furthermore, the model is
extended to multiphase segmentation along the lines of Vese and Chan.
Numerical results of the model applied to the classical piecewise constant
Mumford-Shah functional for two, four and eight phase segmentation are
shown.