  This paper proposes a new variational model by integrating the Allen-Cahn
term with a local binary fitting energy term for segmenting images with
intensity inhomogeneity and noise. An inhomogeneous graph Laplacian
initialization method (IGLIM) is developed to give the initial contour for
two-phase image segmentation problems. To solve the Allen-Cahn equation derived
from the variational model, we adopt the exponential time differencing (ETD)
method for temporal discretization, and the central finite difference method
for spatial discretization. The energy stability of proposed numerical schemes
can be proved. Experiments on various images demonstrate the necessity and
superiority of proper initialization and verify the capability of our model for
two-phase segmentation of images with intensity inhomogeneity and noise.
