  In this paper, we propose a variational multiphase image segmentation model
based on fuzzy membership functions and L1-norm fidelity. Then we apply the
alternating direction method of multipliers to solve an equivalent problem. All
the subproblems can be solved efficiently. Specifically, we propose a fast
method to calculate the fuzzy median. Experimental results and comparisons show
that the L1-norm based method is more robust to outliers such as impulse noise
and keeps better contrast than its L2-norm counterpart. Theoretically, we prove
the existence of the minimizer and analyze the convergence of the algorithm.
