  The median filter scheme is an elegant, monotone discretization of the level
set formulation of motion by mean curvature. It turns out to evolve every level
set of the initial condition precisely by another class of methods known as
threshold dynamics. Median filters are, in other words, the natural level set
versions of threshold dynamics algorithms. Exploiting this connection, we
revisit median filters in light of recent progress on the threshold dynamics
method. In particular, we give a variational formulation of, and exhibit a
Lyapunov function for, median filters, resulting in energy based unconditional
stability properties. The connection also yields analogues of median filters in
the multiphase setting of mean curvature flow of networks. These new multiphase
level set methods do not require frequent redistancing, and can accommodate a
wide range of surface tensions.
