  We consider a total variation type energy which measures the jump
discontinuities different from usual total variation energy. Such a type of
energy is obtained as a singular limit of the Kobayashi-Warren-Carter energy
with minimization with respect to the order parameter. We consider the
Rudin-Osher-Fatemi type energy by replacing relaxation term by this type of
total variation energy. We show that all minimizers are piecewise constant if
the data is continuous in one-dimensional setting. Moreover, the number of
jumps is bounded by an explicit constant involving a constant related to the
fidelity. This is quite different from conventional Rudin-Osher-Fatemi energy
where a minimizer must have no jump if the data has no jumps. The existence of
a minimizer is guaranteed in multi-dimensional setting when the data is
bounded.
