Segmentation of monotone data by Kobayashi-Warren-Carter type total variation energies

We consider a Kobayashi-Warren-Carter (KWC) type total variation energy with a fidelity term. Since the energy is non-convex, the profiles of minimizers are quite different from those of the original Rudin-Osher-Fatemi energy. In one-dimensional setting, we prove that KWC type energy (and its generalization) with fidelity must have a piecewise constant minimizer if the data in fidelity is bounded not necessarily in $BV$. Moreover, we give quantitative estimates of the energy for a monotone data in fidelity. This estimate shows that any minimizer must be piecewise constant with an improved estimate of the number of jumps for a monotone data. We also show the non-uniqueness of minimizers. Since this energy is useful from the point of segmentation or clustering, we compare with results of segmentation by the original Rudin-Osher-Fatemi energy and Mumford-Shah energy.