On segmentation by total variation type energies of Kobayashi-Warren-Carter type with fidelity

Abstract

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 function in the fidelity term 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 has no jumps if the data has no jumps. Our results give an upper bound of the number of segments in a segmentation problem. The existence of a minimizer is guaranteed in multi-dimensional setting when the data is bounded.