Multiphase image segmentation via equally distanced multiple well potential

• We propose models which give true length approximation via ΓΓ-convergence.
• We consider various regularizing functions to give sharp interface.
• We explore the differences and similarities of models.
• We present examples showing the difference in true length vs. weighted length.
• Numerically for noisy images true length Total Variation is the most stable.

Variational models for image segmentation, e.g. Mumford–Shah variational model [47] and Chan–Vese model [21] and [59], generally involve a regularization term that penalizes the length of the boundaries of the segmentation. In practice often the length term is replaced by a weighted length, i.e., some portions of the set of boundaries are penalized more than other portions, thus unbalancing the geometric term of the segmentation functional.In the present paper we consider a class of variational models in the framework of ΓΓ-convergence theory. We propose a family of functionals defined on vector valued functions that involve a multiple well potential of the type arising in diffuse-interface models of phase transitions. A potential with equally distanced wells makes it possible to retrieve the penalization of the true (i.e., not weighted) length of the boundaries as the ΓΓ-convergence parameter tends to zero. We explore the differences and the similarities of behavior of models in the proposed class, followed by some numerical experiments.