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We develop a phase-field approximation of the relaxation of the perimeterfunctional in the plane under a connectedness constraint based on the classicalModica-Mortola functional and the connectedness constraint of (Dondl, Lemenant,Wojtowytsch 2017). We prove convergence of the approximating energies andpresent numerical results and applications to image segmentation.

Approximation of the relaxed perimeter functional under a connectedness constraint by phase-fields