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We provide an approximation result for free-discontinuity functionals of the form where f is quadratic in the gradient-variable and θ is an arbitrary smooth Finsler metric. The approximating functionals are of Ambrosio-Tortorelli type and depend on the Hessian of the edge variable through a suitable nonhomogeneous metric φ. Mathematics Subject Classification. 49J45, 74G65, 68U10.doi:10.1051/cocv/2017027 fatcat:dlx3bbntojadrdvccdere5m65u