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We show how models for transportation networks can be reduced to Mumford-Shah-type image inpainting problems. Classical functional lifting allows to relax the inpainting problem into a convex optimization in a higher-dimensional space. We present a corresponding adaptive finite element discretization with heuristic refinement strategies based on the duality gap and an active-set type approach to deal with the large number of involved nonlocal convex constraints and their update after grid refinement.