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We present an interactive approach for segmenting thin volumetric structures. The proposed segmentation model is based on an anisotropic weighted Total Variation energy with a global volumetric constraint and is minimized using an efficient numerical approach and a convex relaxation. The algorithm is globally optimal w.r.t. the relaxed problem for any volumetric constraint. The binary solution of the relaxed problem equals the globally optimal solution of the original problem. Implemented on today's user-programmable graphics cards, it allows real-time user interaction. The method is applied to and evaluated on the task of articular cartilage segmentation of human knee joints and segmentation of tubular structures like liver vessels and airway trees.