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We present a fully computable a posteriori error estimate for the finite element approximation of the Rudin-Osher-Fatemi problem. We propose an adaptive refinement strategy relying on the fact that the primal-dual gap controls the $L^2$-error between the solution and an FE-approximation and on an accurate conforming discretization of both the primal and the dual problem and reliable iterative solution techniques. Numerical experiments show a significant improvement over approximations on uniformly refined triangulations.