Alternating and parallel overlapping domain decomposition methods for the minimization of the total variation are presented. Their derivation is based on the predual formulation of the total variation minimization problem. In particular, the predual total variation minimization problem is decomposed into overlapping domains yielding subdomain problems in the respective dual space. Subsequently these subdomain problems are again dualized, forming a splitting algorithm for the original total variation minimization problem. The convergence of the proposed domain decomposition methods to a solution of the global problem is proved. In contrast to other works, the analysis is carried out in an infinite dimensional setting. Numerical experiments are shown to support the theoretical results and to demonstrate the effectiveness of the algorithms.