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We consider sequential and parallel decomposition methods for a dual problemof a general total variation minimization problem with applications in severalimage processing tasks, like image inpainting, estimation of optical flow andreconstruction of missing wavelet coefficients. The convergence of thesemethods to a solution of the global problem is analysed in a Hilbert spacesetting and a convergence rate is provided. Thereby, these convergence resulthold not only for exact local minimization but also if the subproblems are justsolved approximately. As a concrete example of an approximate local solutionprocess a surrogate technique is presented and analysed. Further, the obtainedconvergence rate is compared with related results in the literature and shownto be in agreement with or even improve upon them. Numerical experiments arepresented to support the theoretical findings and to show the performance ofthe proposed decomposition algorithms in image inpainting, optical flowestimation and wavelet inpainting tasks.

A General Decomposition Method for a Convex Problem Related to Total Variation Minimization