A primal-dual finite element method for scalar and vectorial total variation minimization

Based on the Fenchel duality we build a primal-dual framework for minimizing a general functional consisting of a combined L1 and L2 data-fidelity term and a scalar or vectorial total variation regularisation term. The minimization is performed over the space of functions of bounded variations and appropriate discrete subspaces. We analyze the existence and uniqueness of solutions of the respective minimization problems. For computing a numerical solution we derive a semi-smooth Newton method on finite element spaces and highlight applications in denoising, inpainting and optical flow estimation.