We present a new split-type algorithm for the minimization of a p-harmonic energy with added data fidelity term. The half-quadratic splitting reduces the original problem to two straightforward problems, that can be minimized efficiently. The minimizers to the two sub-problems can typically be computed pointwise and are easily implemented on massively parallel processors. Furthermore the splitting method allows for the computation of solutions to a large number of more advanced directional regularization problems. In particular we are able to handle robust, non-convex data terms, and to define a 0-harmonic regularization energy where we sparsify directions by means of an L0 norm.