We adapt and import into the TomoPIV scenery a fast algorithm for solving the volume reconstruction problem. Our approach is based on the reformulation of the volume reconstruction task as a constrained optimization problem and the resort to the " Alternating Directions Method of Multipliers " (ADMM). This approach is shown to be an interesting alternative to the " row-action " methods, classically used by the Tomo-PIV community. In particular, our problem formulation allows to: i) take explicitly into account the level of the noise affecting the data; ii) account for both the nonnegativity and the sparsity of the solution. Experiments on a numerical TomoPIV benchmark show that the proposed framework is a serious contender for the state-of-the-art.