Three-directional box-splines are particularly well-suited to interpolate and approximate hexagonally sampled data. In this paper, we propose a computationally efficient end-to-end reconstruction process. First, we introduce a prefiltering step that is based on a quasi-interpolation scheme using low-complexity finite-impulse-response (FIR) filters. Second, we derive a closed analytical expression for three-directional box-splines of any order that leads to a fast evaluation of the spline surface. All operations act locally on the data, and thus are well adapted to applications dealing with large images. To demonstrate the feasibility of our method, we implemented the complete procedure and we present experimental results.