Real-time 3D reconstruction from RGB-D sensor data plays an important role in many robotic applications, such as object modeling and mapping. The popular method of fusing depth information into a truncated signed distance function (TSDF) and applying the marching cubes algorithm for mesh extraction has severe issues with thin structures: not only does it lead to loss of accuracy, but it can generate completely wrong surfaces. To address this, we propose the directional TSDF - a novel representation that stores opposite surfaces separate from each other. The marching cubes algorithm is modified accordingly to retrieve a coherent mesh representation. We further increase the accuracy by using surface gradient-based ray casting for fusing new measurements. We show that our method outperforms state-of-the-art TSDF reconstruction algorithms in mesh accuracy.