Reconstructing thin 3D structures is challenging due to their sparsity, scale variation, and complex geometry. Such structures arise in a wide range of domains, including medical imaging of vascular systems and industrial pipe systems. While recent neural methods perform well on dense surfaces, they often fail to recover fine thin geometries. We propose a reconstruction approach based on local depth projections, which provide an efficient and informative 2D representation of thin structures. Specifically, we traverse the 3D model with a sliding box to generate local orthographic depth projections, which are processed by a neural network to reconstruct missing thin structures in 2D. The local reconstructions are subsequently fused back into the 3D model to produce a coherent and detailed shape. Experiments on pulmonary artery reconstruction from CT volumes and industrial pipeline recovery from synthetic and real scans demonstrate improved preservation of fine structural details over existing methods.