Depth maps play an important role in the representation of 3D information. They are often simultaneously acquired with color images; however, their resolution is significantly lower than that of color images owing to hardware limitations. In this paper, we propose a novel approach to upsample depth maps by using geometric deformation instead of pixel value refinement, which is employed in a majority of existing methods. This approach, known as grid warping, displaces the position of blurred pixels around the edge towards the center of the edge. The displacement vector for warping is obtained from an analysis of the corresponding high-resolution color image. Furthermore, we propose an edge signal and displacement vector modeling for a more effective analysis. The experimental results show that the proposed method significantly improves the quantitative and visual performance, as compared to state-of-the-art methods. The source codes of the proposed method will be available at https://github.com/yym064/DeepGridWarp.