evaluated. From Fig. \[fig:fit plane\](b), we can learn that the surface normal varies significantly with different sampling sizes. For example, the Mean between 3$ \times$3 and 11$\times$11 is 22. Such unstable surface normal negatively affects its effectiveness for learning. Likewise, other 3D geometric constraints demonstrating the ‘local’ relative relations also encounter this problem.

**Virtual Normal.** In order to enforce robust high-order geometric supervision in the 3D space, we propose the virtual normal (VN) to establish 3D geometric connections between regions in a much larger range. The point cloud can be reconstructed from the depth based on the pinhole camera model. For each pixel $p_{i}(u_{i}, v_{i})$, the 3D location $P_{i}(x_{i}, y_{i}, z_{i})$ in the world coordinate can be obtained by the prospective projection. We set the camera coordinate as the world coordinate. Then the 3D coordinate $P_{i}$ is denoted as follows: $$\begin{split}
z_{i} = d_{i}, x_{i} = \frac{d_{i} \cdot (u_{i} - u_{0})}{f_{x}},y_{i} = \frac{d_{i}(v_{i} - v_{0})}{f_{y}}
\end{split}$$ where $d_{i}$ is the depth. $f_{x}$ and $f_{y}$ are the focal length along the $x$ and $y$ coordinate axis respectively. $u_{0}$ and $v_{0}$ are the 2D coordinate of the optical center. We randomly sample $N$ groups points from the depth map, with three points 