deviation $\sigma = [0.0002, ..., 0.01]$, and the mean being $\mu = 0$. The experimental results are illustrated in Fig. \[fig:sn vn robustness results\]. We can learn that our proposed virtual normal is much more robust to the data noise than the surface normal. Other local constraints are also sensitive to data noise.

[ ![Robustness of virtual normal and surface normal against data noise. (a) The ideal surface and noisy surface. (b) The Mean Difference Error (Mean) is applied to evaluate the robustness of virtual normal and surface normal against different noise level. Our proposed virtual normal is more robust. []{data-label="fig:sn vn robustness"}](Fig/surface "fig:"){width="13.00000%"} \[fig:sn vn robustness sphere\] ]{} [ ![Robustness of virtual normal and surface normal against data noise. (a) The ideal surface and noisy surface. (b) The Mean Difference Error (Mean) is applied to evaluate the robustness of virtual normal and surface normal against different noise level. Our proposed virtual normal is more robust. []{data-label="fig:sn vn robustness"}](./Fig/vn_sn_noise "fig:"){width="32.00000%"} \[fig:sn vn robustness results\] ]{}

Most ‘local’ geometric constraints, such as the surface normal, actually enforcing the first-order smoothness of the surface but are less useful for helping the depth map prediction. In contrast, the proposed VN establishes long-range relations 