the depth (red is far, purple is close). Our predicted depth maps have fewer errors in planes (see walls) and have high-quality details in complicated scenes (see the desk and shelf in the last row)](./Fig/NYU_visual_cmp){width="47.00000%"}

. \[fig:visual cmp\]

**KITTI.** In order to demonstrate that our proposed method can still reach the state-of-the-art performance on outdoor scenes, we test our method on the KITTI dataset. Results in Table \[table:errors cmp on KITTI\] show that our method has outperformed all other methods on all evaluation metrics except root mean square (rms) error. The rms error is only slightly behind that of DORN. Note that for outdoor scenes, the rms (log) error, instead of rms, is usually the metric of interest, in which ours is better.

\[table:errors cmp on KITTI\]

![image](./Fig/kitti_loss_cmp){width="90.00000%"}

Ablation Studies
----------------

In this section, we conduct several ablation studies to analyze the details of our approach.

**Effectiveness of VNL.** In this study, in order to prove the effectiveness of the proposed VNL we compare it with two types of pixel-wise depth map supervision, a pair-wise geometric supervision, and a high-order geometric supervision: 1) the ordinary cross-entropy loss (CEL); 2) the $L_{1}$ loss ($L_{1}$); 3) the surface normal loss (SNL); 4) 