\[fig:VNL\_PL\_Samples\] demonstrates that ‘rel’ slumps by $5.6$% with 20K point groups to establish VNL. However, it only drops slightly when the samples for VNL increase from 20K to 100K. Therefore, the performance saturates with more samples, when samples reach a certain number in that the diversity of samples is enough to construct the global geometric constraint. **Lightweight Backbone Network.** We train the network with the MobileNetV2 backbone to evaluate the effectiveness of the proposed geometric constraint on the light network. We train it on the NYUD-Large for $10$ epochs. Results in Table \[table:errors cmp with MobileNetV2\] show that the proposed VNL can improve the performance by $1\%$ - $8\%$. Comparing with previous state-of-the-art methods, we have improved the accuracy by around $29\%$ over all evaluation metrics and achieved a better trade-off between parameters and the accuracy.

\[table:errors cmp with MobileNetV2\]

Recovering 3D Features from Estimated Depth
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We have argued that, with geometric constraints in the 3D space, the network can achieve more accurate depth and also obtain higher-quality 3D information. Here we show the recovered 3D point cloud and the surface normal to support this.

**3D Point Cloud.** Firstly, we compare the reconstructed 3D point cloud from our predicted 