  We introduce VA-DepthNet, a simple, effective, and accurate deep neural
network approach for the single-image depth prediction (SIDP) problem. The
proposed approach advocates using classical first-order variational constraints
for this problem. While state-of-the-art deep neural network methods for SIDP
learn the scene depth from images in a supervised setting, they often overlook
the invaluable invariances and priors in the rigid scene space, such as the
regularity of the scene. The paper's main contribution is to reveal the benefit
of classical and well-founded variational constraints in the neural network
design for the SIDP task. It is shown that imposing first-order variational
constraints in the scene space together with popular encoder-decoder-based
network architecture design provides excellent results for the supervised SIDP
task. The imposed first-order variational constraint makes the network aware of
the depth gradient in the scene space, i.e., regularity. The paper demonstrates
the usefulness of the proposed approach via extensive evaluation and ablation
analysis over several benchmark datasets, such as KITTI, NYU Depth V2, and SUN
RGB-D. The VA-DepthNet at test time shows considerable improvements in depth
prediction accuracy compared to the prior art and is accurate also at
high-frequency regions in the scene space. At the time of writing this paper,
our method -- labeled as VA-DepthNet, when tested on the KITTI depth-prediction
evaluation set benchmarks, shows state-of-the-art results, and is the
top-performing published approach.
