  Generally, convolutional neural networks (CNNs) process data on a regular
grid, e.g. data generated by ordinary cameras. Designing CNNs for sparse and
irregularly spaced input data is still an open research problem with numerous
applications in autonomous driving, robotics, and surveillance. In this paper,
we propose an algebraically-constrained normalized convolution layer for CNNs
with highly sparse input that has a smaller number of network parameters
compared to related work. We propose novel strategies for determining the
confidence from the convolution operation and propagating it to consecutive
layers. We also propose an objective function that simultaneously minimizes the
data error while maximizing the output confidence. To integrate structural
information, we also investigate fusion strategies to combine depth and RGB
information in our normalized convolution network framework. In addition, we
introduce the use of output confidence as an auxiliary information to improve
the results. The capabilities of our normalized convolution network framework
are demonstrated for the problem of scene depth completion. Comprehensive
experiments are performed on the KITTI-Depth and the NYU-Depth-v2 datasets. The
results clearly demonstrate that the proposed approach achieves superior
performance while requiring only about 1-5% of the number of parameters
compared to the state-of-the-art methods.
