  In most computer vision applications, convolutional neural networks (CNNs)
operate on dense image data generated by ordinary cameras. Designing CNNs for
sparse and irregularly spaced input data is still an open problem with numerous
applications in autonomous driving, robotics, and surveillance. To tackle this
challenging problem, we introduce an algebraically-constrained convolution
layer for CNNs with sparse input and demonstrate its capabilities for the scene
depth completion task. We propose novel strategies for determining the
confidence from the convolution operation and propagating it to consecutive
layers. Furthermore, we propose an objective function that simultaneously
minimizes the data error while maximizing the output confidence. Comprehensive
experiments are performed on the KITTI depth benchmark and the results clearly
demonstrate that the proposed approach achieves superior performance while
requiring three times fewer parameters than the state-of-the-art methods.
Moreover, our approach produces a continuous pixel-wise confidence map enabling
information fusion, state inference, and decision support.
