  Homography has an essential relationship with the special linear group and
the embedding Lie algebra structure. Although the Lie algebra representation is
elegant, few researchers have established the connection between homography and
algebra expression in neural networks. In this paper, we propose Warped
Convolution Networks (WCN) to effectively learn and represent the homography by
SL(3) group and sl(3) algebra with group convolution. To this end, six
commutative subgroups within the SL(3) group are composed to form a homography.
For each subgroup, a warping function is proposed to bridge the Lie algebra
structure to its corresponding parameters in homography. By taking advantage of
the warped convolution, homography learning is formulated into several simple
pseudo-translation regressions. By walking along the Lie topology, our proposed
WCN is able to learn the features that are invariant to homography. Moreover,
it can be easily plugged into other popular CNN-based methods. Extensive
experiments on the POT benchmark, S-COCO-Proj, and MNIST-Proj dataset show that
our proposed method is effective for planar object tracking, homography
estimation, and classification.
