Warped convolutional neural networks for large homography transformation with psl(3) algebra

Homography has a fundamental and elegant relationship with the projective special linear group and its embedding Lie algebra structure. However, the integration of homography and algebraic expressions in neural networks remains largely unexplored. In this paper, we propose Warped Convolution Neural Networks to effectively learn and represent the homography by psl(3) algebra with group convolution. Specifically, six commutative subgroups within the PSL(3) group are composed to form a homography. For each subgroup, a warp 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. Our proposed method enables learning features that are invariant to significant homography transformations through exploration along the Lie topology. Moreover, it can be easily plugged into other popular CNN-based methods and empower them with homography representation capability. Through extensive experiments on benchmark datasets such as POT, S-COCO, and MNIST-Proj, we demonstrate the effectiveness of our approach in various tasks like classification, homography estimation, and planar object tracking.