Warped Convolutional Networks: Bridge Homography to $\mathfrak{sl}(3)$ algebra by Group Convolution

Xinrui Zhan, Yang Li, Wenyu Liu, Jianke Zhu

Keywords: SL(3), Homography Learning, Lie algebra, Equivariance, Group Equivariant Architecture

TL;DR: We propose a Warped Convolution Networks to effectively learn the homography on $\mathfrak{sl}(3)$ algebra with group convolution.

Abstract: 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 $\mathfrak{sl}(3)$ 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.

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