Warped Convolutional Networks: Bridge Homography to sl(3) algebra by Group Convolution

TL;DR

This paper introduces Warped Convolution Networks that leverage SL(3) group and sl(3) algebra to improve homography learning, enabling invariant feature extraction and integration with CNNs for tasks like tracking and classification.

Contribution

It establishes a novel connection between homography, Lie groups, and neural networks through warped convolution, enhancing homography estimation and invariance.

Findings

Effective homography learning demonstrated on multiple datasets.

Improved performance in planar object tracking and classification.

Seamless integration with existing CNN architectures.

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 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.