Lie group convolution neural networks with scale-rotation equivariance

The weight-sharing mechanism of convolutional kernels ensures the translation equivariance of convolutional neural networks (CNNs) but not scale and rotation equivariance. This study proposes a SIM(2) Lie group-CNN, which can simultaneously keep scale, rotation, and translation equivariance for image classification tasks. The SIM(2) Lie group-CNN includes a lifting module, a series of group convolution modules, a global pooling layer, and a classification layer. The lifting module transfers the input image from Euclidean space to Lie group space, and the group convolution is parameterized through a fully connected network using the Lie Algebra coefficients of Lie group elements as inputs to achieve scale and rotation equivariance. It is worth noting that the mapping relationship between SIM(2) and its Lie Algebra and the distance measure of SIM(2) are defined explicitly in this paper, thus solving the problem of the metric of features on the space of SIM(2) Lie group, which contrasts with other Lie groups characterized by a single element, such as SO(2). The scale-rotation equivariance of Lie group-CNN is verified, and the best recognition accuracy is achieved on three categories of image datasets. Consequently, the SIM(2) Lie group-CNN can successfully extract geometric features and perform equivariant recognition on images with rotation and scale transformations.