The weight-sharing mechanism of convolutional kernels ensures translation-equivariance of convolution neural networks (CNNs). Recently, rotation-equivariance has been investigated. However, research on scale-equivariance or simultaneous scale-rotation-equivariance is insufficient. This study proposes a Lie group-CNN, which can keep scale-rotation-equivariance for image classification tasks. The 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 Lie-algebra of Lie-group elements as inputs to achieve scale-rotation-equivariance. The Lie group SIM(2) is utilized to establish the Lie group-CNN with scale-rotation-equivariance. Scale-rotation-equivariance of Lie group-CNN is verified and achieves the best recognition accuracy on the blood cell dataset (97.50%) and the HAM10000 dataset (77.90%) superior to Lie algebra convolution network, dilation convolution, spatial transformer network, and scale-equivariant steerable network. In addition, the generalization ability of the Lie group-CNN on SIM(2) on rotation-equivariance is verified on rotated-MNIST and rotated-CIFAR10, and the robustness of the network is verified on SO(2) and SE(2). Therefore, the Lie group-CNN can successfully extract geometric features and performs equivariant recognition on images with rotation and scale transformations.

该研究提出了一种Lie群-CNN模型，利用群卷积模块的全连接网络和Lie-代数实现了尺度旋转等变性，进而成功地从图像中提取几何特征并实现了对图像的等变识别。

尺度-旋转-等变李群卷积神经网络 (Lie Group-CNNs)