  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.
