We propose a framework for rotation and translation covariant deep learning
using $SE(2)$ group convolutions. The group product of the special Euclidean
motion group $SE(2)$ describes how a concatenation of two roto-translations
results in a net roto-translation. We encode this geometric structure into
convolutional neural networks (CNNs) via $SE(2)$ group convolutional layers,
which fit into the standard 2D CNN framework, and which allow to generically
deal with rotated input samples without the need for data augmentation.
We introduce three layers: a lifting layer which lifts a 2D (vector valued)
image to an $SE(2)$-image, i.e., 3D (vector valued) data whose domain is
$SE(2)$; a group convolution layer from and to an $SE(2)$-image; and a
projection layer from an $SE(2)$-image to a 2D image. The lifting and group
convolution layers are $SE(2)$ covariant (the output roto-translates with the
input). The final projection layer, a maximum intensity projection over
rotations, makes the full CNN rotation invariant.
We show with three different problems in histopathology, retinal imaging, and
electron microscopy that with the proposed group CNNs, state-of-the-art
performance can be achieved, without the need for data augmentation by rotation
and with increased performance compared to standard CNNs that do rely on
augmentation.

利用 $SE (2)$ 群卷积提出了一种旋转和平移协变的深度学习框架，可处理旋转输入样本，适用于组织病理学、视网膜成像和电子显微镜等应用，无需数据增强且性能比标准形式的 CNNs 要好。