Convolutional neural networks (CNNs) allow for parameter sharing and translational equivariance by using convolutional kernels in their linear layers. By restricting these kernels to be SO(3)-steerable, CNNs can further improve parameter sharing and equivariance. These equivariant convolutional layers have several advantages over standard convolutional layers, including increased robustness to unseen poses, smaller network size, and improved sample efficiency. Despite this, most segmentation networks used in medical image analysis continue to rely on standard convolutional kernels. In this paper, we present a new family of segmentation networks that use equivariant voxel convolutions based on spherical harmonics, as well as equivariant pooling and normalization operations. These SE(3)-equivariant volumetric segmentation networks, which are robust to data poses not seen during training, do not require rotation-based data augmentation during training. In addition, we demonstrate improved segmentation performance in MRI brain tumor and healthy brain structure segmentation tasks, with enhanced robustness to reduced amounts of training data and improved parameter efficiency. Code to reproduce our results, and to implement the equivariant segmentation networks for other tasks is available at~\url{http://github.com/SCAN-NRAD/e3nn_Unet}.

翻译：卷积神经网络（CNNs）通过在线性层中使用卷积核实现参数共享与平移等变性。将卷积核限制为SO(3)-可转向后，CNNs能够进一步提升参数共享与等变性。这类等变卷积层相较于标准卷积层具有多项优势，包括对未见过姿态的鲁棒性增强、网络规模缩小以及样本效率提升。尽管优势显著，医学图像分析领域的大多数分割网络仍沿用标准卷积核。本文提出一类基于球谐函数实现等变体素卷积，并集成等变池化与归一化操作的新型分割网络家族。这些SE(3)等变体素分割网络对训练中未出现的数据姿态具有鲁棒性，训练过程中无需基于旋转的数据增强。此外，我们在MRI脑肿瘤分割与健康脑结构分割任务中验证了其性能优势：不仅展现出更强的训练数据量缩减鲁棒性，还实现了更优的参数效率。复现结果及将等变分割网络应用于其他任务的代码已发布于~\url{http://github.com/SCAN-NRAD/e3nn_Unet}。