With translation equivariance, convolution neural networks (CNNs) have achieved great success in retinal vessel segmentation. However, some other symmetries of the vascular morphology are not characterized by CNNs, such as rotation and scale symmetries. To embed more equivariance into CNNs and achieve the accuracy requirement for retinal vessel segmentation, we construct a novel convolution operator (FRS-Conv), which is Fourier parameterized and equivariant to rotation and scaling. Specifically, we first adopt a new parameterization scheme, which enables convolutional filters to arbitrarily perform transformations with high accuracy. Secondly, we derive the formulations for the rotation and scale equivariant convolution mapping. Finally, we construct FRS-Conv following the proposed formulations and replace the traditional convolution filters in U-Net and Iter-Net with FRS-Conv (FRS-Nets). We faithfully reproduce all compared methods and conduct comprehensive experiments on three public datasets under both in-dataset and cross-dataset settings. With merely 13.9% parameters of corresponding baselines, FRS-Nets have achieved state-of-the-art performance and significantly outperform all compared methods. It demonstrates the remarkable accuracy, generalization, and clinical application potential of FRS-Nets.

翻译：凭借平移等变性，卷积神经网络（CNNs）在视网膜血管分割中取得了巨大成功。然而，血管形态的某些其他对称性（如旋转和尺度对称性）并未被CNNs表征。为了将更多等变性嵌入CNNs并满足视网膜血管分割的精度要求，我们构建了一种新型卷积算子（FRS-Conv），该算子基于傅里叶参数化且对旋转与尺度具有等变性。具体而言，我们首先采用了一种新的参数化方案，使卷积滤波器能够以高精度任意执行变换。其次，我们推导了旋转与尺度等变卷积映射的公式。最后，我们根据所提公式构建了FRS-Conv，并用其替换U-Net和Iter-Net中的传统卷积滤波器，形成FRS-Nets。我们忠实地复现了所有对比方法，并在三个公开数据集上进行了数据集内与跨数据集下的全面实验。FRS-Nets仅使用对应基线模型13.9%的参数，便取得了最先进的性能，且显著优于所有对比方法。这证明了FRS-Nets卓越的准确性、泛化能力及临床应用潜力。