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.

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