It has been shown that equivariant convolution is very helpful for many types of computer vision tasks. Recently, the 2D filter parametrization technique plays an important role when designing equivariant convolutions. However, the current filter parametrization method still has its evident drawbacks, where the most critical one lies in the accuracy problem of filter representation. Against this issue, in this paper we modify the classical Fourier series expansion for 2D filters, and propose a new set of atomic basis functions for filter parametrization. The proposed filter parametrization method not only finely represents 2D filters with zero error when the filter is not rotated, but also substantially alleviates the fence-effect-caused quality degradation when the filter is rotated. Accordingly, we construct a new equivariant convolution method based on the proposed filter parametrization method, named F-Conv. We prove that the equivariance of the proposed F-Conv is exact in the continuous domain, which becomes approximate only after discretization. Extensive experiments show the superiority of the proposed method. Particularly, we adopt rotation equivariant convolution methods to image super-resolution task, and F-Conv evidently outperforms previous filter parametrization based method in this task, reflecting its intrinsic capability of faithfully preserving rotation symmetries in local image features.

翻译：研究表明，等变卷积对多种计算机视觉任务具有重要意义。近年来，二维滤波器参数化技术在等变卷积设计中扮演着关键角色。然而，现有滤波参数化方法仍存在明显缺陷，其核心问题在于滤波器表示的精度不足。针对这一挑战，本文改进了经典二维滤波器傅里叶级数展开方法，提出了一组新的原子基函数用于滤波参数化。所提参数化方法不仅能在滤波器未旋转时实现零误差精细表示，还能在滤波器旋转时显著缓解栅栏效应导致的质量退化。据此，我们基于该参数化方法构建了新型等变卷积方法F-Conv。理论证明，所提F-Conv在连续域中具有精确等变性，仅在离散化后呈现近似等变。大量实验验证了该方法的优越性。特别地，我们将旋转等变卷积方法应用于图像超分辨率任务，F-Conv在该任务中显著优于先前基于滤波参数化的方法，展现了其内在保持局部图像特征旋转对称性的能力。