Filter-decomposition-based group equivariant convolutional neural networks (CNNs) have shown promising stability and data efficiency for 3D image feature extraction. However, these networks, which rely on parameter sharing and discrete transformation groups, often underperform in modern deep neural network architectures for processing volumetric images with dense 3D textures, such as the common 3D medical images. To address these limitations, this paper presents an efficient non-parameter-sharing continuous 3D affine group equivariant neural network for volumetric images. This network uses an adaptive aggregation of Monte Carlo augmented spherical Fourier-Bessel filter bases to improve the efficiency and flexibility of 3D group equivariant CNNs for volumetric data. Unlike existing methods that focus only on angular orthogonality in filter bases, the introduced spherical Bessel Fourier filter base incorporates both angular and radial orthogonality to improve feature extraction. Experiments on four medical image segmentation datasets and two seismic datasets show that the proposed methods achieve better affine group equivariance and superior segmentation accuracy than existing 3D group equivariant convolutional neural network layers, significantly improving the training stability and data efficiency of conventional CNN layers (at 0.05 significance level). The code is available at https://github.com/ZhaoWenzhao/WMCSFB.

翻译：基于滤波器分解的群等变卷积神经网络在三维图像特征提取方面已展现出良好的稳定性和数据效率。然而，这些依赖参数共享和离散变换群的网络在处理具有密集三维纹理的体图像（如常见的三维医学图像）时，其性能往往逊色于现代深度神经网络架构。为应对这些局限性，本文提出一种用于体图像的高效非参数共享连续三维仿射群等变神经网络。该网络采用蒙特卡洛增强球面傅里叶-贝塞尔滤波器基的自适应聚合，以提高针对体数据的三维群等变卷积神经网络的效率与灵活性。与现有方法仅关注滤波器基的角向正交性不同，本文引入的球面贝塞尔傅里叶滤波器基同时结合了角向与径向正交性，以改善特征提取。在四个医学图像分割数据集和两个地震数据集上的实验表明，所提方法相比现有的三维群等变卷积神经网络层，实现了更好的仿射群等变性及更优的分割精度（在0.05显著性水平下），显著提升了传统卷积神经网络层的训练稳定性和数据效率。代码发布于 https://github.com/ZhaoWenzhao/WMCSFB。