Abstract
Analyzing volumetric data with rotational invariance or equivariance is an active topic in current research. Existing deep-learning approaches utilize either group convolutional networks limited to discrete rotations or steerable convolutional networks with constrained filter structures. This work proposes a novel equivariant neural network architecture that achieves analytical Equivariance to Local Pattern Orientation on the continuous SO(3) group while allowing unconstrained trainable filters - EquiLoPO Network. Our key innovations are a group convolutional operation leveraging irreducible representations as the Fourier basis and a local activation function in the SO(3) space that provides a well-defined mapping from input to output functions, preserving equivariance. By integrating these operations into a ResNet-style architecture, we propose a model that overcomes the limitations of prior methods. A comprehensive evaluation on diverse 3D medical imaging datasets from MedMNIST3D demonstrates the effectiveness of our approach, which consistently outperforms state of the art. This work suggests the benefits of true rotational equivariance on SO(3) and flexible unconstrained filters enabled by the local activation function, providing a flexible framework for equivariant deep learning on volumetric data with potential applications across domains. Our code is publicly available at \url{this https URL}.
Abstract (translated)
分析体积数据具有旋转不变性或等价性是当前研究的一个活跃主题。现有的深度学习方法要么是有限离散旋转的组卷积网络，要么是具有约束滤波器结构的可调节卷积网络。本文提出了一种新颖的等价神经网络架构，可以在连续SO(3)组上实现对局部模式方向的分析等价性，同时允许无约束的训练滤波器 - EquiLoPO网络。我们的关键创新点是一个利用不可约表示作为傅里叶基的组卷积操作，以及SO(3)空间中提供输入到输出函数的良好定义的局部激活函数。通过将这些操作整合到ResNet风格的架构中，我们提出了一个克服了先前方法局限性的模型。对MedMNIST3D等多样3D医疗成像数据集的全面评估表明，我们的方法的有效性得到了充分证明，该方法 consistently超越了最先进的技术水平。这项工作揭示了真旋转等价性对SO(3)的益处以及由局部激活函数实现的可伸缩和不约束滤波器，为在体积数据上实现等价深度学习提供了灵活的框架，具有广泛的应用前景。我们的代码公开可用，通过点击以下链接访问：https:// this https URL。
URL
https://arxiv.org/abs/2404.15979
