We present Clifford-Steerable Convolutional Neural Networks (CS-CNNs), a
novel class of $\mathrm{E}(p, q)$-equivariant CNNs. CS-CNNs process multivector
fields on pseudo-Euclidean spaces $\mathbb{R}^{p,q}$. They cover, for instance,
$\mathrm{E}(3)$-equivariance on $\mathbb{R}^3$ and Poincar\'e-equivariance on
Minkowski spacetime $\mathbb{R}^{1,3}$. Our approach is based on an implicit
parametrization of $\mathrm{O}(p,q)$-steerable kernels via Clifford group
equivariant neural networks. We significantly and consistently outperform
baseline methods on fluid dynamics as well as relativistic electrodynamics
forecasting tasks.

利用 Clifford-Steerable 卷积神经网络 (CS-CNNs) 处理多向量场，对伪欧几里得空间和相对论电动力学预测任务进行了实验，结果显著优于基线方法。

Clifford 可操控卷积神经网络

Clifford-Steerable Convolutional Neural Networks

Rigid motion tracking is paramount in many medical imaging applications where
movements need to be detected, corrected, or accounted for. Modern strategies
rely on convolutional neural networks (CNN) and pose this problem as rigid
registration. Yet, CNNs do not exploit natural symmetries in this task, as they
are equivariant to translations (their outputs shift with their inputs) but not
to rotations. Here we propose EquiTrack, the first method that uses recent
steerable SE(3)-equivariant CNNs (E-CNN) for motion tracking. While steerable
E-CNNs can extract corresponding features across different poses, testing them
on noisy medical images reveals that they do not have enough learning capacity
to learn noise invariance. Thus, we introduce a hybrid architecture that pairs
a denoiser with an E-CNN to decouple the processing of anatomically irrelevant
intensity features from the extraction of equivariant spatial features. Rigid
transforms are then estimated in closed-form. EquiTrack outperforms
state-of-the-art learning and optimisation methods for motion tracking in adult
brain MRI and fetal MRI time series. Our code is available at
github.com/BBillot/equitrack.

等轨迹（EquiTrack）是一种使用可转动的 SE (3) 等轨神经网络 (E-CNN) 进行运动跟踪的方法，通过与去噪器配对，从而在成人脑 MRI 和胎儿 MRI 时间序列中优于现有的学习和优化方法。

医学图像中 SE (3) 等变且噪声不变的 3D 运动追踪

SE(3)-Equivariant and Noise-Invariant 3D Motion Tracking in Medical  Images

This work highlights the significance of equivariant networks as efficient
and high-performance approaches for tomography applications. Our study builds
upon the limitations of Convolutional Neural Networks (CNNs), which have shown
promise in post-processing various medical imaging systems. However, the
efficiency of conventional CNNs heavily relies on an undiminished and proper
training set. To tackle this issue, in this study, we introduce an equivariant
network, aiming to reduce CNN's dependency on specific training sets. We
evaluate the efficacy of equivariant CNNs on spherical signals for tomographic
medical imaging problems. Our results demonstrate superior quality and
computational efficiency of spherical CNNs (SCNNs) in denoising and
reconstructing benchmark problems. Furthermore, we propose a novel approach to
employ SCNNs as a complement to conventional image reconstruction tools,
enhancing the outcomes while reducing reliance on the training set. Across all
cases, we observe a significant decrease in computational costs while
maintaining the same or higher quality of image processing using SCNNs compared
to CNNs. Additionally, we explore the potential of this network for broader
tomography applications, particularly those requiring omnidirectional
representation.

本研究探讨了等变网络在断层扫描成像应用中的重要性，通过解决卷积神经网络在医学成像系统后处理中的局限性，引入了一种能够减少对特定训练集依赖的等变网络，并验证了球面信号上球面卷积神经网络 (SCNNs) 在去噪和重建基准问题中的优越质量和计算效率，同时利用 SCNNs 作为传统图像重建工具的补充方法，提高成像结果质量并降低对训练集的依赖，本研究观察到 SCNNs 与 CNNs 相比，在保持相同或更高的图像处理质量的同时，显著降低了计算成本，并初步探索了其在更广泛断层扫描成像应用中的潜能，特别是那些需要全方位表示的应用。

用于高效和高性能医学图像处理的等变球面 CNN

Equivariant Spherical CNN for Data Efficient and High-Performance  Medical Image Processing

In Reinforcement Learning (RL), Convolutional Neural Networks(CNNs) have been
successfully applied as function approximators in Deep Q-Learning algorithms,
which seek to learn action-value functions and policies in various
environments. However, to date, there has been little work on the learning of
symmetry-transformation equivariant representations of the input environment
state. In this paper, we propose the use of Equivariant CNNs to train RL agents
and study their inductive bias for transformation equivariant Q-value
approximation. We demonstrate that equivariant architectures can dramatically
enhance the performance and sample efficiency of RL agents in a highly
symmetric environment while requiring fewer parameters. Additionally, we show
that they are robust to changes in the environment caused by affine
transformations.

本文提出使用 Equivariant CNNs 训练强化学习智能体并研究其在对称变换方面的归纳偏差，结果表明在高度对称的环境中，使用 Equivariant CNNs 可以显著提高智能体的性能和样本效率，同时还需要更少的参数，而且它们对仿射变换引起的环境变化具有鲁棒性。