Motion correction (MoCo) in radial MRI is a challenging problem due to the unpredictability of subject's motion. Current state-of-the-art (SOTA) MoCo algorithms often use extensive high-quality MR images to pre-train neural networks, obtaining excellent reconstructions. However, the need for large-scale datasets significantly increases costs and limits model generalization. In this work, we propose Moner, an unsupervised MoCo method that jointly solves artifact-free MR images and accurate motion from undersampled, rigid motion-corrupted k-space data, without requiring training data. Our core idea is to leverage the continuous prior of implicit neural representation (INR) to constrain this ill-posed inverse problem, enabling ideal solutions. Specifically, we incorporate a quasi-static motion model into the INR, granting its ability to correct subject's motion. To stabilize model optimization, we reformulate radial MRI as a back-projection problem using the Fourier-slice theorem. Additionally, we propose a novel coarse-to-fine hash encoding strategy, significantly enhancing MoCo accuracy. Experiments on multiple MRI datasets show our Moner achieves performance comparable to SOTA MoCo techniques on in-domain data, while demonstrating significant improvements on out-of-domain data. The code is available at: https://github.com/iwuqing/Moner

翻译：径向MRI中的运动校正（MoCo）因受试者运动的不可预测性而成为一个具有挑战性的问题。当前最先进的MoCo算法通常使用大量高质量MR图像来预训练神经网络，以获得优异的重建效果。然而，对大规模数据集的需求显著增加了成本并限制了模型的泛化能力。本文提出Moner，一种无监督的MoCo方法，能够从欠采样的刚性运动伪影k空间数据中联合求解无伪影MR图像与精确运动参数，且无需训练数据。我们的核心思想是利用隐式神经表示的连续先验来约束这一病态逆问题，从而获得理想解。具体而言，我们将准静态运动模型融入INR中，使其具备校正受试者运动的能力。为稳定模型优化，我们利用傅里叶切片定理将径向MRI重构为反投影问题。此外，我们提出一种新颖的由粗到精的哈希编码策略，显著提升了MoCo的精度。在多个MRI数据集上的实验表明，Moner在域内数据上取得了与最先进MoCo技术相当的性能，同时在域外数据上展现出显著改进。代码发布于：https://github.com/iwuqing/Moner