In recent years, there has been attention on leveraging the statistical modeling capabilities of neural networks for reconstructing sub-sampled Magnetic Resonance Imaging (MRI) data. Most proposed methods assume the existence of a representative fully-sampled dataset and use fully-supervised training. However, for many applications, fully sampled training data is not available, and may be highly impractical to acquire. The development and understanding of self-supervised methods, which use only sub-sampled data for training, are therefore highly desirable. This work extends the Noisier2Noise framework, which was originally constructed for self-supervised denoising tasks, to variable density sub-sampled MRI data. We use the Noisier2Noise framework to analytically explain the performance of Self-Supervised Learning via Data Undersampling (SSDU), a recently proposed method that performs well in practice but until now lacked theoretical justification. Further, we propose two modifications of SSDU that arise as a consequence of the theoretical developments. Firstly, we propose partitioning the sampling set so that the subsets have the same type of distribution as the original sampling mask. Secondly, we propose a loss weighting that compensates for the sampling and partitioning densities. On the fastMRI dataset we show that these changes significantly improve SSDU's image restoration quality and robustness to the partitioning parameters.

翻译：近年来，利用神经网络的统计建模能力重建子采样磁共振成像（MRI）数据引起了广泛关注。现有方法大多假设存在具有代表性的全采样数据集，并采用全监督训练方式。然而，在许多实际应用中，全采样训练数据难以获取甚至缺乏可行性。因此，开发并理解仅使用子采样数据进行训练的自监督方法具有重要意义。本研究将最初针对自监督去噪任务构建的Noisier2Noise框架扩展至变密度子采样MRI数据。我们利用Noisier2Noise框架从理论角度解析了数据欠采样自监督学习（SSDU）方法的性能——该算法虽在实践中表现优异，但此前缺乏理论支撑。在此基础上，我们根据理论推导提出两种SSDU改进方案：首先，建议对采样集进行分区，使各子集保持与原始采样掩膜相同的分布类型；其次，提出一种补偿采样与分区密度的损失加权策略。在fastMRI数据集上的实验表明，这些改进显著提升了SSDU的图像重建质量及其对分区参数的鲁棒性。