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欠采样磁共振成像数据的方法备受关注。现有方法大多假定存在代表性全采样数据集，并采用全监督训练模式。然而在许多应用场景中，全采样训练数据难以获取且采集成本极高，因此亟需开发仅利用欠采样数据训练的自监督方法及其理论理解。本研究将最初针对自监督去噪任务构建的Noisier2Noise框架扩展至可变密度欠采样MRI数据。我们借助Noisier2Noise框架从理论上解释了数据欠采样自监督学习（SSDU）的性能——该方法在实践中表现优异但此前缺乏理论支撑。基于理论分析，我们进一步提出两种SSDU改进方案：其一，对采样集进行划分，使子集保持与原始采样掩码相同的分布类型；其二，提出可补偿采样与划分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划分参数的鲁棒性。