Background Magnetic resonance imaging (MRI) is an effective auxiliary diagnostic method in clinical medicine, but it has always suffered from the problem of long acquisition time. Compressed sensing and parallel imaging are two common techniques to accelerate MRI reconstruction. Recently, deep learning provides a new direction for MRI, while most of them require a large number of data pairs for training. However, there are many scenarios where fully sampled k-space data cannot be obtained, which will seriously hinder the application of supervised learning. Therefore, deep learning without fully sampled data is indispensable. Main text In this review, we first introduce the forward model of MRI as a classic inverse problem, and briefly discuss the connection of traditional iterative methods to deep learning. Next, we will explain how to train reconstruction network without fully sampled data from the perspective of obtaining prior information. Conclusion Although the reviewed methods are used for MRI reconstruction, they can also be extended to other areas where ground-truth is not available. Furthermore, we may anticipate that the combination of traditional methods and deep learning will produce better reconstruction results.

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Multidimensional nuclear magnetic resonance (NMR) spectroscopy is one of the most crucial detection tools for molecular structure analysis and has been widely used in biomedicine and chemistry. However, the development of NMR spectroscopy is hampered by long data collection time. Non-uniform sampling empowers rapid signal acquisition by collecting a small subset of data. Since the sampling rate is lower than that of the Nyquist sampling ratio, undersampling artifacts arise in reconstructed spectra. To obtain a high-quality spectrum, it is necessary to apply reasonable prior constraints in spectrum reconstruction models. The self-learning subspace method has been shown to possess superior advantages than that of the state-of-the-art low-rank Hankel matrix method when adopting high acceleration in data sampling. However, the self-learning subspace method is time-consuming due to the singular value decomposition in iterations. In this paper, we propose a fast self-learning subspace method to enable fast and high-quality reconstructions. Aided by parallel computing, the experiment results show that the proposed method can reconstruct high-fidelity spectra but spend less than 10% of the time required by the non-parallel self-learning subspace method.

Nuclear magnetic resonance with diffusion‐ordered spectroscopy (DOSY) serves as an important analytical tool to non‐destructively separate a molecule from a compound in medicine and chemistry. However, the data acquisition time increases rapidly for multidimensional DOSY. To enable fast DOSY, partial data are acquired with non‐uniform sampling, and the spectrum can be reconstructed with a proper constraint, such as sparsity in the state‐of‐the‐art method. However, the reconstructed spectrum is observed to have isolated artefacts, which can be easily recognised as fake peaks and affect the estimated diffusion coefficients severely. The authors introduce the low‐rank constraint as an effective remedy to remove these artefacts and derive a fast algorithm to solve the reconstruction problem. Results on both synthetic and realistic DOSY spectra show that a better spectrum and more accurate diffusion coefficients can be achieved.

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