Provably bounded prompting prior network for universal compressed sensing magnetic resonance imaging

Compressed sensing magnetic resonance imaging (CSMRI) aims to reconstruct MR images from undersampled $k$-space data. Existing deep unrolling CSMRI methods unfold iterative algorithms into deep neural networks, demonstrating superior reconstruction performance. However, they still face several limitations: ($i$) The prior networks used in deep unrolling methods are often empirically designed, lacking interpretability and hindering further theoretical analysis. ($ii$) These methods require training for each sampling setting (e.g. sampling mode and sampling ratio), which incurs significant storage costs. To address these challenges, we propose PDSNet, a network inspired by a double sparsity model, which is both provable and interpretable. As a prior network, PDSNet is integrated into a deep unrolling framework to solve the universal CSMRI task. This enables our method to use a single model to address the compressed sensing MRI problem across various sampling settings. Specifically, PDSNet is built on a double sparsity model using tight frames, and the thresholds for shrinking frame coefficients are adaptively generated by a dedicated threshold-generating sub-network (TGNet). In TGNet, we introduce an information fusion module that effectively captures both global and regional features. Additionally, a prompt block is designed to learn discriminative information across different sampling settings, enabling high-quality reconstructions for each setting using a single model. Experimental results demonstrate that our method achieves superior reconstruction performance. On the theoretical side, we provide explicit proof that PDSNet satisfies bounded properties and further show that the corresponding iterative algorithm converges to a fixed point.