Model-driven parallel compressive sensing magnetic resonance imaging algorithm based on trainable dual frames and its convergence analysis

[Objective] Parallel compressive sensing magnetic resonance imaging (p-CSMRI) algorithms aim to improve and refine the reconstructed image using partial k-space data sampled from multiple coils. Recently, learning-based model-driven p-CSMRI algorithms have attracted extensive attention because of their superior reconstruction quality. Nevertheless, their prior network architectures are typically designed empirically, lacking model interpretability and hampering the analysis of algorithm convergence. To address this problem, we introduce a provably bounded denoiser based on deep learning and incorporate it as a prior module into a model-driven p-CSMRI network. Moreover, we propose a deep unrolled p-CSMRI algorithm; its convergence can be explicitly analyzed. [Methods] First, to improve the sparse representation capability and learning speed of traditional tight frames, we extend the single tight frame to a dual-frame network. Because the image pixels vary in the spatial domain, a deep threshold network is developed to adaptively extract thresholds from the input images, thereby improving the generalization ability of the dual frames. Based on the dual frames integrated with the elaborated deep threshold network, we introduce a novel provably bounded deep denoiser. Second, we describe a p-CSMRI optimization model based on dual frames. The constructed optimization model is iteratively solved via the half-quadratic splitting solver, and the corresponding iterations are unfolded into a deep neural network that can be trained by end-to-end supervised learning. Finally, under reducing noise level conditions, the convergence of model-driven p-CSMRI algorithms is explicitly proved based on the bounded denoiser theory. The convergence theory of plug-and-play (PnP) imaging methods demonstrates that methods with decreasing noise levels can realize a fixed-point convergence under the assumption of a bounded denoiser. We explicitly prove that the proposed deep denoiser as the prior network is bounded. Based on this bounded property, we develop a model-driven p-CSMRI algorithmic framework with guaranteed convergence. [Results] Theoretically, we explicitly prove that the built deep denoiser as the prior network satisfies the bounded property and perform a convergence analysis of the proposed algorithm under the mild condition of gradually decreasing noise. Simulation experiments carried out on the knee MRI dataset from New York University disclose that, compared with the Modl, VN, and VS-Net algorithms, the proposed method realizes improvements of 1.70, 1.45, and 0.46 dB, respectively, in peak signal-to-noise ratio for reconstructed images under a fourfold acceleration factor. However, a comparative assessment of the proposed model with Modl, VN, and VS-Net algorithms concerning parameter memory and average inference time reveals that the model-driven p-CSMRI method based on the dual frames recommended in this study has a high number of parameters. Furthermore, the image inference time of the proposed method is lower than those of Modl and VN and slightly higher than that of VS-Net. Therefore, our proposed method shows a moderate level of computational complexity. [Conclusions] The model-driven p-CSMRI network algorithm proposed here, based on trainable dual frames, has a theoretical convergence guarantee and demonstrates stable performance in experiments. Moreover, our algorithm proved effective in reconstructing high-quality MR images. This work offers valuable insights into future research and development in the area of magnetic resonance imaging. © 2024 Tsinghua University. All rights reserved.