This work proposes a general learned proximal alternating minimization algorithm, LPAM, for solving learnable two-block nonsmooth and nonconvex optimization problems. We tackle the nonsmoothness by an appropriate smoothing technique with automatic diminishing smoothing effect. For smoothed nonconvex problems we modify the proximal alternating linearized minimization (PALM) scheme by incorporating the residual learning architecture, which has proven to be highly effective in deep network training, and employing the block coordinate decent (BCD) iterates as a safeguard for the convergence of the algorithm. We prove that there is a subsequence of the iterates generated by LPAM, which has at least one accumulation point and each accumulation point is a Clarke stationary point. Our method is widely applicable as one can employ various learning problems formulated as two-block optimizations, and is also easy to be extended for solving multi-block nonsmooth and nonconvex optimization problems. The network, whose architecture follows the LPAM exactly, namely LPAM-net, inherits the convergence properties of the algorithm to make the network interpretable. As an example application of LPAM-net, we present the numerical and theoretical results on the application of LPAM-net for joint multi-modal MRI reconstruction with significantly under-sampled k-space data. The experimental results indicate the proposed LPAM-net is parameter-efficient and has favourable performance in comparison with some state-of-the-art methods.

PET suffers from a low signal-to-noise ratio. Meanwhile, the k-space data acquisition process in MRI is time-consuming by PET-MRI systems. We aim to accelerate MRI and improve PET image quality. This paper proposed a novel joint reconstruction model by diffusion stochastic differential equations based on learning the joint probability distribution of PET and MRI. Compare the results underscore the qualitative and quantitative improvements our model brings to PET and MRI reconstruction, surpassing the current state-of-the-art methodologies. Joint PET-MRI reconstruction is a challenge in the PET-MRI system. This studies focused on the relationship extends beyond edges. In this study, PET is generated from MRI by learning joint probability distribution as the relationship.

Computed Tomography (CT) is pivotal in industrial quality control and medical diagnostics. Sparse-view CT, offering reduced ionizing radiation, faces challenges due to its under-sampled nature, leading to ill-posed reconstruction problems. Recent advancements in Implicit Neural Representations (INRs) have shown promise in addressing sparse-view CT reconstruction. Recognizing that CT often involves scanning similar subjects, we propose a novel approach to improve reconstruction quality through joint reconstruction of multiple objects using INRs. This approach can potentially leverage both the strengths of INRs and the statistical regularities across multiple objects. While current INR joint reconstruction techniques primarily focus on accelerating convergence via meta-initialization, they are not specifically tailored to enhance reconstruction quality. To address this gap, we introduce a novel INR-based Bayesian framework integrating latent variables to capture the inter-object relationships. These variables serve as a dynamic reference throughout the optimization, thereby enhancing individual reconstruction fidelity. Our extensive experiments, which assess various key factors such as reconstruction quality, resistance to overfitting, and generalizability, demonstrate significant improvements over baselines in common numerical metrics. This underscores a notable advancement in CT reconstruction methods.

Undersampling the k-space in MRI allows saving precious acquisition time, yet results in an ill-posed inversion problem. Recently, many deep learning techniques have been developed, addressing this issue of recovering the fully sampled MR image from the undersampled data. However, these learning based schemes are susceptible to differences between the training data and the image to be reconstructed at test time. One such difference can be attributed to the bias field present in MR images, caused by field inhomogeneities and coil sensitivities. In this work, we address the sensitivity of the reconstruction problem to the bias field and propose to model it explicitly in the reconstruction, in order to decrease this sensitivity. To this end, we use an unsupervised learning based reconstruction algorithm as our basis and combine it with a N4-based bias field estimation method, in a joint optimization scheme. We use the HCP dataset as well as in-house measured images for the evaluations. We show that the proposed method improves the reconstruction quality, both visually and in terms of RMSE.