Magnetic Resonance Imaging (MRI) is a powerful technique employed for non-invasive in vivo visualization of internal structures. Sparsity is often deployed to accelerate the signal acquisition or overcome the presence of motion artifacts, improving the quality of image reconstruction. Image reconstruction algorithms use TV-regularized LASSO (Total Variation-regularized LASSO) to retrieve the missing information of undersampled signals, by cleaning the data of noise and while optimizing sparsity. A tuning parameter moderates the balance between these two aspects; its choice affecting the quality of the reconstructions. Currently, there is a lack of general deterministic techniques to choose these parameters, which are oftentimes manually selected and thus hinder the reliability of the reconstructions. Here, we present ALMA (Algorithm for Lagrange Multipliers Approximation), an iterative mathematics-inspired technique that computes tuning parameters for generalized LASSO problems during MRI reconstruction. We analyze quantitatively the performance of these parameters for imaging reconstructions via TV-LASSO in an MRI context on phantoms. Although our study concentrates on TV-LASSO, the techniques developed here hold significant promise for a wide array of applications. ALMA is not only adaptable to more generalized LASSO problems but is also robust to accommodate other forms of regularization beyond total variation. Moreover, it extends effectively to handle non-Cartesian sampling trajectories, broadening its utility in complex data reconstruction scenarios. More generally, ALMA provides a powerful tool for numerically solving constrained optimization problems across various disciplines, offering a versatile and impactful solution for advanced computational challenges.

翻译：磁共振成像（MRI）是一种强大的非侵入性活体内部结构可视化技术。稀疏性常被用于加速信号采集或克服运动伪影，从而提升图像重建质量。图像重建算法采用TV正则化LASSO（全变分正则化LASSO）通过去除噪声并优化稀疏性，来恢复欠采样信号的缺失信息。调节参数控制着这两方面的平衡；其选择直接影响重建质量。目前缺乏通用的确定性方法来选择这些参数，通常依赖人工选取，这影响了重建结果的可靠性。本文提出ALMA（拉格朗日乘子逼近算法），一种受数学启发的迭代技术，用于在MRI重建过程中计算广义LASSO问题的调节参数。我们通过体模实验，在MRI背景下定量分析了这些参数对TV-LASSO图像重建的性能影响。尽管本研究聚焦于TV-LASSO，所开发的技术对更广泛的应用领域具有重要潜力。ALMA不仅能适应更广义的LASSO问题，还可稳健地扩展至全变分之外的其他正则化形式。此外，该方法能有效处理非笛卡尔采样轨迹，拓展了其在复杂数据重建场景中的实用性。更广泛而言，ALMA为跨学科约束优化问题的数值求解提供了强大工具，为先进计算挑战提供了多功能且具有影响力的解决方案。