We introduce a new algorithm for complex image reconstruction with separate regularization of the image magnitude and phase. This optimization problem is interesting in many different image reconstruction contexts, although is nonconvex and can be difficult to solve. In this work, we first describe a novel implementation of the previous proximal alternating linearized minimization (PALM) algorithm to solve this optimization problem. We then make enhancements to PALM, leading to a new algorithm named PALMNUT that combines the PALM together with Nesterov's momentum and a novel approach that relies on uncoupled coordinatewise step sizes derived from coordinatewise Lipschitz-like bounds. Theoretically, we establish that a version of PALMNUT (without Nesterov's momentum) monotonically decreases the objective function, guaranteeing convergence of the cost function value. Empirical results obtained in the context of magnetic resonance imaging demonstrate that PALMNUT has computational advantages over common existing approaches like alternating minimization. Although our focus is on the application to separate magnitude and phase regularization, we expect that the same approach may also be useful in other nonconvex optimization problems with similar objective function structure.

中文翻译：

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PALMUT：一种增强型近端交替线性化最小化算法，适用于幅度和相位的分离正则化

我们引入了一种新的复杂图像重建算法，具有图像幅度和相位的单独正则化。这个优化问题在许多不同的图像重建上下文中都很有趣，尽管它是非凸的并且很难解决。在这项工作中，我们首先描述了先前近端交替线性化最小化 (PALM) 算法的一种新颖实现，以解决此优化问题。然后，我们对 PALM 进行了改进，产生了一种名为 PALMNUT 的新算法，该算法将 PALM 与 Nesterov 的动量相结合，以及一种依赖于从坐标类 Lipschitz 边界导出的非耦合坐标步长的新方法。从理论上讲，我们确定一个版本的 PALMNUT（没有 Nesterov 动量）单调减少目标函数，保证成本函数值的收敛。在磁共振成像背景下获得的实证结果表明，PALMNUT 比交替最小化等常见的现有方法具有计算优势。尽管我们的重点是分离幅度和相位正则化的应用，但我们预计相同的方法也可能适用于具有类似目标函数结构的其他非凸优化问题。