Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization
Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization
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DOI：
10.1364/josaa.19.001334
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发表时间：
2002-07-01
影响因子：
1.9
通讯作者：
Luke, DR
中科院分区：
文献类型：
--
作者：
Bauschke, HH;Combettes, PL;Luke, DR
相位恢复问题在应用物理和工程的各个领域中都是非常重要的。在两个维度上解决这个问题的技术水平在很大程度上依赖于Gerchberg、Saxton和Fienup的开创性工作。尽管这三位研究人员提出的算法得到了广泛应用，但目前的数学理论无法解释它们的显著成功。然而，这些算法的行为、缺点和性能可以从它们在凸优化理论中可能的同行中获得更大的洞察力。二十年前，当误差减少算法被确定为非凸交替投影算法时，朝着这个方向迈出了重要的一步。我们的目的是用数学的方法描述相位恢复问题，并在成熟的数值相位恢复格式和经典的凸优化方法之间建立新的联系。具体地说，Fienup的基本输入输出算法对应于Dykstra算法，Fienup的混合输入输出算法可以看作是Douglas-Rachford算法的一个实例。我们提供了一个理论框架来更好地理解并潜在地改进现有的相位恢复算法。(C)2002年美国光学学会。
The phase retrieval problem is of paramount importance in various areas of applied physics and engineering. The state of the art for solving this problem in two dimensions relies heavily on the pioneering work of Gerchberg, Saxton, and Fienup. Despite the widespread use of the algorithms proposed by these three researchers, current mathematical theory cannot explain their remarkable success. Nevertheless, great insight can be gained into the behavior, the shortcomings, and the performance of these algorithms from their possible counterparts in convex optimization theory. An important step in this direction was made two decades ago when the error reduction algorithm was identified as a nonconvex alternating projection algorithm. Our purpose is to formulate the phase retrieval problem with mathematical care and to establish new connections between well-established numerical phase retrieval schemes and classical convex optimization methods. Specifically, it is shown that Fienup's basic input-output algorithm corresponds to Dykstra's algorithm and that Fienup's hybrid input-output algorithm can be viewed as an instance of the Douglas-Rachford algorithm. We provide a theoretical framework to better understand and, potentially, to improve existing phase recovery algorithms. (C) 2002 Optical Society of America.