Bauschke Heinz H, Combettes Patrick L, Luke D Russell
Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario N1G 2W1, Canada.
J Opt Soc Am A Opt Image Sci Vis. 2003 Jun;20(6):1025-34. doi: 10.1364/josaa.20.001025.
The phase-retrieval problem, fundamental in applied physics and engineering, addresses the question of how to determine the phase of a complex-valued function from modulus data and additional a priori information. Recently we identified two important methods for phase retrieval, namely, Fienup's basic input-output and hybrid input-output (HIO) algorithms, with classical convex projection methods and suggested that further connections between convex optimization and phase retrieval should be explored. Following up on this work, we introduce a new projection-based method, termed the hybrid projection-reflection (HPR) algorithm, for solving phase-retrieval problems featuring nonnegativity constraints in the object domain. Motivated by properties of the HPR algorithm for convex constraints, we recommend an error measure studied by Fienup more than 20 years ago. This error measure, which has received little attention in the literature, lends itself to an easily implementable stopping criterion. In numerical experiments we found the HPR algorithm to be a competitive alternative to the HIO algorithm and the stopping criterion to be reliable and robust.
相位恢复问题在应用物理和工程领域中至关重要，它解决了如何根据模数据和其他先验信息来确定复值函数相位的问题。最近，我们将相位恢复的两种重要方法，即菲纽普的基本输入输出和混合输入输出（HIO）算法，与经典凸投影方法进行了对比，并建议应进一步探索凸优化与相位恢复之间的联系。在此工作的基础上，我们引入了一种新的基于投影的方法，称为混合投影反射（HPR）算法，用于解决在目标域中具有非负约束的相位恢复问题。受HPR算法处理凸约束特性的启发，我们推荐了一种20多年前由菲纽普研究的误差度量。这种误差度量在文献中很少受到关注，但它易于实现停止准则。在数值实验中，我们发现HPR算法是HIO算法的一个有竞争力的替代方案，并且该停止准则可靠且稳健。
