Abstract In this paper, we propose an analytical phase retrieval approach applying only phase modulations in the region isolated from the object to be measured. This approach employs a series of intensity-only measurements and can accurately and conveniently achieve phase recovery in the range of [0, 2π) once a reference phase is calibrated in advance. To improve the robustness of the algorithm, this paper yields an over-determined method (OM) by extending the phase modulation procedure from 3 steps into n steps. The algorithm's exact phase retrieval ability, the influence of key parameters’ disturbance on solving precision, and the theory of robustness improvement are all verified through numerical simulations. It is noteworthy mentioning that the robustness enhancement method performs well–improving the phase retrieval accuracy efficiently. Also, an experiment is undertaken and the results are satisfactory, demonstrating the core of our approach and showing the capability of OM.

中文翻译：

---

仅在隔离区域中应用相位调制的精确相位恢复

摘要 在本文中，我们提出了一种分析相位检索方法，该方法仅在与待测物体隔离的区域中应用相位调制。这种方法采用了一系列仅强度测量，一旦参考相位被提前校准，就可以准确方便地实现 [0, 2π) 范围内的相位恢复。为了提高算法的鲁棒性，本文通过将相位调制过程从 3 步扩展到 n 步，产生了一种超定方法 (OM)。该算法的精确相位反演能力、关键参数扰动对求解精度的影响以及鲁棒性改进理论均通过数值仿真得到验证。值得注意的是，鲁棒性增强方法表现良好——有效地提高了相位检索精度。