Phase retrieval is in general a non-convex and non-linear task and the corresponding algorithms struggle with the issue of local minima. We consider the case where the measurement samples within typically very small and disconnected subsets are coherently linked to each other — which is a reasonable assumption for our objective of antenna measurements. Two classes of measurement setups are discussed which can provide this kind of extra information: multi-probe systems and holographic measurements with multiple reference signals. We propose several formulations of the corresponding phase retrieval problem. The simplest of these formulations poses a linear system of equations similar to an eigenvalue problem where a unique non-trivial null-space vector needs to be found. Accurate phase reconstruction for partially coherent observations is, thus, possible by a reliable solution process and with judgment of the solution quality. Under ideal, noise-free conditions, the required sampling density is less than two times the number of unknowns. Noise and other observation errors increase this value slightly. Simulations for Gaussian random matrices and for antenna measurement scenarios demonstrate that reliable phase reconstruction is possible with the presented approach.

中文翻译：

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部分相干观测的相位检索

相位检索通常是一个非凸和非线性的任务，相应的算法会遇到局部极小值的问题。我们考虑的情况是，通常情况下，非常小的且断开连接的子集中的测量样本会相互关联，这对于我们的天线测量目标来说是一个合理的假设。讨论了两类可提供此类额外信息的测量设置：多探针系统和具有多个参考信号的全息测量。我们提出了相应相位检索问题的几种公式。这些公式中最简单的公式是一个类似于本征值问题的线性方程组，在本征值问题中，需要找到唯一的非平凡的零空间矢量。因此，用于部分相干观测的准确相位重建是 通过可靠的解决方案过程以及对解决方案质量的判断，可以实现这一目标。在理想的无噪声条件下，所需的采样密度小于未知数的两倍。噪声和其他观察误差会稍微增加该值。对高斯随机矩阵和天线测量场景的仿真表明，使用所提出的方法可以进行可靠的相位重建。