In this paper we tackle the problem of recovering the phase of complex linear measurements when only magnitude information is available and we control the input. We are motivated by the recent development of dedicated optics-based hardware for rapid random projections which leverages the propagation of light in random media. A signal of interest $\mathbf{\xi} \in \mathbb{R}^N$ is mixed by a random scattering medium to compute the projection $\mathbf{y} = \mathbf{A} \mathbf{\xi}$, with $\mathbf{A} \in \mathbb{C}^{M \times N}$ being a realization of a standard complex Gaussian iid random matrix. Such optics-based matrix multiplications can be much faster and energy-efficient than their CPU or GPU counterparts, yet two difficulties must be resolved: only the intensity ${|\mathbf{y}|}^2$ can be recorded by the camera, and the transmission matrix $\mathbf{A}$ is unknown. We show that even without knowing $\mathbf{A}$, we can recover the unknown phase of $\mathbf{y}$ for some equivalent transmission matrix with the same distribution as $\mathbf{A}$. Our method is based on two observations: first, conjugating or changing the phase of any row of $\mathbf{A}$ does not change its distribution; and second, since we control the input we can interfere $\mathbf{\xi}$ with arbitrary reference signals. We show how to leverage these observations to cast the measurement phase retrieval problem as a Euclidean distance geometry problem. We demonstrate appealing properties of the proposed algorithm in both numerical simulations and real hardware experiments. Not only does our algorithm accurately recover the missing phase, but it mitigates the effects of quantization and the sensitivity threshold, thus improving the measured magnitudes.

中文翻译：

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不要掉以轻心：用未知算子定相光学随机投影

在本文中，我们解决了当只有幅度信息可用并且我们控制输入时恢复复杂线性测量的相位的问题。我们受到最近开发的用于快速随机投影的基于光学的专用硬件的启发，该硬件利用了光在随机介质中的传播。感兴趣的信号 $\mathbf{\xi} \in \mathbb{R}^N$ 与随机散射介质混合以计算投影 $\mathbf{y} = \mathbf{A} \mathbf{\xi} $，其中 $\mathbf{A} \in \mathbb{C}^{M \times N}$ 是标准复高斯 iid 随机矩阵的实现。这种基于光学的矩阵乘法比它们的 CPU 或 GPU 对应物更快、更节能，但必须解决两个困难：相机只能记录强度 ${|\mathbf{y}|}^2$ , 并且传输矩阵 $\mathbf{A}$ 未知。我们表明，即使不知道 $\mathbf{A}$，我们也可以恢复与 $\mathbf{A}$ 具有相同分布的等效传输矩阵的 $\mathbf{y}$ 的未知相位。我们的方法基于两个观察：首先，共轭或改变 $\mathbf{A}$ 任何行的相位不会改变其分布；其次，由于我们控制输入，我们可以用任意参考信号干扰 $\mathbf{\xi}$。我们展示了如何利用这些观察结果将测量相位检索问题转换为欧几里得距离几何问题。我们在数值模拟和实际硬件实验中证明了所提出算法的吸引人的特性。我们的算法不仅准确地恢复了丢失的相位，