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doi: 10.3934/ipi.2021028

## Phase retrieval from Fourier measurements with masks

 1 Department of Mathematics, Hangzhou Normal University, 2318 Yuhangtang Road, 311121 2 School of Mathematical Sciences, Zhejiang University, 38 Zheda Road, 310027, Hangzhou, China

* Corresponding author: Song Li

Received  November 2019 Revised  March 2021 Published  May 2021

Fund Project: This work is supported by the key project of NSFC under grant number 11531013, 12071426, the fundamental research funds for the central universities (2020XZZX002-03) and the Zhejiang Provincial Natural Science Foundation grant (LQ19A010008)

This paper concerns the problem of phase retrieval from Fourier measurements with random masks. Here we focus on researching two kinds of random masks. Firstly, we utilize the Fourier measurements with real masks to estimate a general signal $\mathit{\boldsymbol{x}}_0\in \mathbb{R}^d$ in noiseless case when $d$ is even. It is demonstrated that $O(\log^2d)$ real random masks are able to ensure accurate recovery of $\mathit{\boldsymbol{x}}_0$. Then we find that such real masks are not adaptable to reconstruct complex signals of even dimension. Subsequently, we prove that $O(\log^4d)$ complex masks are enough to stably estimate a general signal $\mathit{\boldsymbol{x}}_0\in \mathbb{C}^d$ under bounded noise interference, which extends E. Candès et al.'s work. Meanwhile, we establish tighter error estimations for real signals of even dimensions or complex signals of odd dimensions by using $O(\log^2d)$ real masks. Finally, we intend to tackle with the noisy phase problem about an $s$-sparse signal by a robust and efficient approach, namely, two-stage algorithm. Based on the stable guarantees for general signals, we show that the $s$-sparse signal $\mathit{\boldsymbol{x}}_0$ can be stably recovered from composite measurements under near-optimal sample complexity up to a $\log$ factor, namely, $O(s\log(\frac{ed}{s})\log^4(s\log(\frac{ed}{s})))$

Citation: Huiping Li, Song Li. Phase retrieval from Fourier measurements with masks. Inverse Problems & Imaging, doi: 10.3934/ipi.2021028
##### References:

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##### References:
The Number of Masks Needed for Each Case
 complex random masks (4) real random masks (3) $\boldsymbol{x}_0\in\mathbb{C}^{d}/\mathbb{R}^{d}$, $d$ is odd $O(\log^4d)$ $O(\log^2d)$ $\boldsymbol{x}_0\in\mathbb{C}^{d}$, $d$ is even $O(\log^4d)$ not uniquely recovered $\boldsymbol{x}_0\in\mathbb{R}^{d}$, $d$ is even $O(\log^4d)$ $O(\log^2d)$
 complex random masks (4) real random masks (3) $\boldsymbol{x}_0\in\mathbb{C}^{d}/\mathbb{R}^{d}$, $d$ is odd $O(\log^4d)$ $O(\log^2d)$ $\boldsymbol{x}_0\in\mathbb{C}^{d}$, $d$ is even $O(\log^4d)$ not uniquely recovered $\boldsymbol{x}_0\in\mathbb{R}^{d}$, $d$ is even $O(\log^4d)$ $O(\log^2d)$
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