We consider the problem of reconstructing a function \(f\in L^2({\mathbb R})\) given phase-less samples of its Gabor transform, which is defined by

More precisely, given sampling positions \(\Omega \subseteq {\mathbb R}^2\) the task is to reconstruct f (up to global phase) from measurements \(\{|{\mathcal {G}}f(\omega )|: \,\omega \in \Omega \}\). This non-linear inverse problem is known to suffer from severe ill-posedness. As for any other phase retrieval problem, constructive recovery is a notoriously delicate affair due to the lack of convexity. One of the fundamental insights in this line of research is that the connectivity of the measurements is both necessary and sufficient for reconstruction of phase information to be theoretically possible. In this article we propose a reconstruction algorithm which is based on solving two convex problems and, as such, amenable to numerical analysis. We show, empirically as well as analytically, that the scheme accurately reconstructs from noisy data within the connected regime. Moreover, to emphasize the practicability of the algorithm we argue that both convex problems can actually be reformulated as semi-definite programs for which efficient solvers are readily available. The approach is based on ideas from complex analysis, Gabor frame theory as well as matrix completion. As a byproduct, we also obtain improved truncation error for Gabor expensions with Gaussian generators.

我们考虑一个函数 \（f\in L^2（{\mathbb R}）\） 给定其 Gabor 变换的无相样本的问题，它由

更准确地说，给定采样位置 \（\Omega \subseteq {\mathbb R}^2\） 的任务是从测量 \（\{|{\mathcal {G}}f（\omega ）|： \，\omega \in \Omega \}\）。众所周知，这个非线性逆问题存在严重的病态性。与任何其他相检索问题一样，由于缺乏凸性，建设性恢复是一件非常微妙的事情。这一系列研究的基本见解之一是，测量的连通性对于理论上重建相位信息既必要又充分。在本文中，我们提出了一种基于解决两个凸问题的重建算法，因此适用于数值分析。我们从经验和分析上表明，该方案准确地从关联机制内的噪声数据中重建。此外，为了强调该算法的实用性，我们认为这两个凸问题实际上都可以重新表述为半定程序，其中有效的求解器很容易获得。该方法基于复分析、Gabor 框架理论以及矩阵完成的想法。作为副产品，我们还获得了改进的 Gabor 高斯生成器的截断误差。