Non-Convex Phase Retrieval From STFT Measurements
Non-Convex Phase Retrieval From STFT Measurements
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DOI：
10.1109/tit.2017.2745623
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发表时间：
2016-07
影响因子：
2.5
通讯作者：
Tamir Bendory;Yonina C. Eldar;Nicolas Boumal
中科院分区：
文献类型：
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作者：
Tamir Bendory;Yonina C. Eldar;Nicolas Boumal
从傅里叶变换幅度恢复一维信号的问题，称为傅立叶相位恢复，在大多数情况下是不适定的。我们考虑了一个密切相关的问题，即从信号的无相位短时傅立叶变换(STFT)测量中恢复信号。这个问题自然地出现在几个应用中，例如超短激光脉冲表征和印刷术。STFT提供的冗余可以在温和的条件下实现独特的恢复。我们证明了在某些情况下，唯一解可以由矩阵的主特征向量得到，该特征向量构造为简单的最小二乘问题的解。当这些条件不满足时，我们建议利用该矩阵的主特征向量来初始化非凸局部优化算法，并提出了两种方法。第一种是基于最小化经验风险损失函数，第二种是最大化相流形上的二次函数。我们证明了在适当的条件下，所提出的初始化是接近于潜在信号的。然后，我们分析了经验风险损失函数的几何结构，并数值证明了这两种梯度算法即使在测量中具有很小的冗余性也能收敛到基础信号。此外，该算法对噪声具有较强的鲁棒性。
The problem of recovering a one-dimensional signal from its Fourier transform magnitude, called Fourier phase retrieval, is ill-posed in most cases. We consider the closely-related problem of recovering a signal from its phaseless short-time Fourier transform (STFT) measurements. This problem arises naturally in several applications, such as ultra-short laser pulse characterization and ptychography. The redundancy offered by the STFT enables unique recovery under mild conditions. We show that in some cases the unique solution can be obtained by the principal eigenvector of a matrix, constructed as the solution of a simple least-squares problem. When these conditions are not met, we suggest using the principal eigenvector of this matrix to initialize non-convex local optimization algorithms and propose two such methods. The first is based on minimizing the empirical risk loss function, while the second maximizes a quadratic function on the manifold of phases. We prove that under appropriate conditions, the proposed initialization is close to the underlying signal. We then analyze the geometry of the empirical risk loss function and show numerically that both gradient algorithms converge to the underlying signal even with small redundancy in the measurements. In addition, the algorithms are robust to noise.