We study an approach to solving the phase retrieval problem as it arises in a phase-less imaging modality known as ptychography. In ptychography, small overlapping sections of an unknown sample (or signal, say \(x_0\in \mathbb {C}^{d}\)) are illuminated one at a time, often with a physical mask between the sample and light source. The corresponding measurements are the noisy magnitudes of the Fourier transform coefficients resulting from the pointwise product of the mask and the sample. The goal is to recover the original signal from such measurements. The algorithmic framework we study herein relies on first inverting a linear system of equations to recover a fraction of the entries in \(x_0 x_0^*\) and then using non-linear techniques to recover the magnitudes and phases of the entries of \(x_0\). Thus, this paper’s contributions are three-fold. First, focusing on the linear part, it expands the theory studying which measurement schemes (i.e., masks, shifts of the sample) yield invertible linear systems, including an analysis of the conditioning of the resulting systems. Second, it analyzes a class of improved magnitude recovery algorithms and, third, it proposes and analyzes algorithms for phase recovery in the ptychographic setting where large shifts—up to \(50\%\) the size of the mask—are permitted.

我们研究了一种解决相位检索问题的方法，该方法出现在一种称为相位图的无相位成像模式中。在气相色谱中，未知样品（或信号，例如\（x_0 \ in \ mathbb {C} ^ {d} \}）中小的重叠部分每次被照亮一次，通常在样品和光源之间使用物理遮罩。相应的测量值是由蒙版和样本的点积得出的傅里叶变换系数的噪声大小。目的是从此类测量中恢复原始信号。我们在本文中研究的算法框架依赖于首先反转方程的线性系统以恢复\（x_0 x_0 ^ * \）中的一部分项然后使用非线性技术恢复\（x_0 \）项的大小和相位。因此，本文的贡献是三方面的。首先，着重于线性部分，它扩展了研究哪些测量方案（即掩模，样品的位移）产生可逆线性系统的理论，包括对所得系统的条件进行分析。第二，分析了一类改进的幅度恢复算法，第三，提出并分析了在谱图设置中允许较大移位（最大为掩码大小\（50 \％\））的相位恢复算法。