In this paper, we consider the phase retrieval problem for recovering a complex signal, given a number of observations on the magnitude of linear measurements. This problem has direct applications in X-ray crystallography, diffraction imaging and microscopy. Motivated by the extensively studied theory of (tight) wavelet frame and its great success in various applications, we propose a wavelet frame based model for phase retrieval using the balanced approach. A hybrid fidelity term is designed to deal with complicated noises and a hybrid penalty term is constructed for different pursuits of sparsity and smoothness. Consequently, a proximal alternating linearization algorithm is developed and its convergence is analyzed. In particular, our proposed algorithm updates both the internal weights in the hybrid penalty term and the penalty parameter balancing the fidelity and penalty terms in a data-driven way. Extensive numerical experiments show that our method is quite competitive with other existing algorithms. On one hand, our method can reconstruct the truth successfully from a small number of measurements even if the phase retrieval problem is ill-posed. On the other hand, our algorithm is very robust to different types of noise, including Gaussian noise, Poisson noise and their mixtures.

在本文中，鉴于对线性测量幅度的大量观察，我们考虑了用于恢复复杂信号的相位检索问题。该问题直接应用于X射线晶体学，衍射成像和显微镜。基于对（紧）小波框架理论的广泛研究及其在各种应用中的巨大成功，我们提出了一种基于小波框架的模型，该模型用于平衡方法。设计了一个混合保真度项以处理复杂的噪声，并为不同的稀疏性和平滑度构建了一个混合惩罚项。因此，开发了一种近端交替线性化算法，并对其收敛性进行了分析。特别是，我们提出的算法同时更新了混合罚分项中的内部权重和以数据驱动方式平衡保真度和罚分项的罚分参数。大量的数值实验表明，我们的方法与其他现有算法相比具有相当的竞争力。一方面，即使相位检索问题不适当，我们的方法也可以通过少量测量成功地重建真相。另一方面，我们的算法对于不同类型的噪声（包括高斯噪声，泊松噪声及其混合）非常鲁棒。即使相位恢复问题不适当，我们的方法也可以通过少量测量成功地重建真相。另一方面，我们的算法对于不同类型的噪声（包括高斯噪声，泊松噪声及其混合）非常鲁棒。即使相位恢复问题不适当，我们的方法也可以通过少量测量成功地重建真相。另一方面，我们的算法对于不同类型的噪声（包括高斯噪声，泊松噪声及其混合）非常鲁棒。