Phase retrieval: A data-driven wavelet frame based approach

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.