This project aims to improve upon the HIO algorithm by incorporating a low resolution image as a stronger prior than the finite support constraint used by most algorithms. This should yield a phase retrieval method with a higher reconstruction fidelity and convergence rate, because the LR image contains not only the support boundary but also local intensities information, as well as low-k information in the Fourier domain. Figure 1(b) shows the pipeline we have developed. Given a LR prior of reasonable quality, the modified algorithm in general outperforms the conventional HIO in terms of reconstruction quality and convergence rate. For this project, We researched into different methods for optimizing an image to be closer to the given LR image in a dynamic fashion which can be tuned for convergence, and found indeed the alternating direction methods (ADMs) can be employed to implement phase retrieval algorithms, i.e. HIO . In the following sections, we will briefly review how ADM can be used for phase retrieval, and further develop an ADM approach that employs a LR image to enhance the reconstruction fidelity.
 Z. Wen, C. Yang, X. Liu, and Stefano Marchesini, Inverse Prob. 28, 115010 (2012).