Supervised dual tight frame learning with deep thresholding network for phase retrieval

Data-driven tight frames are popular for solving imaging inverse problems. However, the imaging quality is limited by the representation ability of single tight frame and thresholds tuned manually. In this work, a supervised dual tight frame learning framework fused with an elaborated deep thresholding network (DTN) is proposed, and the issue of low-quality reconstructions in previous phase retrieval (PR) algorithms is addressed. To effectively learn dual tight frames, a loss function is formed using the mean square error, tight constraint, dual constraint, and sparse constraint terms. Moreover, to determine the thresholds adaptively, the thresholds are extracted from the frame coefficients via DTN. By an end-to-end supervised learning manner, the dual tight frames and DTN are jointly trained from labels and their counterparts corrupted by Gaussian noise. Using the Gaussian denoiser constructed by dual tight frames, a regularisation model is firstly designed, and then exploited to formulate a PR optimisation problem. The image filtering and image updating steps are performed alternatively for solving this problem. Particularly, the image updating subproblem is tackled by an inertial epigraph solver. The simulation experiments show that the proposed PR algorithm can obtain higher-quality reconstructions compared with the benchmark ones.