DeepCDL-PR: Deep unfolded convolutional dictionary learning with weighted ℓ1-norm for phase retrieval

Phase retrieval (PR) aims to recover the image of interest from the recorded phaseless measurement. Traditional PR algorithms that use hand-crafted priors suffer from low-quality reconstructions at low signal to noise ratios (SNRs). Recent efforts overcome this limitation by using deep priors, but existing algorithms ignore structural priors. To remedy this issue, we propose a deep unfolded convolutional dictionary learning with the weighted ℓ1-norm, termed DeepCDL, for PR. By doing so, deep priors and structural priors can be utilized. Concretely, we exploit the weighted ℓ1-norm to formulate a convolutional dictionary learning (CDL)-based minimization problem, and then unfold the corresponding iterative algorithm into a deep network architecture. Moreover, we design a data-driven weight generator to generate crucial weights in the weighted ℓ1-norm from representation coefficients. For the PR task, we first utilize structural priors to formulate a PR minimization problem, and then propose an iterative algorithm to deal with the formulated problem. The proposed DeepCDL method is utilized to solve the convolutional dictionary learning subproblem with the weighted ℓ1-norm, and an inertial epigraph method employing the inertial technique is proposed to tackle the image updating subproblem. Furthermore, the proposed PR iterative algorithm is unfolded into a feed-forward network dubbed as DeepCDL-PR, where DeepCDL serves as a prior module and the unfolded inertial epigraph method acts as an image updating module. Experiments demonstrate that DeepCDL-PR can recover higher-quality images at various noise levels, compared with previous PR algorithms.