Net-BDR: Untrained Neural Networks Embedded Background Douglas Rachford Method for Fourier Phase Retrieval

Fourier phase retrieval (FPR) is a fundamental yet severely ill-posed inverse problem in physical optics. Although the Background Douglas Rachford (BDR) method alleviates ill-posedness by incorporating background information as a structural prior, its robustness degrades substantially under intense noise. To achieve robustness to strong noise and reduce the background information requirements, we introduce an implicit signal prior by reparameterizing the target signal through an untrained neural network (UNN). By doing so, we propose Net-BDR, an algorithm that integrates the UNN's implicit prior with the background's structural prior. This approach requires no pre-training and implicitly regularizes the solution space through the UNN's architecture. Specifically, Net-BDR employs an adaptive weighting strategy to embed the UNN into the BDR framework, dynamically balancing these dual priors while significantly enhancing computational efficiency. Furthermore, we theoretically prove that both Net-BDR and Adaptive Net-BDR converge to a stationary point of the objective function. Experiments demonstrate that Adaptive Net-BDR not only improves the reconstruction quality by at least 2-3 dB in PSNR under intense noise but also reduces the dependency on background information.