A deep-learning single-shot phase retrieval approach with <i>ABCD</i> matrices

The phase retrieval problem centers on reconstructing a signal or image from the magnitude of its Fourier transform, a task complicated by the inherent loss of phase information. Single-shot intensity measurements in Fourier domain are typically insufficient for accurate recovery. To address this challenge, we introduce a novel phase retrieval framework based on a generalized ABCD matrix transform, which unifies Fourier, fractional Fourier and linear canonical transform domains. Our approach is implemented within a self-supervised, end-to-end deep learning architecture that integrates a local implicit image function (LIIF) with a super-resolution module. This design enables direct reconstruction from a single intensity measurement, eliminating the need for multiple acquisitions or iterative refinement commonly required in classical settings. We evaluate the proposed method on the DIV2K and Set12 benchmark datasets using selected parameter configurations. Extensive experiments, including an ablation study, demonstrate that our framework consistently outperforms existing algorithms in reconstruction fidelity, underscoring its robustness and practical effectiveness.