Structured Random Models for Phase Retrieval with Optical Diffusers

Phase retrieval is a nonlinear inverse problem that arises in a wide range of imaging modalities, from electron microscopy to Fourier ptychography. In particular, the reconstruction is facilitated when the sensing matrix is i.i.d. random, enabling strong theoretical guarantees and efficient reconstruction algorithms. However, its applicability is restricted by excessive computational costs. In this paper, we propose structured random models for phase retrieval, where we emulate a dense random matrix by a cascade of structured transforms and random diagonal matrices. We reduce the complexity from quadratic to log-linear at no cost in reconstruction performance. Through a spectral method initialization followed by gradient descent, robust reconstruction is obtained at an oversampling ratio as low as 2.8. Moreover, we observe that the reconstruction performance is solely determined by the singular-value distribution of the forward matrix. This class of models can directly be implemented with basic optical elements such as lenses and diffusers, paving the way for large-scale phase imaging with robust reconstruction guarantees.