In this work, we study the robust phase retrieval problem where the task is to recover an unknown signal $\theta^* \in \mathbb{R}^d$ in the presence of potentially arbitrarily corrupted magnitude-only linear measurements. We propose an alternating minimization approach that incorporates an oracle solver for a non-convex optimization problem as a subroutine. Our algorithm guarantees convergence to $\theta^*$ and provides an explicit polynomial dependence of the convergence rate on the fraction of corrupted measurements. We then provide an efficient construction of the aforementioned oracle under a sparse arbitrary outliers model and offer valuable insights into the geometric properties of the loss landscape in phase retrieval with corrupted measurements. Our proposed oracle avoids the need for computationally intensive spectral initialization, using a simple gradient descent algorithm with a constant step size and random initialization instead. Additionally, our overall algorithm achieves nearly linear sample complexity, $\mathcal{O}(d \, \mathrm{polylog}(d))$.

The phase retrieval problem in the presence of noise aims to recover the signal vector of interest from a set of quadratic measurements with infrequent but arbitrary corruptions, and it plays an important role in many scientific applications. However, the essential geometric structure of the nonconvex robust phase retrieval based on the $\ell_1$-loss is largely unknown to study spurious local solutions, even under the ideal noiseless setting, and its intrinsic nonsmooth nature also impacts the efficiency of optimization algorithms. This paper introduces the smoothed robust phase retrieval (SRPR) based on a family of convolution-type smoothed loss functions. Theoretically, we prove that the SRPR enjoys a benign geometric structure with high probability: (1) under the noiseless situation, the SRPR has no spurious local solutions, and the target signals are global solutions, and (2) under the infrequent but arbitrary corruptions, we characterize the stationary points of the SRPR and prove its benign landscape, which is the first landscape analysis of phase retrieval with corruption in the literature. Moreover, we prove the local linear convergence rate of gradient descent for solving the SRPR under the noiseless situation. Experiments on both simulated datasets and image recovery are provided to demonstrate the numerical performance of the SRPR.

An artifact-free computational approach to extract the phase of light from noisy intensity signals improves imaging of transparent objects, such as biological cells, under low light conditions. Deep neural networks are trained to operate on these two frequency bands, before a final algorithm recombines them into a full-band phase image. This method avoids the tendency of automatic phase extraction programs to over-represent low frequencies. The retrieval of phase of electromagnetic fields is one of the most important problems in optics as it allows the shape of transparent objects, including cells, to be quantified using visible light. Phase is a quantity that relates to the wave nature of light; it is not directly detectable by our eyes or common cameras, and yet carries important information about objects the light went through.

Phase retrieval (PR) algorithms have become an important component in many modern computational imaging systems. For instance, in the context of ptychography and speckle correlation imaging PR algorithms enable imaging past the diffraction limit and through scattering media, respectively. Unfortunately, traditional PR algorithms struggle in the presence of noise. Recently PR algorithms have been developed that use priors to make themselves more robust. However, these algorithms often require unrealistic (Gaussian or coded diffraction pattern) measurement models and offer slow computation times. These drawbacks have hindered widespread adoption. In this work we use convolutional neural networks, a powerful tool from machine learning, to regularize phase retrieval problems and improve recovery performance. We test our new algorithm, prDeep, in simulation and demonstrate that it is robust to noise, can handle a variety system models, and operates fast enough for high-resolution applications.