We address the problem of phase retrieval (PR) from quantized measurements. The goal is to reconstruct a signal from quadratic measurements encoded with a finite precision, which is indeed the case in practical applications. We develop an iterative projected-gradient-type algorithm that recovers the signal subject to ensuring consistency with the measurement, meaning that the recovered signal, when encoded, must yield the same set of measurements that one started with. The algorithm involves rank-1 projection, which stems from the idea of lifting, originally proposed in the context of PhaseLift. The consistency criterion is enforced using a one-sided quadratic cost. We also determine the probability with which different vectors lead to the same set of quantized measurements, which makes it impossible to resolve them. Naturally, this probability depends on how correlated such vectors are, and how coarsely/finely the measurements are quantized. The proposed algorithm is also capable of incorporating a sparsity constraint on the signal. An analysis of the cost function reveals that it is bounded probabilistically, both above and below, by functions that are dependent on how well-correlated the estimate is with the ground-truth. We also derive the Cramér-Rao lower bound (CRB) on the achievable reconstruction accuracy. A comparison with the state-of-the-art algorithms shows that the proposed algorithm has a higher reconstruction accuracy and is about 2 to 3dB away from the CRB. The edge, in terms of the reconstruction signal-to-noise ratio, over the competing algorithms is higher (about 5 to 6dB) when the quantization is coarse, thereby making the proposed scheme particularly attractive in such scenarios. We also demonstrate a concrete application of the proposed method to frequency-domain optical-coherence tomography (FDOCT). 

We propose a new approach for learning end-to-end reconstruction operators based on unpaired training data for ill-posed inverse problems. The proposed method combines the classical variational framework with iterative unrolling and essentially seeks to minimize a weighted combination of the expected distortion in the measurement space and the Wasserstein-1 distance between the distributions of the reconstruction and the ground-truth. More specifically, the regularizer in the variational setting is parametrized by a deep neural network and learned simultaneously with the unrolled reconstruction operator. The variational problem is then initialized with the output of the reconstruction network and solved iteratively till convergence. Notably, it takes significantly fewer iterations to converge as compared to variational methods, thanks to the excellent initialization obtained via the unrolled operator. The resulting approach combines the computational efficiency of end-to-end unrolled reconstruction with the well-posedness and noise-stability guarantees of the variational setting. Moreover, we demonstrate with the example of image reconstruction in X-ray computed tomography (CT) that our approach outperforms state-of-the-art unsupervised methods and that it outperforms or is at least on par with state-of-the-art supervised data-driven reconstruction approaches.

We consider the problem of reconstructing a complex-valued signal from its phase-only measurements. This framework can be considered as a generalization of the well-known one-bit compressed sensing paradigm where the underlying signal is known to be sparse. In contrast, the proposed formalism does not rely on the assumption of sparsity and hence applies to a broader class of signals. The optimization problem for signal reconstruction is formulated by first splitting the linear measurement vector into its phase and magnitude components and subsequently using the non-negativity property of the magnitude component as a constraint. The resulting optimization problem turns out to be a quadratic program (QP) and is solved using two algorithms: (i) alternating directions method of multipliers; and (ii) projected gradient-descent with Nesterov's momentum. Due to the inherent scale ambiguity of the phase-only measurement, the underlying signal can be reconstructed only up to a global scale-factor. We obtain high accuracy for reconstructing 1-D synthetic signals in the absence of noise. We also show an application of the proposed approach in reconstructing images from the phase of their measurement coefficients. The underlying image is recovered up to a peak signal-to-noise ratio exceeding 30 dB in several examples, indicating an accurate reconstruction.

We consider signal denoising via transform-domain shrinkage based on a novel risk criterion called the minimum probability of error (MPE), which measures the probability that the estimated parameter lies outside an epsilon-neighborhood of the true value. The underlying parameter is assumed to be deterministic. The MPE, similar to the mean-squared error (MSE), depends on the ground-truth parameter, and therefore, has to be estimated from the noisy observations. The optimum shrinkage parameter is obtained by minimizing an estimate of the MPE. When the probability of error is integrated over epsilon, it leads to the expected l(1) distortion. The proposed MPE and l(1) distortion formulations are applicable to various noise distributions by invoking a Gaussian mixture model approximation. Within the realm of MPE, we also develop a specific extension to subband shrinkage. The denoising performance of MPE turns out to be better than that obtained using the minimum MSE-based approaches formulated within Stein's unbiased risk estimation (SURE) framework, especially in the low signal-to-noise ratio (SNR) regime. Performance comparisons with three benchmarking algorithms carried out on electrocardiogram signals and standard test signals taken from the Wavelab toolbox show that the MPE framework results in SNR gains particularly for low input SNR.