Phase Retrieval via Wirtinger Flow: Theory and Algorithms

Phase retrieval (PR) from coded diffraction patterns (CDP) is an inverse problem that consists of estimating a phase image from phaseless measurements acquired with optical systems that include coding masks into the setups. Specifically, CDP contains only magnitude information because current optical sensors cannot measure the phase information of the optical field. Several computational algorithms have been proposed to handle the PR problem. Recently, deep recovery networks in PR have been developed, learning a deep model that acts as a non-linear mapping from CDP to the phase image. However, traditional deep learning approaches in PR lack interpretability in their neural architectures. This paper proposes an unrolling approach where each deep block layer is interpreted as an iteration of a traditional PR algorithm; mainly, the proposed method incorporates two branches used as a proximal operator to estimate magnitude and phase information separately. Simulations show that the proposed unrolled network for PR achieves better reconstruction results at recovering the phase compared to their competitive algorithms.

Phase retrieval (PR) arises from the lack of phase information in the measures recorded by optical sensors. Phase masks that modulate the optical field and reduce ambiguities in the PR problem by producing redundancy in coded diffraction patterns (CDPs) have been included in these diffractive optical systems. Several algorithms have been developed to solve the PR problem from CDPs. Also, deep neural networks (DNNs) are used for solving inverse problems in computational imaging by considering physical constraints in propagation models. However, traditional algorithms based on non-convex formulation include an initialization stage that requires a high number of iterations to properly estimate the optical field. This work proposes an end-to-end (E2E) approach for addressing the PR problem, which jointly learns the spectral initialization and network parameters. Mainly, the proposed deep network approach contains an optical layer that simulates the propagation model in diffractive optical systems, an initialization layer that approximates the underlying optical field from CDPs, and a double branch DNN that improves the obtained initial guess by separately recovering phase and amplitude information. Simulation results show that the proposed E2E approach for PR requires fewer snapshots and iterations than the state of the art.

High-throughput computational imaging requires efficient processing algorithms to retrieve multi-dimensional and multi-scale information. In computational phase imaging, phase retrieval (PR) is required to reconstruct both amplitude and phase in complex space from intensity-only measurements. The existing PR algorithms suffer from the tradeoff among low computational complexity, robustness to measurement noise and strong generalization on different modalities. In this work, we report an efficient large-scale phase retrieval technique termed as LPR. It extends the plug-and-play generalized-alternating-projection framework from real space to nonlinear complex space. The alternating projection solver and enhancing neural network are respectively derived to tackle the measurement formation and statistical prior regularization. This framework compensates the shortcomings of each operator, so as to realize high-fidelity phase retrieval with low computational complexity and strong generalization. We applied the technique for a series of computational phase imaging modalities including coherent diffraction imaging, coded diffraction pattern imaging, and Fourier ptychographic microscopy. Extensive simulations and experiments validate that the technique outperforms the existing PR algorithms with as much as 17dB enhancement on signal-to-noise ratio, and more than one order-of-magnitude increased running efficiency. Besides, we for the first time demonstrate ultra-large-scale phase retrieval at the 8K level (7680times 4320 pixels) in minute-level time.

Phase-retrieval from coded diffraction patterns (CDP) is important to X-ray crystallography, diffraction tomography and astronomical imaging, yet remains a hard, non-convex inverse problem. We show that CDP recovery can be reformulated exactly as the minimization of a continuous-variable XY Hamiltonian and solved by gain-based photonic networks. The coupled-mode equations we exploit are the natural mean-field dynamics of exciton-polariton condensate lattices, coupled-laser arrays and driven photon Bose-Einstein condensates, while other hardware such as the spatial photonic Ising machine can implement the same update rule through high-speed digital feedback, preserving full optical parallelism. Numerical experiments on images, two- and three-dimensional vortices and unstructured complex data demonstrate that the gain-based solver consistently outperforms the state-of-the-art Relaxed-Reflect-Reflect (RRR) algorithm in the medium-noise regime (signal-to-noise ratios 10-40 dB) and retains this advantage as problem size scales. Because the physical platform performs the continuous optimisation, our approach promises fast, energy-efficient phase retrieval on readily available photonic hardware.

General optical detection devices rely on converting photons to electrons (current), and do not allow for direct recording of the phase due to the high oscillation frequency. As so, the missing phase can only be recovered from the intensity measurements. The emerging non-convex phase retrieval algorithm, represented by the Wirtinger Flow (WF) algorithm requires multiple-shot coded diffraction patterns (CDPs) for accurate recovery. To achieve the real-time acquisition for multiple CDPs, this paper proposes a 4f imaging system based on an electrically tunable lens (ETL), which can be used for real-time acquisition of multiple-shot CDPs, and can take the advantages of highspeed, high-resolution, extended depth-of-field, high-sensitivity and low-cost imaging. In this paper, the performance of 4f-ETL based imaging system for phase retrieval with multiple CDPs is compared under different iteration times, different object size, different numbers of masks and different noise levels. Numerical experiments demonstrate the effectiveness and superiority of our proposed ETL-based imaging system, and ETL allows variable-distance focusing of imaging and display systems without mechanical structures, which reduces the mechanical complexity and power consumption, improves acquisition speed.

Phase retrieval in optical imaging refers to the recovery of a complex signal from phaseless data acquired in the form of its diffraction patterns. These patterns are acquired through a system with a coherent light source that employs a diffractive optical element (DOE) to modulate the scene, resulting in coded diffraction patterns (CDPs) at the sensor. Recently, the hybrid approach of a model-driven network or deep unfolding has emerged as an effective alternative to conventional model- and learning-based phase-retrieval techniques because it allows for bounding the complexity of algorithms while also retaining their efficacy. Additionally, such hybrid approaches have shown promise in improving the design of DOEs that follow theoretical uniqueness conditions. There are opportunities to exploit novel experimental setups and resolve even more complex DOE phase-retrieval applications. This article presents an overview of algorithms and applications of deep unfolding for bootstrapped—regardless whether near, middle, or far zones—phase retrieval.

Phase retrieval is an inverse problem that consists in estimating a scene from diffraction intensities. This problem appears in optical imaging, which has three main diffraction zones where the measurements can be acquired, i.e., near, middle and far. Recent works have theoretically solved this inverse problem for the far zone, creating redundancy in the measurement process by including a coded aperture, which allows to modulate the scene and acquire coded diffraction patterns (CDP). However, in the state-of-the-art, the PR problem has not been theoretically studied for CDP at the near and middle zones. Moreover, the structure of the coded aperture is selected at random, leading to suboptimal estimations. Indeed, some of the coding elements employed in the literature are unfeasible because they increase the power of the scene in the modulation process. This paper provides theoretical guarantees for the recovery of a scene from CDP acquired at the three diffraction zones using admissible modulations. Based on the theoretical results, it is established that the image reconstruction quality directly depends on the coded aperture structure; therefore, a design strategy is proposed. In fact, when the scene can be sparsely represented in some basis, its support can be better estimated for a suitable choice of the coding elements in the modulation process. Experimental results show that the scene is successfully recovered by using designed coded apertures with up to 40% less measurements compared to non-designed ensembles.

The three-dimensional (3D) shape reconstruction problem of an object is a task of high interest in autonomous vehicles, detection of moving objects, and precision agriculture. A common methodology to recover the 3D shape of an object is using its optical phase. However, this approach involves solving a non-convex computationally demanding inverse problem known as phase retrieval (PR) in a setup that records coded diffraction patterns (CDP). Usually, the acquisition of several snapshots from the scene is required to solve the PR problem. This work proposes a single-shot 3D shape estimation technique using the optical phase of the object from CDP. The presented approach consists on accurately estimating the optical phase of the object by low-passfiltering the leading eigenvector of a carefully constructed matrix. Then, the estimated phase is used to infer the 3D object shape. It is important to mention that the estimation procedure does not involve a full time demanding reconstruction of the objects. Numerical results on synthetic data demonstrate that the proposed methodology closely estimates the 3D surface of an object with a normalized Mean-Square-Error of up to 0.27, under both noiseless and noisy scenarios. Additionally, the proposed method requires up to 60% less measurements to accurately estimate the 3D surface compared to a state-of-the-art methodology.

A transmission matrix describes the input-output relationship of a complex wavefront as it passes through/reflects off a multiple-scattering medium, such as frosted glass or a painted wall. Knowing a medium's transmission matrix enables one to image through the medium, send signals through the medium, or even use the medium as a lens. The double phase retrieval method is a recently proposed technique to learn a medium's transmission matrix that avoids difficult-to-capture interferometric measurements. Unfortunately, to perform high resolution imaging, existing double phase retrieval methods require (1) a large number of measurements and (2) an unreasonable amount of computation. In this work we focus on the latter of these two problems and reduce computation times with two distinct methods: First, we develop a new phase retrieval algorithm that is significantly faster than existing methods, especially when used with an amplitude-only spatial light modulator (SLM). Second, we calibrate the system using a phase-only SLM, rather than an amplitude-only SLM which was used in previous double phase retrieval experiments. This seemingly trivial change enables us to use a far faster class of phase retrieval algorithms. As a result of these advances, we achieve a 100x reduction in computation times, thereby allowing us to image through scattering media at state-of-the-art resolutions. In addition to these advances, we also release the first publicly available transmission matrix dataset. This contribution will enable phase retrieval researchers to apply their algorithms to real data. Of particular interest to this community, our measurement vectors are naturally i.i.d. subgaussian, i.e., no coded diffraction pattern is required.

Knowledge of the structure of vitreous floaters is crucial to evaluate the need for surgical removal of these floaters. We simulated the phase retrieval of microstructures simulating vitreous floaters by an algorithm PhaseLift and investigate the effects of various parameters on the retrieved phase. The object under test was modulated and the coded diffraction patterns were calculated. Next, PhaseLift was used to retrieve the phase. In the current study, we simulate the effect of Gaussian and Poison noise on the phase retrieval of pure phase objects. We apply an iterative algorithm PhaseLift for phase retrieval as this algorithm requires a very few modulating masks and is able to retrieve the phase of an object from very sparse data. Both types of noise are added to the intensity measurements and their effect on the retrieved phase is assessed in terms of the root-mean-square error. We conclude that Poisson noise compromises the accuracy of phase retrieval more compared to Gaussian noise given that the levels of both types of noise are equal. We also conclude that the thermal noise having nearly Gaussian distribution is of less importance compared to the shot noise having Poisson distribution and dominating at low-light levels.

In the current study, we investigate the effect of uniform white noise, Poisson noise and a constant background on the phase retrieval of pure phase objects. We also study the influence of the aforementioned factors on phase retrieval at different bit depths of intensity measurements. An algorithm called PhaseLift is used for phase retrieval as it requires a small number of modulating masks and can retrieve the phase of an object from sparse intensity measurements of low bit depth. A test object is modulated by eight random masks generated from a single mask and the phase of the object is retrieved from coded diffraction patterns. Different levels of uniform white noise, Poisson noise and constant background are superimposed on the diffraction patterns and the root-mean-square error (RMSE) of the retrieved object is calculated at each level. The results suggest that Poisson noise and a constant background at the same level cause similar RMSE compared to uniform white noise. Lowering the bit depth from 18-bits to 14-bits resulted in the decrease of the RMSE caused by Poisson noise and a constant background. We conclude that the effects of noise and constant background can be reduced by lowering the bit depth.

Fourier phase retrieval with a single mask by Douglas-Rachford algorithms.

Phase retrieval (PR) consists of recovering the phase information from captured intensity measurements, known as coded diffraction patterns (CDPs). Non-convex algorithms for addressing the PR problem require a proper initialization that is refined through a gradient descent approach. These PR algorithms have proven to be robust for different scenarios. Despite deep models showing surprising results in this area, these approaches lack interpretability in their neural architectures. This work proposes unrolling the initialization and iterative reconstruction algorithm for the PR problem using the near-field model based on a non-convex formulation; resulting in an interpretable deep neural network (DNN) that can be trained in an end-to-end (E2E) manner. Furthermore, the proposed method can jointly optimize the phase mask for the CDP acquisition and the DNN parameters. Simulation results demonstrate that the proposed E2E method provides high-quality reconstruction using a learned phase mask from a single projection. Also, the proposed method is tested over an experimental optical setup that incorporates the learned phase mask via an only-phase spatial light modulator.

Coded diffraction imaging via double sparse regularization model

In recent years, phase retrieval methods recovering the phase of an object from coded diffraction patterns have gained popularity. A numerical phase retrieval method called PhaseLift that recovers the phase of an object from a very limited number of coded diffraction patterns was recently proposed. Performance of PhaseLift has been analyzed for different types and the number of masks modulating an object. We present a unique application of PhaseLift that uses four rotations of a single mask, modulating only the amplitude of an object. In simulations, a phase screen with the root-mean-square (RMS) value 0.294 μm was used as the test object. The RMS value of the retrieved phase screen after smoothing was 0.257 μm. In experiments, the RMS value of a wavefront measured with a Shack–Hartmann wavefront sensor was 0.094 while that of the retrieved wavefront after smoothing was 0.054 μm. While PhaseLift is able to recover a wavefront using this kind of modulation, a serious limitation to applicability of this method is its high computational cost and time.