Phase Retrieval Problem Research Articles

Analysis of crystallographic phase retrieval using iterative projection algorithms.

For protein crystals in which more than two thirds of the volume is occupied by solvent, the featureless nature of the solvent region often generates a constraint that is powerful enough to allow direct phasing of X-ray diffraction data. Practical implementation relies on the use of iterative projection algorithms with good global convergence properties to solve the difficult nonconvex phase-retrieval problem. In this paper, some aspects of phase retrieval using iterative projection algorithms are systematically explored, where the diffraction data and density-value distributions in the protein and solvent regions provide thesole constraints. The analysis is based on the addition of random error to thephases of previously determined protein crystal structures, followed by evaluation of the ability to recover the correct phase set as the distance from the solution increases. The properties of the difference-map (DM), relaxed-reflect-reflect (RRR) and relaxed averaged alternating reflectors (RAAR) algorithms are compared. All of these algorithms prove to be effective for crystallographic phase retrieval, and the useful ranges of the adjustable parameter which controls their behavior are established. When these algorithms converge to the solution, the algorithm trajectory becomes stationary; however, the density function continues to fluctuate significantly around its mean position. It is shown that averaging over the algorithm trajectory in the stationary region, following convergence, improves the density estimate, with this procedure outperforming previous approaches for phase or density refinement.

Variable Splitting and Fusing for Image Phase Retrieval.

Phase Retrieval is defined as the recovery of a signal when only the intensity of its Fourier Transform is known. It is a non-linear and non-convex optimization problem with a multitude of applications including X-ray crystallography, microscopy and blind deconvolution. In this study, we address the problem of Phase Retrieval from the perspective of variable splitting and alternating minimization for real signals and seek to develop algorithms with improved convergence properties. An exploration of the underlying geometric relations led to the conceptualization of an algorithmic step aiming to refine the estimate at each iteration via recombination of the separated variables. Following this, a theoretical analysis to study the convergence properties of the proposed method and justify the inclusion of the recombination step was developed. Our experiments showed that the proposed method converges substantially faster compared to other state-of-the-art analytical methods while demonstrating equivalent or superior performance in terms of quality of reconstruction and ability to converge under various setups.

Regularization by denoising diffusion process meets deep relaxation in phase

Fourier phase retrieval is one of the representative inverse problems where a signal needs to be recovered using only the measured magnitude of its Fourier transform. Deep learning-based algorithms for solving Fourier phase retrieval have been widely studied. These methods provide better reconstruction than the conventional algorithms, such as alternating projection approaches and convex relaxation methods. However, it is difficult to recover the phase information of 256 × 256 images accurately, and they often cannot provide fine details and textures. Recently, diffusion models have been used to solve Fourier phase retrieval problems. They offer realistic reconstruction results, but due to the nature of generative models, they often create non-existent features in the actual images. To address these issues, we introduced a novel algorithm inspired by regularization by denoising diffusion, a variational diffusion sampling for reconstructing the images from the measurements. In particular, the optimization problem in the convex relaxation approach for phase retrieval is interpreted as an additional constraint during the variational sampling process to estimate the phase from the given Fourier magnitude measurement. The proposed method stands out by leveraging not only pre-trained diffusion models as image priors but also the classical optimization approach as the regularization. This novel combination ensures not just accurate phase reconstruction, but also performance guarantees. Our experiments demonstrate that the proposed algorithm consistently provides state-of-the-art performance across various datasets of 256 × 256 images. We further showed the effectiveness of the new regularization for the performance gain in the phase estimation.

Phase retrieval and phaseless inverse scattering with background information

Abstract We consider the problem of finding a compactly supported potential in the multidimensional Schrödinger equation from its differential scattering cross section (squared modulus of the scattering amplitude) at fixed energy. In the Born approximation this problem simplifies to
the phase retrieval problem of reconstructing the potential from the absolute value of its Fourier transform on a ball. To compensate for the missing phase information we use the method of a priori known background scatterers. In particular, we propose an iterative scheme for finding the potential from measurements of a single differential scattering cross section corresponding to the sum of the unknown potential and a known background potential, which is sufficiently disjoint. If this condition is relaxed, then we give similar results for finding the potential from additional monochromatic measurements of the differential scattering cross section of the unknown potential
without the background potential. The performance of the proposed algorithms is demonstrated in numerical examples. In the present work we significantly advance theoretically and numerically studies of [Agaltsov, Hohage, Novikov, Inverse Problems 35, 24001, 2019] and [Novikov, Sivkin, Inverse Problems, 37(5), 2021].

Arithmetic progressions and holomorphic phase retrieval

AbstractWe study the determination of a holomorphic function from its absolute value. Given a parameter , we derive the following characterization of uniqueness in terms of rigidity of a set : if is a vector space of entire functions containing all exponentials , then every is uniquely determined up to a unimodular phase factor by if and only if is not contained in an arithmetic progression . Leveraging this insight, we establish a series of consequences for Gabor phase retrieval and Pauli‐type uniqueness problems. For instance, is a uniqueness set for the Gabor phase retrieval problem in , provided that is a suitable perturbation of the integers.

Atomic resolution coherent x-ray imaging with physics-based phase retrieval

Coherent x-ray imaging and scattering from accelerator based sources such as synchrotrons continue to impact biology, medicine, technology, and materials science. Many synchrotrons around the world are currently undergoing major upgrades to increase their available coherent x-ray flux by approximately two orders of magnitude. The improvement of synchrotrons may enable imaging of materials in operando at the atomic scale which may revolutionize battery and catalysis technologies. Current algorithms used for phase retrieval in coherent x-ray imaging are based on the projection onto sets method. These traditional iterative phase retrieval methods will become more computationally expensive as they push towards atomic resolution and may struggle to converge. Additionally, these methods do not incorporate physical information that may additionally constrain the solution. In this work, we present an algorithm which incorporates molecular dynamics into Bragg coherent diffraction imaging (BCDI). This algorithm, which we call PRAMMol (Phase Retrieval with Atomic Modeling and Molecular Dynamics) combines statistical techniques with molecular dynamics to solve the phase retrieval problem. We present several examples where our algorithm is applied to simulated coherent diffraction from 3D crystals and show convergence to the correct solution at the atomic scale.

No existence of a linear algorithm for the one-dimensional Fourier phase retrieval

Fourier phase retrieval, which aims to reconstruct a signal from its Fourier magnitude, is of fundamental importance in fields of engineering and science. In this paper, we provide a theoretical understanding of algorithms for the one-dimensional Fourier phase retrieval problem. Specifically, we demonstrate that if an algorithm exists which can reconstruct an arbitrary signal x∈CN in Poly(N)log⁡(1/ϵ) time to reach ϵ-precision from its magnitude of discrete Fourier transform and its initial value x(0), then P=NP. This partially elucidates the phenomenon that, despite the fact that almost all signals are uniquely determined by their Fourier magnitude and the absolute value of their initial value |x(0)|, no algorithm with theoretical guarantees has been proposed in the last few decades. Our proofs employ the result in computational complexity theory that the Product Partition problem is NP-complete in the strong sense.

Generalized phase retrieval in quaternion Euclidean spaces

Recently, quaternionic Fourier analysis has received increasing attention due to its applications in signal analysis and image processing. This paper addresses quaternionic generalized phase retrieval (QGPR) problem in quaternion Euclidean spaces . We introduce the concept of QGPR which aims to reconstruct a signal in from the quadratic measurements , where each is an self‐adjoint quaternion matrix. We characterize QGPR sequences in terms of their real Jacobian matrices, prove that the set of QGPR sequences is an open set in some sense, and present some phaselift‐based sufficient conditions on QGPR which gives a method to construct QGPR sequences.

Phase retrieval in incoherent diffractive imaging using higher-order photon correlation functions

To obtain spatial information about an arbitrary atomic distribution in x-ray structure analysis, e.g. in molecules or proteins, the standard method is to measure the intensity in the far field, i.e. the first-order photon correlation function of the coherently scattered x-ray photons (coherent diffractive imaging). Recently, it was suggested to record alternatively the incoherently scattered photons and measure the second-order photon correlation function to reconstruct the geometry of the unknown atomic distribution (incoherent diffractive imaging). Yet, besides various advantages of the latter method, both techniques suffer from the so-called phase retrieval problem. Lately, an ab-initio phase retrieval algorithm to reconstruct the phase of the so-called structure factor of the scattering objects based on the third-order photon correlation function was reported. The algorithm makes use of the closure phase, which contains important, yet incomplete phase information, well-known from triple correlations and their bispectrum in speckle masking and astronomy applications. Here, we provide a detailed analysis of the underlying scheme and quantities in the context of x-ray structure analysis. In particular, we explicitly calculate for the first time the third-order photon correlation function for single photon emitters in a full quantum mechanical treatment and discuss the uniqueness of the closure phase equations constructed from. In this context, we recapitulate the sign problem of the closure phase and how it can be lifted using redundant information. We further show how the algorithm can be improved using even higher-order photon correlation functions produced by single photon emitters, e.g. the fourth-order correlation function, delivering new phase relations appearing in the four-point correlations.

Uniqueness of phase retrieval with short-time linear canonical transform

In this paper, we study the problem of phase retrieval with short-time linear canonical transform (STLCT). The relation between signal, window and their STLCT is provided through Fourier transform. Based on this theorem, a uniqueness result is established for all square integrable functions. For nonseparable real continuous signal, we prove the uniqueness theorems under some weaker conditions. In complex bandlimited and cardinal [Formula: see text]-spline spaces, uniqueness results are provided with magnitude-only STLCT.

Phase retrieval from integrated intensity of auto-convolution

Ultra-fast optical pulses are the most ephemeral sensing paradigm ever devised, examining events over incredibly brief timescales with broadband illumination. A consequence of sensing at timescales lower than a picosecond is that pulse characterization cannot be done with traditional analog-to-digital samplers and must be ascertained from integrating intensity sensors. Techniques for pulse characterization have been constructed using combinations of time-invariant and time-variant filter responses to create non-linear but invertible intensity datasets (Walmsley & Dorrer, 2009). In this paper, we develop a novel high-order phase retrieval technique to perform pulse characterization from a single-pixel integrating sensor measuring integrated intensity of auto-convolution (IIAC). We examine gradient descent’s ability to recover signals as a function of signal dimension and measurement count, and we demonstrate the effective use of iterative hard tensor thresholding as an initializer. Finally, we demonstrate IIAC recovery in a laboratory setting to recover the time profile of a complex laser pulse. We assert that the IIAC recovery solution demonstrated here simultaneously provides the optics community with a pulse characterization technique that scales to low-power microscopy systems and provides the optimization community with a physically motivated high-order phase retrieval problem enhanced by low-rank tensor processing.

Phaseless Sampling on Square-Root Lattices

AbstractDue to its appearance in a remarkably wide field of applications, such as audio processing and coherent diffraction imaging, the short-time Fourier transform (STFT) phase retrieval problem has seen a great deal of attention in recent years. A central problem in STFT phase retrieval concerns the question for which window functions $$g \in {L^2({\mathbb R}^d)}$$ g ∈ L 2 ( R d ) and which sampling sets $$\Lambda \subseteq {\mathbb R}^{2d}$$ Λ ⊆ R 2 d is every $$f \in {L^2({\mathbb R}^d)}$$ f ∈ L 2 ( R d ) uniquely determined (up to a global phase factor) by phaseless samples of the form $$\begin{aligned} |V_gf(\Lambda )| = \left\{ |V_gf(\lambda )|: \lambda \in \Lambda \right\} , \end{aligned}$$ | V g f ( Λ ) | = | V g f ( λ ) | : λ ∈ Λ , where $$V_gf$$ V g f denotes the STFT of f with respect to g. The investigation of this question constitutes a key step towards making the problem computationally tractable. However, it deviates from ordinary sampling tasks in a fundamental and subtle manner: recent results demonstrate that uniqueness is unachievable if $$\Lambda $$ Λ is a lattice, i.e $$\Lambda = A{\mathbb Z}^{2d}, A \in \textrm{GL}(2d,{\mathbb R})$$ Λ = A Z 2 d , A ∈ GL ( 2 d , R ) . Driven by this discretization barrier, the present article centers around the initiation of a novel sampling scheme which allows for unique recovery of any square-integrable function via phaseless STFT-sampling. Specifically, we show that square-root lattices, i.e., sets of the form $$\begin{aligned} \Lambda = A \left( \sqrt{{\mathbb Z}} \right) ^{2d}, \ \sqrt{{\mathbb Z}} = \{ \pm \sqrt{n}: n \in {\mathbb N}_0 \}, \end{aligned}$$ Λ = A Z 2 d , Z = { ± n : n ∈ N 0 } , guarantee uniqueness of the STFT phase retrieval problem. The result holds for a large class of window functions, including Gaussians

On a Frank-Wolfe approach for abs-smooth functions

We propose an algorithm which appears to be the first bridge between the fields of conditional gradient methods and abs-smooth optimization. Our problem setting is motivated by various applications that lead to nonsmoothness, such as ℓ 1 regularization, phase retrieval problems, or ReLU activation in machine learning. To handle the nonsmoothness in our problem, we propose a generalization to the traditional Frank-Wolfe gap and prove that first-order minimality is achieved when it vanishes. We derive a convergence rate for our algorithm which is identical to the smooth case. Although our algorithm necessitates the solution of a subproblem which is more challenging than the smooth case, we provide an efficient numerical method for its partial solution, and we identify several applications where our approach fully solves the subproblem. Numerical and theoretical convergence is demonstrated, yielding several conjectures.

A Data Centric Approach Towards Server Time Series Model Compression

Effective model compression plays a pivotal role in mitigating the computational and interpretational challenges inherent in the domain of time series forecasting. In this study, we introduce an innovative data-centric methodology tailored to identify a representative data subset from the entirety of the dataset. This chosen representative segment forms the cornerstone for the training of proficient time series forecasting models. Furthermore, our investigation unveils a compelling outcome of this approach—a substantial reduction in the size of time series forecasting models when trained with this selected representative data segment. This model compression strategy results in a remarkable 56.31% decrease in storage consumption, a discovery of considerable significance for optimizing resources and enhancing scalability in time series forecasting. By distilling the dataset to its fundamental components through our data-centric approach, we aim to enhance both computational efficiency and the interpretability of the resultant models. This paper introduces a pioneering technique to tackle the challenges associated with data volume and model complexity in the field of time series forecasting, offering potential pathways for more efficient and insightful modelling in this domain.

What conjugate phase retrieval complex vectors can do in quaternion Euclidean spaces

Abstract Quaternion algebra ℍ {\mathbb{H}} is a noncommutative associative algebra. In recent years, quaternionic Fourier analysis has received increasing attention due to its applications in signal analysis and image processing. This paper addresses conjugate phase retrieval problem in the quaternion Euclidean space ℍ M {\mathbb{H}^{M}} with M ≥ 2 {M\geq 2} . Write ℂ η = { ξ : ξ = ξ 0 + β ⁢ η , ξ 0 , β ∈ ℝ } {\mathbb{C}_{\eta}=\{\xi:\xi=\xi_{0}+\beta\eta,\,\xi_{0},\,\beta\in\mathbb{R}\}} for η ∈ { i , j , k } {\eta\in\{i,\,j,\,k\}} . We remark that not only ℂ η M {\mathbb{C}_{\eta}^{M}} -vectors cannot allow traditional conjugate phase retrieval in ℍ M {\mathbb{H}^{M}} , but also ℂ i M ∪ ℂ j M {\mathbb{C}_{i}^{M}\cup\mathbb{C}_{j}^{M}} -complex vectors cannot allow phase retrieval in ℍ M {\mathbb{H}^{M}} . We are devoted to conjugate phase retrieval of ℂ i M ∪ ℂ j M {\mathbb{C}_{i}^{M}\cup\mathbb{C}_{j}^{M}} -complex vectors in ℍ M {\mathbb{H}^{M}} , where “conjugate” is not the traditional conjugate. We introduce the notions of conjugation, maximal commutative subset and conjugate phase retrieval. Using the phase lifting techniques, we present some sufficient conditions on complex vectors allowing conjugate phase retrieval. And some examples are also provided to illustrate the generality of our theory.

Dual-constrained physics-enhanced untrained neural network for lensless imaging.

An untrained neural network (UNN) paves a new way to realize lensless imaging from single-frame intensity data. Based on the physics engine, such methods utilize the smoothness property of a convolutional kernel and provide an iterative self-supervised learning framework to release the needs of an end-to-end training scheme with a large dataset. However, the intrinsic overfitting problem of UNN is a challenging issue for stable and robust reconstruction. To address it, we model the phase retrieval problem into a dual-constrained untrained network, in which a phase-amplitude alternating optimization framework is designed to split the intensity-to-phase problem into two tasks: phase and amplitude optimization. In the process of phase optimization, we combine a deep image prior with a total variation prior to retrain the loss function for the phase update. In the process of amplitude optimization, a total variation denoising-based Wirtinger gradient descent method is constructed to form an amplitude constraint. Alternative iterations of the two tasks result in high-performance wavefield reconstruction. Experimental results demonstrate the superiority of our method.

Motion-resolved, reference-free holographic imaging via spatiotemporally regularized inversion

Holography is a powerful technique that records the amplitude and phase of an optical field simultaneously, enabling a variety of applications such as label-free biomedical analysis and coherent diffraction imaging. Holographic recording without a reference wave has been long pursued because it obviates the high experimental requirements of conventional interferometric methods. However, due to the ill-posed nature of the underlying phase retrieval problem, reference-free holographic imaging is faced with an inherent tradeoff between imaging fidelity and temporal resolution. Here, we propose a general computational framework, termed spatiotemporally regularized inversion (STRIVER), to achieve motion-resolved, reference-free holographic imaging with high fidelity. Specifically, STRIVER leverages signal priors in the spatiotemporal domain to jointly eliminate phase ambiguities and motion artifacts, and, when combined with diversity measurement schemes, produces a physically reliable, time-resolved holographic video from a series of intensity-only measurements. We experimentally demonstrate STRIVER in near-field ptychography, where dynamic holographic imaging of freely swimming paramecia is performed at a framerate-limited speed of 112 fps. The proposed method can be potentially extended to other measurement schemes, spectral regimes, and computational imaging modalities, pushing the temporal resolution toward higher limits.

The performance of the amplitude-based model for complex phase retrieval

Abstract This paper aims to study the performance of the amplitude-based model $\widehat{{\boldsymbol x}} \in \mathop{\mathrm{argmin}}\limits _{{\boldsymbol x}\in \mathbb{C}^{d}}\sum _{j=1}^{m}\left (|\langle{\boldsymbol a}_{j},{\boldsymbol x}\rangle |-b_{j}\right )^{2}$, where $b_{j}:=|\langle{\boldsymbol a}_{j},{\boldsymbol x}_{0}\rangle |+\eta _{j}$ and ${\boldsymbol x}_{0}\in \mathbb{C}^{d}$ is a target signal. The model is raised in phase retrieval as well as in absolute value rectification neural networks. Many efficient algorithms have been developed to solve it in the past decades. However, there are very few results available regarding the estimation performance in the complex case under noisy conditions. In this paper, we present a theoretical guarantee on the amplitude-based model for the noisy complex phase retrieval problem. Specifically, we show that $\min _{\theta \in [0,2\pi )}\|\widehat{{\boldsymbol x}}-\exp (\mathrm{i}\theta )\cdot{\boldsymbol x}_{0}\|_{2} \lesssim \frac{\|{\boldsymbol \eta }\|_{2}}{\sqrt{m}}$ holds with high probability provided the measurement vectors ${\boldsymbol a}_{j}\in \mathbb{C}^{d},$ $j=1,\ldots ,m,$ are i.i.d. complex sub-Gaussian random vectors and $m\gtrsim d$. Here ${\boldsymbol \eta }=(\eta _{1},\ldots ,\eta _{m})\in \mathbb{R}^{m}$ is the noise vector without any assumption on the distribution. Furthermore, we prove that the reconstruction error is sharp. For the case where the target signal ${\boldsymbol x}_{0}\in \mathbb{C}^{d}$ is sparse, we establish a similar result for the nonlinear constrained $\ell _{1}$ minimization model. To accomplish this, we leverage a strong version of restricted isometry property for an operator on the space of simultaneous low-rank and sparse matrices.

A dimension-reduced variational approach for solving physics-based inverse problems using generative adversarial network priors and normalizing flows

We propose a novel modular inference approach combining two different generative models — generative adversarial networks (GAN) and normalizing flows — to approximate the posterior distribution of physics-based Bayesian inverse problems framed in high-dimensional ambient spaces. We dub the proposed framework GAN-Flow. The proposed method leverages the intrinsic dimension reduction and superior sample generation capabilities of GANs to define a low-dimensional data-driven prior distribution. Once a trained GAN-prior is available, the inverse problem is solved entirely in the latent space of the GAN using variational Bayesian inference with normalizing flow-based variational distribution, which approximates low-dimensional posterior distribution by transforming realizations from the low-dimensional latent prior (Gaussian) to corresponding realizations of a low-dimensional variational posterior distribution. The trained GAN generator then maps realizations from this approximate posterior distribution in the latent space back to the high-dimensional ambient space. We also propose a two-stage training strategy for GAN-Flow wherein we train the two generative models sequentially. Thereafter, GAN-Flow can estimate the statistics of posterior-predictive quantities of interest at virtually no additional computational cost. The synergy between the two types of generative models allows us to overcome many challenges associated with the application of Bayesian inference to large-scale inverse problems, chief among which are describing an informative prior and sampling from the high-dimensional posterior. GAN-Flow does not involve Markov chain Monte Carlo simulation, making it particularly suitable for solving large-scale inverse problems. We demonstrate the efficacy and flexibility of GAN-Flow on various physics-based inverse problems of varying ambient dimensionality and prior knowledge using different types of GANs and normalizing flows. Notably, one of the applications we consider involves a 65,536-dimensional inverse problem of phase retrieval wherein an object is reconstructed from sparse noisy measurements of the magnitude of its Fourier transform.

Target Localization Based on Distributed Array Networks With Magnitude-Only Measurements

The source localization problem based on distributed array networks is formulated into a group sparsity-based phase retrieval problem where only the magnitude of received signals is available. Under such a framework, a 2-D localization method is proposed, where unlike traditional methods, random phase errors at array sensors will not affect estimation results. In addition, to deal with the off-grid problem for sparsity-based solutions, a model for off-grid bias is proposed and an efficient two-step method is developed accordingly to solve the 2-D off-grid problem. Simulation results show that the proposed solutions can solve the problem effectively.