Phase Retrieval Research Articles

Advanced Phase-Retrieval for Stepping-Free X-Ray Dark-Field Computed Tomography.

Grating-based phase- and dark-field-contrast X-ray imaging is a novel technology that aims to extend conventional attenuation-based X-ray imaging by unlocking two additional contrast modalities. The so called phase-contrast and dark-field channels provide enhanced soft tissue contrast and additional microstructural information. Accessing this additional information comes at the expense of a more intricate measurement setup and necessitates sophisticated data processing. A big challenge for translating grating-based dark-field computed tomography to medical applications lies in minimizing the data acquisition time. While a continuously moving detector is ideal, it prohibits conventional phase stepping techniques that require multiple projections under the same angle with different grating positions. One solution to this problem is the so-called sliding window processing approach that is compatible with continuous data acquisition. However, conventional sliding window techniques lead to crosstalk-artifacts between the three image channels, if the projection of the sample moves too fast on the detector within a processing window. In this work we introduce a new interpretation of the phase retrieval problem for continuous acquisitions as a demodulation problem. In this interpretation, we identify the origin of the crosstalk-artifacts as partially overlapping modulation side bands. Furthermore, we present three algorithmic extensions that improve the conventional sliding-window-based phase retrieval and mitigate crosstalk-artifacts. The presented algorithms are tested in a simulation study and on experimental data from a human-scale dark-field CT prototype. In both cases they achieve a substantial reduction of the occurring crosstalk-artifacts.

Blind deep-learning based preprocessing method for Fourier ptychographic microscopy

Fourier ptychographic microscopy (FPM) is a technique for tackling the trade-off between the resolution and the imaging field of view by combining the techniques from aperture synthesis and phase retrieval to estimate the complex object from a series of low-resolution intensity images captured under angle-varied illumination. The captured images are commonly corrupted by multiple noise, leading to the degradation of the reconstructed image quality. Typically speaking, the noise model and noise level of the experimental images are unknown, and the traditional image denoising methods have limited effect. In this paper we model the FPM forward imaging process corrupted by noise and divide the noise in the captured images into two parts: the signal-dependent part and the signal-independent part. Based on the noise model we propose a novel blind deep-learning based Fourier ptychographic microscopy preprocessing method, termed BDFP, for removing these two components of noise. First, from a portion of the captured low-resolution images, a set of blocks corresponding to the smooth area of the object are extracted to model signal-independent noise. Second, under the assumption that the signal-dependent noise follows a Poisson distribution, we add Poisson noise and signal-independent noise blocks to clean images to form a paired training dataset, which is then used for training a deep convolutional neural network (CNN) model to reduce both signal-dependent noise and signal-independent noise. The proposed blind preprocessing method, combining with typical FPM reconstruction algorithms, is tested on simulated data and experimental images. Experimental results show that our preprocessing method can significantly reduce the noise in the captured images and bring about effective improvements in reconstructed image quality.

Stable phase retrieval and perturbations of frames

A frame ( x j ) j ∈ J (x_j)_{j\in J} for a Hilbert space H H is said to do phase retrieval if for all distinct vectors x , y ∈ H x,y\in H the magnitudes of the frame coefficients ( | ⟨ x , x j ⟩ | ) j ∈ J (|\langle x, x_j\rangle |)_{j\in J} and ( | ⟨ y , x j ⟩ | ) j ∈ J (|\langle y, x_j\rangle |)_{j\in J} distinguish x x from y y (up to a unimodular scalar). A frame which does phase retrieval is said to do C C -stable phase retrieval if the recovery of any vector x ∈ H x\in H from the magnitude of the frame coefficients is C C -Lipschitz. It is known that if a frame does stable phase retrieval then any sufficiently small perturbation of the frame vectors will do stable phase retrieval, though with a slightly worse stability constant. We provide new quantitative bounds on how the stability constant for phase retrieval is affected by a small perturbation of the frame vectors. These bounds are significant in that they are independent of the dimension of the Hilbert space and the number of vectors in the frame.

Water layer and radiation damage effects on the orientation recovery of proteins in single-particle imaging at an X-ray free-electron laser

The noise caused by sample heterogeneity (including sample solvent) has been identified as one of the determinant factors for a successful X-ray single-particle imaging experiment. It influences both the radiation damage process that occurs during illumination as well as the scattering patterns captured by the detector. Here, we investigate the impact of water layer thickness and radiation damage on orientation recovery from diffraction patterns of the nitrogenase iron protein. Orientation recovery is a critical step for single-particle imaging. It enables to sort a set of diffraction patterns scattered by identical particles placed at unknown orientations and assemble them into a 3D reciprocal space volume. The recovery quality is characterized by a “disconcurrence” metric. Our results show that while a water layer mitigates protein damage, the noise generated by the scattering from it can introduce challenges for orientation recovery and is anticipated to cause problems in the phase retrieval process to extract the desired protein structure. Compared to these disadvantageous effects due to the thick water layer, the effects of radiation damage on the orientation recovery are relatively small. Therefore, minimizing the amount of residual sample solvent should be considered a crucial step in improving the fidelity and resolution of X-ray single-particle imaging experiments.

Imaging of excised cochleae by micro-CT: staining, liquid embedding, and image modalities.

Assessing the complex three-dimensional (3D) structure of the cochlea is crucial to understanding the fundamental aspects of signal transduction in the inner ear and is a prerequisite for the development of novel cochlear implants. X-ray phase-contrast computed tomography offers destruction-free 3D imaging with little sample preparation, thus preserving the delicate structure of the cochlea. The use of heavy metal stains enables higher contrast and resolution and facilitates segmentation of the cochlea. For μ-CT of small animal and human cochlea, we explore the heavy metal osmium tetroxide (OTO) as a radiocontrast agent and delineate laboratory from synchrotron CT. We investigate how phase retrieval can be used to improve the image quality of the reconstructions, both for stained and unstained specimens. Image contrast for soft tissue in an aqueous solution is insufficient under the in-house conditions, whereas the OTO stain increases contrast for lipid-rich tissue components, such as the myelin sheaths in nervous tissue, enabling contrast-based rendering of the different components of the auditory nervous system. The overall morphology of the cochlea with the three scalae and membranes is very well represented. Further, the image quality of the reconstructions improves significantly when a phase retrieval scheme is used, which is also suitable for non-ideal laboratory settings. With highly brilliant synchrotron radiation (SR), we achieve high contrast for unstained whole cochleae at the cellular level. The OTO stain is suitable for 3D imaging of small animal and human cochlea with laboratory , and relevant pathologies, such as a loss of sensory cells and neurons, can be visualized. With SR and optimized phase retrieval, the cellular level can be reached even for unstained samples in aqueous solution, as demonstrated by the high visibility of single hair cells and spiral ganglion neurons.

Long-range Fourier ptychographic imaging of the dynamic object with a single camera.

Fourier ptychographic imaging technology is a new imaging method proposed in recent years. This technology captures multiple low-resolution images, and synthesizes them into a high-resolution image in the Fourier domain by a phase retrieval algorithm, breaking through the diffraction limit of the lens. In the field of macroscopic Fourier ptychographic imaging, most of the existing research generally focus on high-resolution imaging of static objects, and applying Fourier ptychographic imaging technology to dynamic objects is a hot research area now. At present, most of the researches are to use camera arrays combined with multiplexed lighting, deep learning or other algorithms, but the implementation of these methods is complicated or costly. Based on the diffraction theory of Fourier optics, this paper proposes that by expanding and focusing the illumination area, we can apply Fourier ptychographic imaging technology with a single camera to moving objects within a certain range. Theoretical analysis and experiments prove the feasibility of the proposed method. We successfully achieve high-resolution imaging of the dynamic object, increasing the resolution by about 2.5 times. This paper also researches the impact of speckles in the illuminated area on imaging results and proposes a processing method to reduce the impact of speckles.

Flexible dynamic quantitative phase imaging based on division of focal plane polarization imaging technique.

This paper proposes a flexible and accurate dynamic quantitative phase imaging (QPI) method using single-shot transport of intensity equation (TIE) phase retrieval achieved by division of focal plane (DoFP) polarization imaging technique. By exploiting the polarization property of the liquid crystal spatial light modulator (LC-SLM), two intensity images of different defocus distances contained in orthogonal polarization directions can be generated simultaneously. Then, with the help of the DoFP polarization imaging, these images can be captured with single exposure, enabling accurate dynamic QPI by solving the TIE. In addition, our approach gains great flexibility in defocus distance adjustment by adjusting the pattern loaded on the LC-SLM. Experiments on microlens array, phase plate, and living human gastric cancer cells demonstrate the accuracy, flexibility, and dynamic measurement performance for various objects. The proposed method provides a simple, flexible, and accurate approach for real-time QPI without sacrificing the field of view.

Bivariate retrieval from intensity of cross-correlation

Pulse characterization in ultra-fast optics presents a powerful motivation to study phase retrieval problems of high order. Frequency- and time-resolved techniques for pulse characterization both construct measurements that depend on the intensity of the cross-correlation between two unknown signals undergoing known modulations. The problem of recovering these signals has been traditionally studied and solved with alternating minimization, but recently Wirtinger gradient techniques were demonstrated to invert frequency-resolved measurements on a symmetric signal pair (Pinilla et al., 2019). In this paper, we construct a generalized Wirtinger gradient and Hessian to solve a wide breadth of problems including signal recovery from time- and frequency-resolved measurements. We further demonstrate that both measurement paradigms are special cases of low-rank phase retrieval but with a special structure that disrupts spectral initializers. To combat this problem, we present a tensor-based iterative hard thresholding initializer that, when paired with a Wirtinger gradient descent, is capable of recovering unknown signals with fewer measurements than matrix-based alternating minimization or spectral initialization methods. Finally, we employ Wirtinger gradient descent to recover signals from real-world DSCAN (Wilhelm et al., 2021) measurements and compare results with the existing state-of-the-art.

Online dynamic flat-field correction for MHz microscopy data at European XFEL.

The high pulse intensity and repetition rate of the European X-ray Free-Electron Laser (EuXFEL) provide superior temporal resolution compared with other X-ray sources. In combination with MHz X-ray microscopy techniques, it offers a unique opportunity to achieve superior contrast and spatial resolution in applications demanding high temporal resolution. In both live visualization and offline data analysis for microscopy experiments, baseline normalization is essential for further processing steps such as phase retrieval and modal decomposition. In addition, access to normalized projections during data acquisition can play an important role in decision-making and improve the quality of the data. However, the stochastic nature of X-ray free-electron laser sources hinders the use of standard flat-field normalization methods during MHz X-ray microscopy experiments. Here, an online (i.e. near real-time) dynamic flat-field correction method based on principal component analysis of dynamically evolving flat-field images is presented. The method is used for the normalization of individual X-ray projections and has been implemented as a near real-time analysis tool at the Single Particles, Clusters, and Biomolecules and Serial Femtosecond Crystallography (SPB/SFX) instrument of EuXFEL.

An InSAR Deformation Phase Retrieval Method Combined with Reference Phase in Mining Areas

The acquisition of precise deformation data, including the entirety of the subsidence basin resulting from subterranean mining operations, assumes critical significance in the context of surface impairment monitoring during the course of mining activities. In light of the constraints associated with InSAR technology when applied to the surveillance of expansive deformation gradient mining regions, an innovative approach is advanced herein for InSAR deformation phase retrieval. This approach integrates a reference phase, derivable through a variety of means, including pre-existing models or measurements. Initially, the reference deformation phase is subjected to subtraction from the wrapped InSAR deformation phase, culminating in the derivation of the wrapped phase indicative of the residual phase. Notably, it is posited that the fringe density characterizing the wrapped phase of the residual phase is theoretically diminished in comparison to that of the InSAR wrapped phase. This reduction in complexity in phase unwrapping ensues as a direct consequence. Subsequent to this, the phase retrieval process is effectuated through the summation of the reference phase and the unwrapped phase pertaining to the residual phase. The study harnesses Sentinel-1A and ALOS PALSAR-2 data, employing the PIM-predicted outcomes and GNSS-RTK monitoring outcomes as reference phases for the execution of phase retrieval experiments in two designated study areas. The computation of subsidence is subsequently realized through the combination of the displacement vector depression angle model and the retrieved phase, with the accuracy thereof corroborated through the utilization of leveling data. The experimental findings underscore the efficacy of the reference phase retrieval methodology in securing a more precise deformation phase characterization within expansive deformation gradient mining regions, thereby demonstrating the suitability of this methodological approach.

Dual-view transport of intensity phase imaging flow cytometry.

In this work, we design multi-parameter phase imaging flow cytometry based on dual-view transport of intensity (MPFC), which integrates phase imaging and microfluidics to a microscope, to obtain single-shot quantitative phase imaging on cells flowing in the microfluidic channel. The MPFC system has been proven with simple configuration, accurate phase retrieval, high imaging contrast, and real-time imaging and has been successfully employed not only in imaging, recognizing, and analyzing the flowing cells even with high-flowing velocities but also in tracking cell motilities, including rotation and binary rotation. Current results suggest that our proposed MPFC provides an effective tool for imaging and analyzing cells in microfluidics and can be potentially used in both fundamental and clinical studies.

Compressive phase retrieval: Optimal sample complexity with deep generative priors

AbstractAdvances in compressive sensing (CS) provided reconstruction algorithms of sparse signals from linear measurements with optimal sample complexity, but natural extensions of this methodology to nonlinear inverse problems have been met with potentially fundamental sample complexity bottlenecks. In particular, tractable algorithms for compressive phase retrieval with sparsity priors have not been able to achieve optimal sample complexity. This has created an open problem in compressive phase retrieval: under generic, phaseless linear measurements, are there tractable reconstruction algorithms that succeed with optimal sample complexity? Meanwhile, progress in machine learning has led to the development of new data‐driven signal priors in the form of generative models, which can outperform sparsity priors with significantly fewer measurements. In this work, we resolve the open problem in compressive phase retrieval and demonstrate that generative priors can lead to a fundamental advance by permitting optimal sample complexity by a tractable algorithm. We additionally provide empirics showing that exploiting generative priors in phase retrieval can significantly outperform sparsity priors. These results provide support for generative priors as a new paradigm for signal recovery in a variety of contexts, both empirically and theoretically. The strengths of this paradigm are that (1) generative priors can represent some classes of natural signals more concisely than sparsity priors, (2) generative priors allow for direct optimization over the natural signal manifold, which is intractable under sparsity priors, and (3) the resulting non‐convex optimization problems with generative priors can admit benign optimization landscapes at optimal sample complexity, perhaps surprisingly, even in cases of nonlinear measurements.

Optical phase retrieving of a projected object by employing a differentiation of a single pattern of two-beam interference

In this work, we present a new approach to retrieve the optical phase map of an object which is projected by a single differentiated two-beam interference pattern. This approach is based on the differentiation of the intensity equation of the two-beam interference with respect to the carrier’s phase angle. Therefore, two interference patterns which are shifted by a very small phase angle can be obtained. Then, these two patterns are projected on the object. By exploiting the definition of the mathematical differentiation, the optical phase object’s variations are retrieved from the recorded intensity distributions of both projected patterns. According to this method, the extracted optical phase angles are raised as an inverse “sin” function. This means that the unwrapping process of this function limits the recovered phase angles between − π/2 and π/2. So, the unwrapping process of these unusual wrapped phase angles is explained. The proposed method is applied on (a) two objects which are simulated by combinations of multiple Gaussian functions and (b) a 3D real object. It is found that the inclination of the projected interference pattern on the object redistributes the intensity distribution due to the Lamber’s “cos” aw of illumination. This effect is considered in the retrieving process of the object’s phase map. The limitations of the presented method are discussed and the obtained results are found promising.

A novel phase-shifting segmented phase coding method for absolute phase retrieval

The phase-coding method is widely used during 3D measurement for attaining absolute phase retrieval. However, traditional the phase-coding method still suffers from low accuracy and insufficient reliability. To address these issues, a phase-shifting segmented phase coding method was proposed in this work for absolute phase retrieval in complex object measurements, which specifically improved upon the segmented method of phase coding. Coding patterns with different phase-shifting were obtained by embedding the codeword into the phase-shifting phase. These patterns were also combined and arranged to create novel segmented coding patterns. In addition, this segmented encoding can be self-connected based on its different phase-shifting information to obtain the complete fringe order, achieving thus absolute phase retrieval of over 36 codewords. To solve the problem of the periodic misalignment of the wrapped phase and fringe order, the complementary phase coding method was encoded by the bidirectional coding method, and the separation method of the bidirectional coding was improved. From experimental results, it was demonstrated that the proposed method has high measurement accuracy and reliability.

Low-cost printed holography for the generation ofstructured light.

Recently, the study of structured light fields has attracted great interest, which includes their generation and characterization techniques, as well as their application. Most of these techniques rely on the use of expensive devices, such as liquid crystal spatial light modulators or digital micromirror devices that also require specialized knowledge and software. In this work, we present a scheme for producing low-cost amplitude holograms for the generation of structured light fields. We demonstrate the feasibility of this technique by creating a variety of paraxial modes, such as the well-known Laguerre-Gaussian and Hermite-Gaussian beams. We also demonstrate the potential of our technique in solving the phase retrieval problem to generate 2D and 3D holographic images of objects. Finally, we compare our proposal with the typical generation techniques using digital micromirror devices. Our proposal will pave the path for the generation of structured light beams in more affordable ways for the application in undergrad laboratories.

Improving the stability of phase retrieval algorithm in speckle autocorrelation imaging by Huber penalty sparsity constraint

One of the most widely used methods for reconstructing images through a scattering medium is the Hybrid Input Output (HIO) phase retrieval algorithm. However, the stability and noise robustness of the HIO method are poor, as they are affected by initial values and noise during the imaging process. This paper presents an optimized HIO phase retrieval algorithm that utilizes the Huber penalty as a sparsity constraint to effectively enhance the quality and stability of the reconstruction algorithm. The introduced constraint is designed to guide the iteration process away from local optimal solutions, ensuring superior recovery images even with random initial values. Additionally, an adaptive step size parameter is incorporated to control the iterations and convergence steps, resulting in stronger robustness against noise effects. Moreover, the error criterion for image restoration is defined as the sum of the Fourier transform amplitude of the gradient. Experimental results demonstrate that the proposed phase retrieval algorithm achieves superior quality in the recovered images, while significantly improving noise immunity and stability for randomly selected initial values.

Encoded-image-based authentication utilizing diffractive-imaging scheme and secret-key-assisted phase retrieval

Traditional optical encryption and authentication systems required complex holographic setups to capture complex-valued wavefronts. To address this limitation and create more compact architectures, there is growing interest in combining phase retrieval algorithms (PRAs) with optical authentication. However, most of the existing authentication systems based on PRAs heavily rely on random phase masks for encryption, which raises security concerns. In this paper, a secure method for encoded-image-based authentication is proposed, utilizing a diffractive-imaging scheme and secret-key-assisted phase retrieval. The method employs a simplified optical diffractive-imaging scheme to encrypt an input secret image and an aperture, resulting in the generation of a noise-like image for authentication. Both the encoded image for authentication and the aperture key are essential for decryption and authentication. The decryption process employs a secret-key-assisted phase retrieval algorithm within the framework of diffractive-imaging-based encryption. This work provides an alternative approach to authenticate users by utilizing a noise-like image and an aperture key. By providing an enlarged key space and enabling authentication without visual disclosure of the secret image, this method ensures a high level of security. Simulation results validate the feasibility and effectiveness of the proposed method.

Common-path quantitative phase imaging by propagation through a sinusoidal intensity mask

We present a new quantitative approach for phase retrieval that is derived from the analysis of light intensity propagation through a phase object and a sinusoidal intensity mask positioned into the light path. Unlike the Transport of Intensity Equation (TIE), we deduce a new algorithm that results in a simpler expression since it does not include the Laplacian of the phase. In practice, the method is implemented in a common path optics where each optical component of the assembly remains fixed and phase unwrapping is not required. We introduce the analytical expressions for this algorithm and show experimental results including the phase retrieval of a phase object and of cheek cell samples.

Finite alphabet phase retrieval

We consider the finite alphabet phase retrieval problem: recovering a signal whose entries lie in a small alphabet of possible values from its Fourier magnitudes. This problem arises in the celebrated technology of X-ray crystallography to determine the atomic structure of biological molecules. Our main result states that for generic values of the alphabet, two signals have the same Fourier magnitudes if and only if several partitions have the same difference sets. Thus, the finite alphabet phase retrieval problem reduces to the combinatorial problem of determining a signal from those difference sets. Notably, this result holds true when one of the letters of the alphabet is zero, namely, for sparse signals with finite alphabet, which is the situation in X-ray crystallography.

Single-shot blind deconvolution in coherent diffraction imaging with coded aperture

We present a method for single-shot blind deconvolution in coherent diffraction imaging. Coherent diffraction imaging is a technique for non-interferometric quantitative phase imaging without reference light. In our method based on coherent diffraction imaging, a complex amplitude object is illuminated with coherent light, and light from the object is captured through unknown aberrating media and a coded aperture located on the pupil plane to reduce estimated variables on the aberrated pupil function. Both the amplitude and the phase of the object are recovered from the single captured intensity image by a phase retrieval algorithm in which the coded aperture is utilized as a support to estimate the sparse aberrated pupil function. We numerically and experimentally demonstrate the proposed method with complex amplitude objects under severe aberrating conditions. In the experiment, we quantitatively evaluate its performance with ptychography, which is a method for multi-shot coherent diffraction imaging. Our method enables quantitative phase imaging through turbulence by using simple and reference-free optical hardware without any invasive process.