Phase Retrieval Algorithm Research Articles

Enhancing target recognition rate in atmospheric turbulence using orbital angular momentum spectra of vortex beams

Abstract Traditional methods for extracting and recognizing targets from laser echo signals typically involve complex processing and require extensive data. Vortex beams carry orbital angular momentum (OAM), and upon reflection from a target, the distribution of the OAM spectrum carries features related to the target, thereby enriching the dimensions of target recognition. Using the OAM spectrum simplifies the recognition process but faces challenges like atmospheric turbulence that affect beam transmission and target recognition accuracy. Our study employs the Gerchberg–Saxton phase retrieval (GS) algorithm to mitigate the effects of atmospheric turbulence on the beams. Using OAM spectrum data, we achieved effective target recognition with various shapes under atmospheric turbulence through a back-propagation neural network (BPNN). Simulations revealed a recognition rate increase from 76.25% to 96% post-compensation by the GS algorithm. We also found that the highest recognition rate occurs at a target ratio of 0.2. After compensation with the GS algorithm at a target ratio of 0.1, the recognition rate for each shape increased to 99%. This demonstrates the effectiveness of utilizing the OAM spectrum for recognizing diverse target shapes, with the GS algorithm further improving recognition rates. These findings can be applied to intelligent transportation and robotic vision.

Deep learning phase retrieval in x-ray single-particle imaging for biological macromolecules

Abstract Phase retrieval is an important optimization problem that occurs, for example, in the analysis of coherent diffraction patterns from isolated proteins. All iterative algorithms employed for phase retrieval in this context require some a priori knowledge of the object, usually in the form of a support that describes the extent of the particle. Phase retrieval is a time-consuming task that can often fail, particularly if the support is too loose or of bad quality. This paper presents a neural network that can produce low-resolution estimates of the phased object in a fraction of the time it takes for a full phase retrieval. It can also successfully be used as support for further analysis. Our network is trained on simulated data from biological macromolecules and is thus tailored to the type of data seen in a typical CDI experiment. Other approaches to support finding require very accurate data without missing regions or the full phase-retrieval algorithm to be run for a long time. Our network could speed up offline analysis and provide real-time feedback during data collection.

Secure authentication and encryption via diffraction imaging-based encoding and vector decomposition

Abstract Traditional optical encryption systems have security risks due to their linearity and usually encounter problems such as the heavy burden of key transmission and storage. This paper proposes a novel security-enhanced optical image authentication and encryption framework that combines diffractive imaging-based encryption with the vector decomposition algorithm (VDA). Chaotic random phase masks (CRPMs) are used to encrypt data for authentication via VDA, and a pair of complementary binary matrix keys are utilized to extract information from the encrypted data to generate ciphertext. During the authentication and decryption processes, a sparse reference image is reconstructed from the ciphertext for verification. If the authentication is successful, image decryption can be executed using a key-assisted phase retrieval algorithm. The employment of nonlinear VDA, an additional layer of authentication, and the use of CRPMs and binary matrix keys enhance security and address key burden concerns. Simulation results demonstrate the feasibility, effectiveness, and security of the scheme.

Adaptive support constraint based on pixel difference distribution for twin-image removal in in-line holography

A tight support domain can enhance the reconstruction effect of phase retrieval algorithms using a support domain constraint. Hence, an adaptive support method based on pixel difference distribution to calculate a more accurate support domain is proposed in this paper. The support domain is continuously updated at each iteration of the phase retrieval algorithm, resulting in a support domain that closely corresponds to the shape and size of the object. Simulations are conducted to investigate the reconstruction effect of complex-valued objects as well as the influence of varying the binarization threshold and the difference window size on the convergence performance in the absence of noise. The changing dynamics of the support domain during the iterative process are also showcased. The experiments demonstrate that the adaptive support method can improve the reconstruction image quality and image contrast under actual conditions. Both simulations and experiments affirm the feasibility of this approach.

Quantitative phase imaging with two in-line holograms.

Quantitative phase imaging (QPI) has emerged as a practical technique for acquiring structural information from phase objects. Digital holography can realize phase detection, but it is limited by a spatial bandwidth product or affected by the overlap of conjugate images. The phase retrieval algorithm serves as an effective tool for QPI dealing with intensity patterns. Traditional phase retrieval algorithms heavily rely on strong support constraints or high data redundancy to accurately reconstruct the sample image. However, in single-frame phase retrieval algorithms, the precise acquisition of support constraints is notably challenging. The multiple-measurement spends much time on data acquisition and is unsuitable for dynamic sample observation. In this paper, we propose a novel, to the best of our knowledge, quantitative phase imaging method that utilizes only two in-line holograms. We have developed a phase retrieval algorithm based on ptychography, which eliminates twin-image and separates illumination background. The proposed method achieves high data utilization efficiency and can be employed for dynamic imaging.

Two-plus-two fringe projection profilometry based on phase-shifted coding

In fringe projection profilometry based on temporal phase unwrapping, determining a fringe order map commonly requires a large number of fringes. To reduce the fringe number, this paper proposes a concise absolute phase retrieval algorithm just by projecting four fringes. The first two orthogonal fringes with relatively large frequency can collect reliable height information. The second two fringes are designed the same as the first two, but the only difference is that each 2π-phase of them is shifted by a unique amount, which can robustly label a large number of fringe orders. For decoding the fringes, we develop an average intensity one-time extraction algorithm, which allows for the rapid acquisition of the two pairs of alternating current components. From this, the wrapped phase containing height information and the stair-coded phase providing fringe orders can be directly extracted by arctangent operation in a point-to-point manner. Furthermore, we also develop a universal fringe order correction algorithm that can simultaneously correct the common errors and the misalignment between the wrapped phase and fringe orders. Experiment results demonstrate that this method achieves comparable accuracy and adaptability to the phase-coding method, while utilizing two fewer fringes.

Phase recovery from Fresnel incoherent correlation holography using differential Zernike fitting.

Fresnel incoherent correlation holography (FINCH) was created to improve imaging resolution and 3D imaging capabilities using spatially incoherent illumination. The optical setup of a FINCH-based interferometer is closely related to a radial shearing interferometer, which measures the radial phase difference of an input wavefront. By using phase retrieval methodologies from lateral shearing interferometry, namely, differential Zernike fitting (DZF), we show that FINCH-based and radial shearing interferometry can be used for phase retrieval and adaptive optics (AO). In this paper, we describe the phase retrieval algorithm using least squares-based DZF and demonstrate a simple adaptive optics loop with an aberrated point spread function using wave optics simulation. We find that FINCH-based phase retrieval has the advantages of fast phase retrieval measurements, thanks to well-studied least squares-based phase reconstruction methods, improved resolution compared to the Shack-Hartmann-based wavefront sensing, and the simplified optical setup of radial shearing interferometry.

High‐resolution incoherent interference imaging without phase measurement

AbstractAiming for long‐distance, high‐resolution, passive imaging, we use fiber coupling to replace the spatial coupling of the segmented planar imaging detector for electro‐optical reconnaissance imaging system and propose a novel incoherent interference imaging method, which avoids the problem that the resolution of the imaging system is limited by the size of the silicon wafer. The method uses a phase retrieval algorithm and does not need to measure the phase in the system, effectively avoiding the intrinsic jitter problem of fiber optics. The feasibility of the method in achieving target image reconstruction was verified through simulation; furthermore, an indoor two‐dimensional discrete sampling imaging experiment for a simple four‐rod target was conducted, and an outdoor one‐dimensional feature identification experiment for a two‐rod target was also performed. The results showed that the resolution exceeds the diffraction limit and the reconstruction error of target size is less than 3.5%.

Optical information hiding system with pixel-free expansion visual cryptography

Abstract We propose an optical information hiding system with pixel-free expansion visual cryptography (PEVC). In the optical concealment process, initially, a PEVC scheme is utilized to encode the secret image. This approach generates visual keys of the same size as the secret image, addressing the pixel expansion issue inherent in visual cryptography encoding schemes. As a result, it significantly reduces both storage space requirements and the network bandwidth occupied during the transmission process and exhibits a higher hidden capacity. Furthermore, PEVC is combined with the optical phase retrieval algorithm for hiding, embedding the visual keys into the phase keys. In this process, wavelength and diffraction distance are introduced as keys, enhancing the security of the system. The phase keys can be fabricated into diffractive optical elements for physical preservation and transmission in tangible form. Simulation experiments and optical experimental results indicate that the system is applicable in practical scenarios and possesses excellent security and exhibits a higher hidden capacity.

Phase retrieval from a single diffraction intensity pattern by generating the support constraint using deep learning

Phase retrieval is an essential technique for imaging applications in many fields such as biology and materials science. The fundamental challenge lies in reconstructing objects from measured intensity patterns. A prevalent solution involves an iterative strategy that alternates projections between the object and detector planes. However, reconstructing objects from a single diffraction intensity pattern remains challenging due to the absence of constraints at the object plane. Here, we propose a learning-based approach that generates reliable object support in real time from its autocorrelation support. With the generated support as the constraint for the object plane, we show that the convergence of iterative phase retrieval algorithms can be markedly enhanced. Compared with the end-to-end learning-based methods, the proposed method has a unique character that the training features are binary supports, instead of the type and content of objects, so that our approach has better generalization. Furthermore, one of the advantages of our method is that it uses simulated data for training, eliminating the need for experimental data collection.

A Framework for Iterative Phase Retrieval Technique Integration into Atmospheric Adaptive Optics—Part I: Wavefront Sensing in Strong Scintillations

The objective of this study, which is divided into two parts, is twofold: to address long-standing challenges in the sensing of atmospheric turbulence-induced wavefront aberrations under strong scintillation conditions via a comparative analysis of several basic scintillation-resistant wavefront sensing (SR-WFS) architectures and iterative phase retrieval (IPR) techniques (Part I, this paper), and to develop a framework for the potential integration of SR-WFS techniques into practical closed-loop non-astronomical atmospheric adaptive optics (AO) systems (Part II). In this paper, we consider basic SR-WFS mathematical models and phase retrieval algorithms, tradeoffs in sensor design and phase retrieval technique implementation, and methodologies for WFS parameter optimization and performance assessment. The analysis is based on wave-optics numerical simulations imitating realistic turbulence-induced phase aberrations and intensity scintillations, as well as optical field propagation inside the SR-WFSs. Several potential issues important for the practical implementation of SR-WFS and IPR techniques, such as the requirements for phase retrieval computational grid resolution, tolerance with respect to optical element misalignments, and the impact of camera noise and input light non-monochromaticity, are also considered. The results demonstrate that major wavefront sensing requirements desirable for AO operation under strong intensity scintillations can potentially be achieved by transitioning to novel SR-WFS architectures, based on iterative phase retrieval techniques.

Thermal vibrations in the inversion of dynamical electron scattering

Relativistic forward scattering of electrons at finite temperature involves the incoherent superposition of diffraction patterns formed by different snapshots of thermal atomic displacements. In experiments, thermal vibrations lead to thermal diffuse scattering (TDS), partly dominating diffraction patterns of thick specimens. This study sheds light on the effects of TDS on solutions to the inverse scattering problem using combined real- and diffraction-space information acquired in a scanning transmission electron microscope (STEM) to retrieve the object's phase. Using frozen phonon multislice within the Einstein approximation, realistic ground truth data of 20-nm-thick SrTiO3 is generated and subjected to contemporary inverse multislice schemes to retrieve the projected Coulomb potential slicewise. We first classify phase retrieval algorithms as to their assumptions on periodicity along the incident beam direction, as well as pixelwise and parametrized reconstruction methods. It is found that pixelwise object reconstructions are capable of retrieving structural details qualitatively while being prone to contain TDS-related artifacts which can result in unphysical potentials. For pixelwise reconstructions of multiple independent specimen slices, we observe that the origin of TDS, i.e., thermal atomic displacements, starts to emerge naturally. However, the quantitative assessment tends to too small mean squared thermal displacements, also when reconstructing multiple object modes. Using an atomistically parametrized inversion strategy which exploits the explicit separation of thermal vibrations and potentials, temperature and chemistry of the specimen can be retrieved quantitatively with high accuracy. Published by the American Physical Society 2024

Ptychographic phase retrieval via a deep-learning-assisted iterative algorithm.

Ptychography is a powerful computational imaging technique with microscopic imaging capability and adaptability to various specimens. To obtain an imaging result, it requires a phase-retrieval algorithm whose performance directly determines the imaging quality. Recently, deep neural network (DNN)-based phase retrieval has been proposed to improve the imaging quality from the ordinary model-based iterative algorithms. However, the DNN-based methods have some limitations because of the sensitivity to changes in experimental conditions and the difficulty of collecting enough measured specimen images for training the DNN. To overcome these limitations, a ptychographic phase-retrieval algorithm that combines model-based and DNN-based approaches is proposed. This method exploits a DNN-based denoiser to assist an iterative algorithm like ePIE in finding better reconstruction images. This combination of DNN and iterative algorithms allows the measurement model to be explicitly incorporated into the DNN-based approach, improving its robustness to changes in experimental conditions. Furthermore, to circumvent the difficulty of collecting the training data, it is proposed that the DNN-based denoiser be trained without using actual measured specimen images but using a formula-driven supervised approach that systemically generates synthetic images. In experiments using simulation based on a hard X-ray ptychographic measurement system, the imaging capability of the proposed method was evaluated by comparing it with ePIE and rPIE. These results demonstrated that the proposed method was able to reconstruct higher-spatial-resolution images with half the number of iterations required by ePIE and rPIE, even for data with low illumination intensity. Also, the proposed method was shown to be robust to its hyperparameters. In addition, the proposed method was applied to ptychographic datasets of a Simens star chart and ink toner particles measured at SPring-8 BL24XU, which confirmed that it can successfully reconstruct images from measurement scans with a lower overlap ratio of the illumination regions than is required by ePIE and rPIE.

Accurate two-step random phase retrieval approach without pre-filtering based on hyper ellipse fitting

In this work, we propose a hyper ellipse fitting-based high-precision random two-frame phase shifting algorithm to improve the accuracy of phase retrieval. This method includes a process of Gram-Schmidt orthonormalization, followed by a hyper ellipse fitting procedure. The Gram-Schmidt orthonormalization algorithm constructs a quadrature fringe pattern relative to the original fringe pattern. These two quadrature fringe patterns are then fed into the hyper ellipse fitting procedure, which reconstructs the phase map and refines the background light to produce the final accurate phase of interest. Due to the hyper ellipse fitting procedure, the demodulation results are significantly improved in many cases. This method allows us to design a two-shot phase reconstruction algorithm without the need for least squares iteration or pre-filtering, effectively mitigating residual background to the greatest extent. It combines the advantages of both the Gram-Schmidt orthonormalization method and the Lissajous ellipse fitting method, making our hyper ellipse fitting approach a simple, flexible, and accurate phase retrieval algorithm. Experiments show that by using the weighted least squares method and adjusting the weights, this method can prioritize data points with more significant information or higher reliability, ensuring more accurate estimation of the ellipse parameters.

A novel laser interferometric method for thin film thickness measurement in gas–liquid intermittent flow

Abstract In horizontal intermittent flow, the long bubbles move toward the center of the pipe due to inertia, forming the thin liquid film above the long bubbles. Accurate measurement of the liquid film thickness is crucial for heat and mass transfer. In this paper, laser interferometric technology is innovatively introduced to measure the film thickness of the intermittent flow, and the thin liquid film is detected with a resolution of 100 nm. Considering the curvature of the circular pipe wall, which leads to divergent reflected light, the effect of the pipe wall on the interference pattern is explored by the ray tracing technique. A two-dimensional interpolation phase retrieval algorithm based on light intensity is proposed to reconstruct the thickness of the liquid film, and the average error is less than 1.86%. Benefiting from the exceptionally high resolution, research is conducted on the thin liquid film at the top of the horizontal intermittent flow, revealing its dependence on the sub-regimes and gas–liquid velocities.

Speckle tracking phase-contrast computed tomography at an inverse Compton X-ray source.

Speckle-based X-ray imaging (SBI) is a phase-contrast method developed at and for highly coherent X-ray sources, such as synchrotrons, to increase the contrast of weakly absorbing objects. Consequently, it complements the conventional attenuation-based X-ray imaging. Meanwhile, attempts to establish SBI at less coherent laboratory sources have been performed, ranging from liquid metal-jet X-ray sources to microfocus X-ray tubes. However, their lack of coherence results in interference fringes not being resolved. Therefore, algorithms were developed which neglect the interference effects. Here, we demonstrate phase-contrast computed tomography employing SBI in a laboratory-setting with an inverse Compton X-ray source. In this context, we investigate and compare also the performance of the at synchrotron conventionally used phase-retrieval algorithms for SBI, unified modulated pattern analysis (UMPA) with a phase-retrieval method developed for low coherence systems (LCS). We successfully retrieve a full computed tomography in a phantom as well as in biological specimens, such as larvae of the greater wax moth (Galleria mellonella), a model system for studies of pathogens and infections. In this context, we additionally demonstrate quantitative phase-contrast computed tomography using SBI at a low coherent set-up.

Fast Generation of Low PAPR and Low Sidelobe Noise Waveform

Noise radar waveform, as an important waveform in the field of low-probability-of-intercept radars, has been widely studied. However, the general noise waveform has a high peak average power ratio (PAPR), which leads to a reduction of signal-to-noise ratio. Thus, it is necessary to control the noise waveform PAPR to a suitable range. In this paper, the PAPR value reduction problem of low sidelobe noise waveform is studied. First, the proposed new algorithm is given a general overview. Then, the Combined Spectral Shaping and Peak-to-Average Power Ratio Reduction (COSPAR) algorithm is adjusted according to the alternate projection method and phase retrieval algorithm, and an improved COSPAR (ICOSPAR) algorithm with a faster execution speed and higher waveform generation efficiency is derived. The specific method performs waveform projection between the spectral constraint domain and the PAPR constraint domain and then evaluates the computational load of the ICOSPAR algorithm. Simulation results show that the ICOSPAR algorithm is reliable and efficient in generating low PAPRs and low sidelobe noise waveforms.

Single-shot lensless masked imaging with enhanced self-calibrated phase retrieval.

Single-shot lensless imaging with a binary amplitude mask enables a low-cost and miniaturized configuration for wave field recovery. However, the mask only allows a part of the wave field to be captured, and thus the inverse decoding process becomes a highly ill-posed problem. Here we propose an enhanced self-calibrated phase retrieval (eSCPR) method to realize single-shot joint recovery of mask distribution and the sample's wavefront. In our method, a sparse regularized phase retrieval (SrPR) algorithm is designed to calibrate the mask distribution. Then, a denoising regularized phase retrieval (DrPR) algorithm is constructed to reconstruct the wavefront of the sample. Compared to conventional single-shot methods, our method shows robust and flexible image recovery. Experimental results of different samples are given to demonstrate the superiority of our method.

Subgradient-projection-based stable phase-retrieval algorithm for X-ray ptychography.

X-ray ptychography is a lensless imaging technique that visualizes the nano-structure of a thick specimen which cannot be observed with an electron microscope. It reconstructs a complex-valued refractive index of the specimen from observed diffraction patterns. This reconstruction problem is called phase retrieval (PR). For further improvement in the imaging capability, including expansion of the depth of field, various PR algorithms have been proposed. Since a high-quality PR method is built upon a base PR algorithm such as ePIE, developing a well performing base PR algorithm is important. This paper proposes an improved iterative algorithm named CRISP. It exploits subgradient projection which allows adaptive step size and can be expected to avoid yielding a poor image. The proposed algorithm was compared with ePIE, which is a simple and fast-convergence algorithm, and its modified algorithm, rPIE. The experiments confirmed that the proposed method improved the reconstruction performance for both simulation and real data.

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.