Phase Retrieval Algorithm Research Articles

Understanding and correcting wavenumber error in interference pattern structured illumination imaging.

The impacts of uncertainty in mirror movements in mechanically scanned interference pattern structured illumination imaging (IPSII) are discussed. It is shown that uncertainty in IPSII mirror movements causes errors in both the phase and amplitude of the Fourier transform of the resulting imaging. Finally, we demonstrate that iterative phase retrieval algorithms can improve the quality of IPSII images by correcting the phase errors caused by mirror movement uncertainties.

Normalized and orthogonal phase retrieval algorithm based on Lissajous ellipse fitting

A normalized and orthogonal phase retrieval algorithm based on Lissajous ellipse fitting (NOLEF) is proposed in this study, which can extract phase distributions under random phase shifts with high accuracy. In this algorithm, the parameters related to the phase can be calculated via Lissajous ellipse fitting after the normalization and orthogonalization processes. Compared with other high-accuracy phase-shifting algorithms, the proposed method has a shorter calculation time and provides stable and accurate phase extraction with random phase-shifting interferograms, in which the accuracy is not affected by the number of fringes and phase shifts; it is also suitable for interferograms with different backgrounds and modulations. This method's feasibility and performance at different number of fringes and noise levels are verified via experimental and simulation analyses.

Single-shot phase reconstruction based on beam splitting encoding and averaging

Coherent modulation imaging (CMI) can effectively improve the convergence performance of coherent diffraction imaging by introducing a pre-characterized wave modulator. However, traditional CMI algorithms suffer from a low signal-to-noise ratio (SNR), with insufficient information redundancy inheriting from a single diffraction pattern. Additionally, the strong modulation capability of the modulator with a small basic pitch is preferred; however, it leads to the difficulty of fabrication and measurement with a limited aperture size of the detector. To overcome those obstacles, this study proposes a revised CMI algorithm based on beam splitting encoding and averaging. A diffraction pattern array was recorded after the incident wave was split by grating and modulated by a weak scattering modulator simultaneously. This approach differed from the previous grating-based single-shot phase retrieval algorithm because the diffraction array was not segmented and used integrally during the iteration process, which guarantees the capability of diffraction-limited resolution in theory. Additionally, an average process was employed in the image plane of the object to improve SNR significantly. The performance of the revised algorithm was demonstrated by simulations and experiments and can be applied as a universal single-shot phase retrieval algorithm to various fields practically with fast convergence speed and high SNR.

Synthetic aperture interference light (SAIL) microscopy for high-throughput label-free imaging.

Quantitative phase imaging (QPI) is a valuable label-free modality that has gained significant interest due to its wide potentials, from basic biology to clinical applications. Most existing QPI systems measure microscopic objects via interferometry or nonlinear iterative phase reconstructions from intensity measurements. However, all imaging systems compromise spatial resolution for the field of view and vice versa, i.e., suffer from a limited space bandwidth product. Current solutions to this problem involve computational phase retrieval algorithms, which are time-consuming and often suffer from convergence problems. In this article, we presented synthetic aperture interference light (SAIL) microscopy as a solution for high-resolution, wide field of view QPI. The proposed approach employs low-coherence interferometry to directly measure the optical phase delay under different illumination angles and produces large space-bandwidth product label-free imaging. We validate the performance of SAIL on standard samples and illustrate the biomedical applications on various specimens: pathology slides, entire insects, and dynamic live cells in large cultures. The reconstructed images have a synthetic numeric aperture of 0.45 and a field of view of 2.6 × 2.6 mm2. Due to its direct measurement of the phase information, SAIL microscopy does not require long computational time, eliminates data redundancy, and always converges.

Numerical and Experimental Investigation of Phaseless Spherical Near-Field Antenna Measurements

This article presents a thorough study of spherical probe-corrected phaseless near-field measurements with the two-scans technique. Such technique is based on retrieving the antenna under test radiation pattern from the measurement of the near-field amplitude signals on two spheres of different radii. The postprocessing of this type of measurements results in a highly nonlinear algorithm, prone to get trapped in local minima and provide incorrect solutions when the measurement conditions are not properly selected. Through a series of numerical simulations, the influence of different measurement parameters on the phaseless technique is analyzed. It will be shown how both the relative and absolute values of the measurement spheres highly affect the convergence of the phase retrieval algorithm. The type of AUT and its radiation pattern characteristic also play a fundamental role in the feasibility of phaseless measurements. Other parameters such as sampling rate, noise, probe correction, polar truncation, and measurement offsets are also investigated. The conducted study allows to extract a set of guidelines to improve the accuracy of phaseless spherical near-field algorithms. In addition, purely phaseless antenna measurement examples are given to demonstrate the algorithm capabilities and limitations, and to validate the developed numerical investigations.

Virtual speckle-based X-ray phase-contrast and dark-field imaging with digital phantoms

We demonstrate two versatile, flexible, and accurate frameworks based on numerical and Monte Carlo approaches to simulate the X-ray speckle-based (SBI) technique for lab-based systems. The established tools can reproduce experimental setups in a cone-beam geometry and with polychromatic sources. Furthermore, they are computationally efficient to enable a fast virtual multi-modal tomography of digitized inhomogeneous phantoms. The proposed methods were evaluated and validated by analytical and experimental data for various samples. The Monte Carlo approach provides a realistic and accurate simulation, which is useful in diffuser design and dosimetry studies, while the numerical method is very efficient for parametric and tomographic studies. These approaches will be used for the optimization of lab-based X-ray SBI setups and generating sample images for enhancing phase retrieval algorithms.

Simulated Annealing Applied to HIO Method for Phase Retrieval

Phase retrieval utilizing Fourier amplitudes plays a significant role in image recovery. Iterative phase retrieval algorithms have been developed to retrieve phase information that cannot be recorded by detectors directly. However, iterative algorithms face the problem of being trapped in local minima due to the nonconvexity of phase retrieval, and most existing works addressed this by optimizing in multiple runs parallelly to improve the possibility that one of these could reach the global minimum. Alternatively, we propose in this work to increase the probability of reaching the global minimum with one arbitrary initial distribution by adapting simulated annealing in the standard hybrid input-output (HIO) algorithm. Numerical and experimental results demonstrate that the proposed method reconstructs images with mean square errors 50.12% smaller than those reconstructed by HIO. More importantly, the proposed method can be applied to any HIO-based algorithm with multiple runs to further improve the performance.

Cryptanalysis of compressive interference-based optical encryption using a U-net deep learning network

We present a deep learning (DL) framework based on a U-type convolutional neural network (U-net) to analysis the security of optical interference-based encryption system with sparsity constraints. In the optical cryptosystem, two phase-only masks (POMs) can be obtained by using phase retrieval algorithm (PRA) with the constraints of complementary binary amplitude masks (BAMs). By integrating the interference technique and phase encoding, the encryption scheme achieves a high security level and is immune to conventional attack methods. Here, we investigate and construct a neural network architecture for evaluating the security of the cryptosystem based on the idea of U-net. In this method, we achieve the approximate equivalent model of the compressive interference-based optical cryptosystem by using a U-net model, which has been pre-trained by a series of ciphertext–plaintext pairs. It can be found that the cryptosystem can be broken by using the proposed DL network. The plaintexts can be retrieved from their corresponding ciphertexts without the use of security keys, such as the diffraction distances, the BAMs, etc. The results of numerical experiments are presented to show the good performance of our proposed DL-based method.

Phase-shifting digital holography with an unknown reference beam

In digital holography, errors of the reference field degrade the quality of the reconstructed object field. In this paper, we propose an effective method in phase-shifting digital holography in which the reference field does not need to be known and perfect. The unknown complex amplitudes of both reference and object fields are derived simultaneously. The method employs only five digital holograms and a single execution of a phase retrieval algorithm. So, the required measurements and algorithm executions in this method are fewer than those in other methods; it suggests a simpler and faster method. The effectiveness of the suggested method is indicated by simulation, under noise-free and noisy conditions. Moreover, the capability of the method to extract full information about the phase singularities in both fields is demonstrated.

Transport of intensity equation-based terahertz lensless full-field phase imaging.

Terahertz (THz) phase imaging is widely spreading in various scenarios, among which full-field phase distributions are commonly retrieved by digital holography or ptychography. In this Letter, the transport of the intensity equation reconstruction method is applied into the THz band. An algorithm named the lensless US-transport of intensity equation (TIE) is proposed to accommodate to an in-line configuration. The object phase is retrieved by primarily conducting iterations between the axial intensity derivative and the phase distribution at the recording plane and subsequent backward diffraction propagation. This method is applicable to both isolated and extended weakly absorbing samples with higher reconstruction quality and remarkably less time cost than holographic phase retrieval algorithms. It can also be attempted in other non-interferometric geometries or using low-cost partially coherent THz sources, which significantly broaden the application scope of THz phase imaging.

Projection methods for high numerical aperture phase retrieval

We develop for the first time a mathematical framework in which the class of projection algorithms can be applied to high numerical aperture (NA) phase retrieval. Within this framework, we first analyze the basic steps of solving the high-NA phase retrieval problem by projection algorithms and establish the closed forms of all the relevant projection operators. We then study the geometry of the high-NA phase retrieval problem and the obtained results are subsequently used to establish convergence criteria of projection algorithms in the presence of noise. Making use of the vectorial point-spread-function (PSF) is, on the one hand, the key difference between this paper and the literature of phase retrieval mathematics which deals with the scalar PSF. The results of this paper, on the other hand, can be viewed as extensions of those concerning projection methods for low-NA phase retrieval. Importantly, the improved performance of projection methods over the other classes of phase retrieval algorithms in the low-NA setting now also becomes applicable to the high-NA case. This is demonstrated by the accompanying numerical results which show that available solution approaches for high-NA phase retrieval are outperformed by projection methods.

Inversion of coherent surface scattering images via deep learning network

Coherent surface scattering imaging is a powerful tool for imaging a surface/interface of a thin nanostructure deposited on an opaque substrate. A mathematical conversion of an object image from a scattering pattern is essential for coherent surface scattering imaging to visualize structures of specimens. It has been achieved by using phase retrieval algorithms requiring oversampling in scattering patterns and employing alternating projection approaches. It is a computationally challenging and time-consuming process. In this paper, we demonstrate CSSI-NN, which is a deep learning neural network model to predict images of objects from scattering intensities in coherent surface scattering imaging. This model allowing for the instant outcome from scattering patterns would be tremendously beneficial not only for effective experiments but also for data analysis of phase retrieval.

Quantitative phase imaging of single particles from a cryoEM micrograph

We show that de-focused single particle micrographs recorded using a cryo-electron microscope (cryoEM) system may be processed like a Fresnel zone in-line hologram to obtain physically meaningful quantitative phase maps associated with individual particles. In particular, a region-of-interest (ROI) of the de-focused micrograph surrounding a particle can be numerically back-propagated, in order to determine accurate de-focus information for single particles based on the sparsity-of-gradient merit function. Further with the knowledge of de-focus information, an iterative Fresnel zone phase retrieval algorithm using image sparsity constraints can accurately estimate the quantitative phase information associated with a single particle. The proposed methodology which can correct for both de-focus and spherical aberrations is a deviation from the image processing chain currently used in single particle cryoEM reconstructions. Our illustrations as presented here suggest that the phase retrieval approach applies uniformly to de-focused image data recorded using the traditional CCD detectors as well as the newer direct electron detectors.

Three-dimensional coherent X-ray diffraction imaging via deep convolutional neural networks

As a critical component of coherent X-ray diffraction imaging (CDI), phase retrieval has been extensively applied in X-ray structural science to recover the 3D morphological information inside measured particles. Despite meeting all the oversampling requirements of Sayre and Shannon, current phase retrieval approaches still have trouble achieving a unique inversion of experimental data in the presence of noise. Here, we propose to overcome this limitation by incorporating a 3D Machine Learning (ML) model combining (optional) supervised learning with transfer learning. The trained ML model can rapidly provide an immediate result with high accuracy which could benefit real-time experiments, and the predicted result can be further refined with transfer learning. More significantly, the proposed ML model can be used without any prior training to learn the missing phases of an image based on minimization of an appropriate ‘loss function’ alone. We demonstrate significantly improved performance with experimental Bragg CDI data over traditional iterative phase retrieval algorithms.

Computer-Free, All-Optical Reconstruction of Holograms Using Diffractive Networks

Reconstruction of in-line holograms of unknown objects in general suffers from twin-image artifacts due to the appearance of an out-of-focus image overlapping with the desired image to be reconstructed. Computer-based iterative phase retrieval algorithms and learning-based methods have been used for the suppression of such image artifacts in digital holography. Here we report an all-optical hologram reconstruction method that can instantly retrieve the image of an unknown object from its in-line hologram and eliminate twin-image artifacts without using a digital processor or a computer. Multiple transmissive diffractive layers are trained using deep learning so that the diffracted light from an arbitrary input hologram is processed all-optically, through light-matter interaction, to reconstruct the image of an unknown object at the speed of light propagation and without the need for any external power. This passive all-optical processor composed of spatially-engineered transmissive layers forms a diffractive network, which successfully generalizes to reconstruct in-line holograms of unknown, new objects and exhibits improved diffraction efficiency as well as extended depth-of-field at the hologram recording distance. This all-optical hologram processor and the underlying design framework can find numerous applications in coherent imaging and holographic display-related applications owing to its major advantages in terms of image reconstruction speed and computer-free operation.

Hyperuniform scalar random fields for lensless, multispectral imaging systems.

We propose a novel framework for the systematic design of lensless imaging systems based on the hyperuniform random field solutions of nonlinear reaction-diffusion equations from pattern formation theory. Specifically, we introduce a new class of imaging point-spread functions (PSFs) with enhanced isotropic behavior and controllable sparsity. We investigate PSFs and modulated transfer functions for a number of nonlinear models and demonstrate that two-phase isotropic random fields with hyperuniform disorder are ideally suited to construct imaging PSFs with improved performances compared to PSFs based on Perlin noise. Additionally, we introduce a phase retrieval algorithm based on non-paraxial Rayleigh-Sommerfeld diffraction theory and introduce diffractive phase plates with PSFs designed from hyperuniform random fields, called hyperuniform phase plates (HPPs). Finally, using high-fidelity object reconstruction, we demonstrate improved image quality using engineered HPPs across the visible range. The proposed framework is suitable for high-performance lensless imaging systems for on-chip microscopy and spectroscopy applications.

Single-shot imaging through scattering medium with a deterministic phase-retrieval algorithm

Imaging through scattering layers has a wide range of applications. However, even a thin layer can scatter the light into a random speckle pattern thus difficult to image. Speckle correlation-based methods enable noninvasive imaging of objects hidden behind such layers whereas, still suffer from inaccurate solutions and losing the orientation information of the object. In this paper, we proposed a novel deterministic phase-retrieval algorithm to fully reconstruct the object from its speckle pattern and experimentally demonstrated that the new method can retrieve a high-quality image from a single-shot speckle pattern. Compared to other phase retrieval techniques, it maintains the orientation information correctly in a single reconstruction and is more robust in noisy conditions.

Accurate and stable two-step LED position calibration method for Fourier ptychographic microscopy.

.Significance: Fourier ptychography microscopy (FPM) is a computational optical imaging technology that employs angularly varying illuminations and a phase retrieval algorithm to achieve a wide field of view and high-resolution imaging simultaneously. In the FPM, LED position error will reduce the quality of the reconstructed high-resolution image. To correct the LED positions, current methods consider each of the LED positions as independent and use an optimization algorithm to get each of the positions. When the positional misalignment is large or the search position falls into a local optimal value, the current methods may lack stability and accuracy.Aim: We improve the model of the LED position and propose an accurate and stable two-step correction scheme (tcFPM) to calibrate the LED position error.Approach: The improved LED positions model combines the overall offset, which represents the relative deviation of the LED array and the optical axis, with the slight deviation of each LED’s independent position. In the tcFPM, the overall offset of the LED array is corrected at first, which obtains an approximate value of the overall offset of the LED array. Then the position of each LED is precisely adjusted, which obtains the slight offset of each LED.Results: This LED position error model is more in line with the actual situation. The simulation and experimental results show that the method has high accuracy in correcting the LED position. Furthermore, the reconstruction process of tcFPM is more stable and significantly improves the quality of the reconstruction results, which is compared with some LED position error correction methods.Conclusions: An LED position error correction technology is proposed, which has a stable iterative process and improves the reconstruction accuracy of complex amplitude.

Kaczmarz holography: holographic projection in the Fraunhofer region

The Kaczmarz algorithm is an iterative method for solving linear equations in the form of Ax = b. It is widely used in computed tomography and digital signal processing (DSP) but has yet to be adopted in computer-generated holography. Phase retrieval algorithms, such as Gerchberg–Saxton or Fienup, are significantly more popular in this field. However, we propose a unique and alternative approach to projecting a replay field through discrete Fourier transform matrices and have shown that there are legitimate benefits to implementing this approach. The gradient descent iteration mechanism adopted by Kaczmarz, for instance provides finer granularity control over the individual pixels in the replay field. We consequently demonstrate the quality of the image is significantly improved when compared with Gerchberg–Saxton.

Robust phase retrieval with green noise binary masks

Phase retrieval with pre-defined optical masks can provide extra constraints and thus achieve improved performance. Recent progress in optimization theory demonstrates the superiority of random masks in enhancing the accuracy of phase retrieval algorithms. However, traditional approaches only focus on the randomness of the masks but ignore their non-bandlimited nature. When using these masks for phase retrieval, the intensity measurements contain many significant high-frequency components that the phase retrieval algorithm cannot take care of and thus leads to degraded performance. Based on the concept of digital halftoning, this paper proposes a green noise binary masking scheme that can significantly reduce the high-frequency contents of the masks while fulfilling the randomness requirement. The resulting intensity measurements will contain data concentrated in the mid-frequency band and around zero frequency areas which can be fully utilized in the phase retrieval optimization process. Our experimental results show that the proposed green noise binary masking scheme consistently outperforms the traditional ones when using in binary coded diffraction pattern phase retrieval systems.