Phase Retrieval Method Research Articles

Two-step differential phase-shifting phase retrieval using generative adversarial network

Phase-shifting interferometry (PSI) is a high-accuracy optical measurement technique which requires accurate phase-shift estimation or sufficient phase-shifting interferograms to ensure phase-retrieval accuracy. Therefore, obtaining the highest phase retrieval accuracy from fewer interferograms remains a challenge. Based on this, we propose a two-step differential phase-shifting phase retrieval method using generative adversarial network, which can achieve higher phase-retrieval accuracy from two arbitrary phase-shifting interferograms, compared to representative two-step arbitrary phase-shifting phase retrieval methods. A phase retrieval generative adversarial network (PR-GAN) which can be trained with artificial training data for phase retrieval from experimental interferograms is employed. Experimental results demonstrate that accurate phase results can be obtained, even when the phase shift is approximately π or the fringe number is relatively small.

Ab initio spatial phase retrieval via intensity triple correlations.

Second-order intensity correlations from incoherent emitters can reveal the Fourier transform modulus of their spatial distribution, but retrieving the phase to enable completely general Fourier inversion to real space remains challenging. Phase retrieval via the third-order intensity correlations has relied on special emitter configurations which simplified an unaddressed sign problem in the computation. Without a complete treatment of this sign problem, the general case of retrieving the Fourier phase from a truly arbitrary configuration of emitters is not possible. In this paper, a general method for ab initio phase retrieval via the intensity triple correlations is described. Simulations demonstrate accurate phase retrieval for clusters of incoherent emitters which could be applied to imaging stars or fluorescent atoms and molecules. With this work, it is now finally tractable to perform Fourier inversion directly and reconstruct images of arbitrary arrays of independent emitters via far-field intensity correlations alone.

Single-shot quantitative phase contrast imaging based on deep learning.

Quantitative differential phase-contrast (DPC) imaging is one of the commonly used methods for phase retrieval. However, quantitative DPC imaging requires several pairwise intensity measurements, which makes it difficult to monitor living cells in real-time. In this study, we present a single-shot quantitative DPC imaging method based on the combination of deep learning (DL) and color-encoded illumination. Our goal is to train a model that can generate an isotropic quantitative phase image (i.e., target) directly from a single-shot intensity measurement (i.e., input). The target phase image was reconstructed using a linear-gradient pupil with two-axis measurements, and the model input was the measured color intensities obtained from a radially asymmetric color-encoded illumination pattern. The DL-based model was trained, validated, and tested using thirteen different cell lines. The total number of training, validation, and testing images was 264 (10 cells), 10 (1 cell), and 40 (2 cells), respectively. Our results show that the DL-based phase images are visually similar to the ground-truth phase images and have a high structural similarity index (>0.98). Moreover, the phase difference between the ground-truth and DL-based phase images was smaller than 13%. Our study shows the feasibility of using DL to generate quantitative phase imaging from a single-shot intensity measurement.

Phase microscopy using band-limited image and its Fourier transform constraints.

In this Letter, we present a new, to the best of our knowledge, form of single-exposure quantitative phase microscopy based on the phase retrieval method by recording the band-limited image and its Fourier image simultaneously. Applying the intrinsic physical constraints of microscopy systems in the phase retrieval algorithm, we remove the inherent ambiguities of the reconstruction and achieve a rapid iterative convergence. In particular, this system does not require tight support of the object and the oversampling needed in coherent diffraction imaging. We have demonstrated that, in both simulations and experiments, the phase can be rapidly retrieved from a single-exposure measurement using our algorithm. The presented phase microscopy provides a promising technique for real-time quantitative biological imaging.

Comparison of denoising tools for the reconstruction of nonlinear multimodal images.

Biophotonic multimodal imaging techniques provide deep insights into biological samples such as cells or tissues. However, the measurement time increases dramatically when high-resolution multimodal images (MM) are required. To address this challenge, mathematical methods can be used to shorten the acquisition time for such high-quality images. In this research, we compared standard methods, e.g., the median filter method and the phase retrieval method via the Gerchberg-Saxton algorithm with artificial intelligence (AI) based methods using MM images of head and neck tissues. The AI methods include two approaches: the first one is a transfer learning-based technique that uses the pre-trained network DnCNN. The second approach is the training of networks using augmented head and neck MM images. In this manner, we compared the Noise2Noise network, the MIRNet network, and our deep learning network namely incSRCNN, which is derived from the super-resolution convolutional neural network and inspired by the inception network. These methods reconstruct improved images using measured low-quality (LQ) images, which were measured in approximately 2 seconds. The evaluation was performed on artificial LQ images generated by degrading high-quality (HQ) images measured in 8 seconds using Poisson noise. The results showed the potential of using deep learning on these multimodal images to improve the data quality and reduce the acquisition time. Our proposed network has the advantage of having a simple architecture compared with similar-performing but highly parametrized networks DnCNN, MIRNet, and Noise2Noise.

Lensless masked imaging with self-calibrated phase retrieval.

Lensless imaging with a mask is an attractive topic as it enables a compact configuration to acquire wavefront information of a sample with computational approaches. Most existing methods choose a customized phase mask for wavefront modulation and then decode the sample's wave field from modulated diffraction patterns. Different from phase masks, lensless imaging with a binary amplitude mask facilitates a cheaper fabrication cost, but high-quality mask calibration and image reconstruction have not been well resolved. Here we propose a self-calibrated phase retrieval (SCPR) method to realize a joint recovery of a binary mask and sample's wave field for a lensless masked imaging system. Compared with conventional methods, our method shows a high-performance and flexible image recovery without the help of an extra calibration device. Experimental results of different samples demonstrate the superiority of our method.

High-precision phase retrieval method for speckle suppression based on optimized modulation masks.

Traditional methods of coherent diffraction imaging using random masks result in an insufficient difference between the diffraction patterns, making it challenging to form a strong amplitude constraint, causing significant speckle noise in the measurement results. Hence, this study proposes an optimized mask design method combining random and Fresnel masks. Increasing the difference between diffraction intensity patterns enhances the amplitude constraint, suppresses the speckle noise effectively, and improves the phase recovery accuracy. The numerical distribution of the modulation masks is optimized by adjusting the combination ratio of the two mask modes. The simulation and physical experiments show that the reconstruction results of PSNR and SSIM using the proposed method are higher than those using random masks, and the speckle noises are effectively reduced.

One-Shot Phase Retrieval Method for Interferometry Using a Multi-Stage Phase-Shifting Network

In this letter, we proposed a Multi-Stage phase-shifting network (MSPS-Net) based on Multi-Stage Progressive Image Restoration for one-shot phase retrieval interferometry. This network generates three interferograms with specific phase shifts from interferograms recorded in a single frame excellently, metrics such as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PSNR</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SSIM</i> indicate that the generated phase shift interferograms have high similarity to the real one. Further, By analyzing different types of aberrations, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RMS</i> value of error map retrieved by the proposed method is less than 1/200λ, which shows higher accuracy than latest phase-shifting network method built by Encoder and Decoder. All the experiment results show that our solution is available for the problem of high-accuracy dynamic phase measurement and has excellent generalization ability.

Orbital angular momentum in the near-field of a fork grating.

Light beams with Orbital Angular Momentum (OAM) are explored in applications from microscopy to quantum communication, while the Talbot effect revives in applications from atomic systems to x-ray phase contrast interferometry. We evidence the topological charge of an OAM carrying THz beam in the near-field of a binary amplitude fork-grating by means of the Talbot effect, which we show to persist over several fundamental Talbot lengths. We measure and analyze the evolution of the diffracted beam behind the fork grating in Fourier domain to recover the typical donut-shaped power distribution, and we compare experimental data to simulations. We isolate the inherent phase vortex using the Fourier phase retrieval method. To complement the analysis, we assess the OAM diffraction orders of a fork grating in the far-field using a cylindrical lens.

Holographic phase-retrieval method of near-field antenna pattern measurement for bolometer-array-equipped millimeter-wave telescopes

Wide field-of-view millimeter-wave telescopes with a bolometric detector array have been developed for cosmic microwave background radiation observations. For the purpose of laboratory verification of these telescopes, several studies have demonstrated near-field antenna measurements using a phase-sensitive detector that replaces a few representative pixels of the focal-plane detector array. We present a holographic phase-retrieval method that enables near-field measurements with the bolometric detector array as it is. We place a reference emitter at a fixed position and scan a signal emitter at the telescope aperture. These two emitters are phase-locked and generate interference patterns (holograms) on the focal plane, from which the amplitude and phase of the aperture field can be retrieved. We experimentally demonstrated this method with a crossed-Dragone telescope with a field-of-view that is 18 deg × 9 deg. In the demonstration, we placed a phase-sensitive detector at three detector positions on the focal plane. The antenna patterns calculated from the hologram, neglecting the directly measured phase information, were consistent with those calculated from both intensity and phase measurements at the −60-dB level at 180 GHz. Applying this method, the antenna patterns for all of the bolometric detectors on the focal plane can theoretically be measured simultaneously.

Investigating the robustness of a deep learning-based method for quantitative phase retrieval from propagation-based x-ray phase contrast measurements under laboratory conditions

Objective. Quantitative phase retrieval (QPR) in propagation-based x-ray phase contrast imaging of heterogeneous and structurally complicated objects is challenging under laboratory conditions due to partial spatial coherence and polychromaticity. A deep learning-based method (DLBM) provides a nonlinear approach to this problem while not being constrained by restrictive assumptions about object properties and beam coherence. The objective of this work is to assess a DLBM for its applicability under practical scenarios by evaluating its robustness and generalizability under typical experimental variations. Approach. Towards this end, an end-to-end DLBM was employed for QPR under laboratory conditions and its robustness was investigated across various system and object conditions. The robustness of the method was tested via varying propagation distances and its generalizability with respect to object structure and experimental data was also tested. Main results. Although the end-to-end DLBM was stable under the studied variations, its successful deployment was found to be affected by choices pertaining to data pre-processing, network training considerations and system modeling. Significance. To our knowledge, we demonstrated for the first time, the potential applicability of an end-to-end learning-based QPR method, trained on simulated data, to experimental propagation-based x-ray phase contrast measurements acquired under laboratory conditions with a commercial x-ray source and a conventional detector. We considered conditions of polychromaticity, partial spatial coherence, and high noise levels, typical to laboratory conditions. This work further explored the robustness of this method to practical variations in propagation distances and object structure with the goal of assessing its potential for experimental use. Such an exploration of any DLBM (irrespective of its network architecture) before practical deployment provides an understanding of its potential behavior under experimental settings.

Quantitative phase imaging comparison of digital holographic microscopy and transport of intensity equation phase through simultaneous measurements of live cells

Quantitative phase imaging (QPI) of live cells allows label-free assessment of cell morphology and quantitative estimation of cell thickness, and therefore volume. Of the techniques available for QPI, digital holographic microscopy (DHM) is an accurate and reliable high-resolution phase imaging technique, but assembly and operation of a DHM capable setup can require significant optical engineering experience and specialized equipment that may not be practical for many labs. Transport of intensity equation (TIE) imaging is a QPI method that is compatible with conventional bright field microscopy, has a lower implementation cost without the need for specialized optics or alignment, and has significantly reduced spatial and temporal coherence requirements when compared to DHM. However, TIE is susceptible to different forms of noise than DHM and its accuracy in live cell imaging has not been well studied. Here, we present a microscope capable of performing simultaneous DHM and TIE phase reconstructions to compare both methods and validate TIE as an accurate phase retrieval method both for a static sample array and for live cells.

Plug-and-play algorithm for imaging through scattering media under ambient light interference.

Imaging through scattering media is a fascinating subject in the computational imaging domain. The methods based on speckle correlation imaging have found tremendous versatility. However, a darkroom condition without any stray light is required because the speckle contrast is easily disturbed by ambient light, which can lead to the reduction in object reconstruction quality. Here, we report a plug-and-play (PnP) algorithm to restore the object through scattering media under the non-darkroom environment. Specifically, the PnPGAP-FPR method is established via the generalized alternating projection (GAP) optimization framework, Fienup phase retrieval (FPR) method, and FFDNeT. The proposed algorithm is demonstrated experimentally and shows significant effectiveness and flexible scalability, which describe the potential for its practical applications.

Recent Advances in Coherent Optical Communications for Short-Reach: Phase Retrieval Methods

Short-reach transmission systems traditionally utilize intensity modulation (IM) at the transmitter and direct detection (DD) at the receiver due to their cost-effectiveness, small footprint, and low power consumption. However, with the exponential increase in bandwidth demand, coherent optical communication systems have become necessary for long-haul distances, requiring application-specific integrated circuits (ASIC) and advanced digital signal processing (DSP) algorithms coupled with high-speed digital-to-analog and analog-to-digital converters to achieve Tbit/s speeds. As coherent technology matures, it will eventually become feasible for short-reach transmission. In this context, self-coherent systems have emerged as an intermediary solution, offering advantages over traditional IM/DD systems. While comprehensive review studies exist on self-coherent transceivers, they do not cover recent advances in phase retrieval methods for short-reach optical communications. This review article highlights recent developments in cost-effective self-coherent detection for short-reach systems through comparing the benefits of single sideband (SSB) transmission and Kramers-Kronig detection to carrier-assisted phase retrieval, the Gerchberg-Saxton (GS) algorithm, and the transport of intensity equation (TIE) method.

Posterior regularization method for phase removal of shale nano-structure imaging in space domain

X-Ray computed tomography is a non-destructive method that is used, among many applications, to study the size, shape, 3D structures and interconnections of pores in shale. We use phase retrieval methods to deal with the “edge enhancement” effect caused by phase shift. The process of phase retrieval can be described by the transport-of-intensity equation (TIE). But this is an ill-posed problem. The existing methods focus on phase retrieval in the frequency domain. To tackle the ill-posedness, we propose a new method whose main idea is to solve this problem in space domain with a regularization technique. We study a synthetic shale model and simulate the projection data. Then we apply three methods to retrieve the phase: conventional method in frequency domain, direct solving method and iterative Tikhonov regularization method in space domain. Finally, we use the standard filtered back-projection (FBP) method to present the outcome. By analyzing the results, we find advantages of the new method: more stability and fewer artifacts under noise perturbations. The study shows that relative errors of the new method are nearly 1% of that of the traditional method based on frequency domain, and hence the new method is promising for the practical data processing.

Untrained deep learning-based phase retrieval for fringe projection profilometry

Fringe projection profilometry (FPP) based on deep learning shows potential for challenging 3-D sensing tasks, e.g., bio-medicine, reverse engineering, and face recognition, etc. Supervised deep learning has been introduced to retrieve the desired phase for the 3-D reconstruction, which relies on lots of advanced training to construct the fringe-to-phase transformation. The traditional deep learning-based method becomes unreliable for scenes that are different from the training ones, which restricts it to be applied for actual applications. In this paper, an untrained deep learning-based phase retrieval method is proposed. By adding a camera to the traditional FPP system, scene-independent physical constraints such as phase, structure and 3-D consistency are constructed to optimize the fringe-to-phase transformation. The proposed deep learning-based method can retrieve the desired phase from a single fringe pattern without advance training. Both theoretical analyses and experimental results demonstrate its accurateness and robustness. The proposed method also shows potential for single-shot 3-D sensing applications such as high-speed 3-D measurement.

Single-Shot Multi-Frequency 3D Shape Measurement for Discontinuous Surface Object Based on Deep Learning

It is challenging to stably and rapidly achieve accurate absolute phase retrieval for isolated objects with a single-shot pattern in fringe projection profilometry (FPP). In this context, a single-shot multi-frequency absolute phase retrieval (SAPR-DL) method based on deep learning is proposed, which only needs to capture one fringe image to obtain the full-field precise absolute phase. Specifically, a low-frequency deformed fringe image is loaded into the trained one-to-two deep learning framework (DLFT) to predict unit-frequency and high-frequency deformed fringe images. Then, three fringe images with different frequencies are loaded into the trained deep learning phase retrieval framework (DLPR) to calculate the corresponding absolute phase. The experimental results prove that the proposed SAPR-DL method can obtain the three-dimensional (3D) shape measurement of multiple complex objects by collecting a single-shot fringe image, showing great prospects in advancing scientific and engineering applications.

Imaging optimisation process of human lung tissue using synchrotron radiation microcomputed tomography in PLS‐II

AbstractSynchrotron radiation microcomputed tomography (SR‐μCT) can provide non‐destructive three‐dimensional investigation for soft tissue with high resolution. The aim of this study is to evaluate technical improvements in microcomputed tomographic imaging of human lung tissue using a synchrotron radiation source at the 6C biomedical beamline. SR‐μCT images of excised human lung tissue (6 mm vertically and 10 mm horizontally sized) were explored at the 6C biomedical beamline in the Pohang accelerator laboratory (Pohang, Republic of Korea) using phase retrieval methods with a modified Bronnikov algorithm. Experimental data collection was performed with changes in the following conditions: the presence of a graphite attenuator, sample to detector distance, the presence of different scintillators and increasing pixel sizes. We identified that the resolution of lung tissue images was sufficiently enhanced when the graphite attenuator was not applied, 550 mm of sample to detector distance, with Cadmium tungsten (CdWO4) scintillator and twice binning of the pixel data and could be comparable with images from pathologic examination. Refinement of SR‐μCT techniques for soft tissue originating from the human body would advance the potential clinical application of synchrotron radiation in the near future.

Deep Learning-Based Phase Retrieval Scheme for Minimum-Phase Signal Recovery

We propose a deep learning-based phase retrieval method to accurately reconstruct the optical field of a single-sideband minimum-phase signal from the directly detected intensity waveform. Our method relies on a fully convolutional Neural Network (NN) model to realize non-iterative and robust phase retrieval. The NN is trained so that it performs full-field reconstruction and jointly compensates for transmission impairments. Compared to the recently proposed Kramers-Kronig (KK) receiver, our method avoids the distortions introduced by the nonlinear operations involved in the KK phase-retrieval algorithm and hence does not require digital upsampling. We validate the proposed phase-retrieval method by means of extensive numerical simulations in relevant system settings, and we compare the performance of the proposed scheme with the conventional KK receiver operated with a 4-fold digital upsampling. The results show that the 7% hard-decision forward error correction (HD-FEC) threshold at BER 3.8e-3 can be achieved with up to 2.8 dB lower carrier-to-signal power ratio (CSPR) value and 1.8 dB better receiver sensitivity compared to the conventional 4-fold upsampled KK receiver. We also present a comparative analysis of the complexity of the proposed scheme with that of the KK receiver, showing that the proposed scheme can achieve the 7% HD-FEC threshold with 1.6 dB lower CSPR, 0.4 dB better receiver sensitivity, and 36% lower complexity.

Simulated wrapped phase optimizes phase retrieval in phase-shifting interferometry.

Phase retrieval is crucial in phase-shifting interferometry and other phase measurement techniques. However, in noisy wrapped phase maps with high steepness, discontinuities arise and cause phase unwrapping errors. To solve this problem, this Letter presents a phase retrieval method based on a simulated wrapped phase. By establishing the correspondence between the simulated and measured interferograms, the difference in wrapped phases between them can be obtained. The difference in wrapped phase map, which has sparse and wide interference fringes, has a higher reliability of phase unwrapping. The proposed method not only possesses high phase retrieval accuracy but it also simplifies the processing of interferograms. Furthermore, the layout of all interferometric systems, the parameters of optical components, and the model of the measured object are known, so the proposed method can be used as a reference for phase retrieval.