Phase Retrieval Algorithm Research Articles

Fast and robust phase retrieval for masked coherent diffractive imaging

Conventional phase retrieval algorithms for coherent diffractive imaging (CDI) require many iterations to deliver reasonable results, even using a known mask as a strong constraint in the imaging setup, an approach known as masked CDI. This paper proposes a fast and robust phase retrieval method for masked CDI based on the alternating direction method of multipliers (ADMM). We propose a plug-and-play ADMM to incorporate the prior knowledge of the mask, but note that commonly used denoisers are not suitable as regularizers for complex-valued latent images directly. Therefore, we develop a regularizer based on the structure tensor and Harris corner detector. Compared with conventional phase retrieval methods, our technique can achieve comparable reconstruction results with less time for the masked CDI. Moreover, validation experiments on real in situ CDI data for both intensity and phase objects show that our approach is more than 100 times faster than the baseline method to reconstruct one complex-valued image, making it possible to be used in challenging situations, such as imaging dynamic objects. Furthermore, phase retrieval results for single diffraction patterns show the robustness of the proposed ADMM.

Single-scan multiplane phase retrieval with a radiation of terahertz quantum cascade laser

Terahertz phase retrieval from a set of axially separated diffractive intensity distributions is a promising single-beam computational imaging technique that ensures the obtention of high spatial resolutions and phase wavefronts, but remains restricted by time-consuming data acquisition processes. In this work, we have adopted an approach, relying on the radiation of a quantum cascade laser and the implementation of an express single-scan measurement of intensity distributions through the continuous on-the-go displacement of a high-sensitivity antenna-coupled microbolometer sensor array. In addition to the simplicity of this practical implementation and the minimization of measurement times, such an approach overcomes the problem of preliminary optimal selections of transverse intensity distributions used in the iterative phase retrieval algorithm and guarantees the required data diversity for high-quality wavefront reconstruction.

Coherent modulation imaging realized by coaxial shift of an amplitude mask

As a lensless imaging technology, coherent modulation imaging (CMI) can reconstruct the complex amplitude of the object from the measured intensity distribution. In this paper, a new CMI scheme is proposed. A random binary amplitude mask is introduced between the light source and the object. The incident light is modulated by the mask and the speckle field is generated to illuminate the object. The object can then be accurately recovered with the iterative phase retrieval algorithm by means of moving the mask several times in the axial direction and recording the corresponding diffraction pattern. Compared to the existing CMI schemes, the experimental system of the proposed method is simple and compact. No spatial light modulator or phase plate is required. Both numerical simulations and experiments are performed to verify the feasibility of the method. The influence of moving times, moving interval and pixel size of the mask on the reconstruction accuracy are analyzed in detail. In addition, the image registration encountered in the experiment is discussed.

Characterization of surface defects using a phase retrieval technique in a high-power laser system.

In the framework of high-power lasers, surface defects on optics can generate strong light intensification and induce damage sites on downstream optics. To evaluate this intensification during high-energy laser shots, a three-step method is proposed. First, a dedicated measurement bench is designed to measure the intensification induced by defects on a wide variety of optics, including amplifier slabs, KDP crystals, mirrors, gratings, and vacuum windows, for propagation distances up to 2000 mm. A multi-resolution single-beam multiple-intensity reconstruction phase retrieval algorithm is then used to reconstruct a model of the defect, in both amplitude and phase, from a set of intensification measurements. Finally, the impact of the modeled defect on downstream optics is evaluated with a simulation of the high-power laser system. This method is experimentally validated through a case study of damage identified on one of the Laser Mégajoule (LMJ) beams, characterized with the method presented in this paper. The long-distance impact on the LMJ beam is estimated by simulation and compared to a direct near-field measurement.

Asymmetric double-image encryption via wavelength multiplexing.

In this paper, we propose an asymmetric double-image encryption via wavelength multiplexing. First, a novel iterative phase retrieval algorithm, to the best of our knowledge, is developed to encode two images into one complex-valued function via wavelength multiplexing. Then, the function is encoded by equal modulus decomposition (EMD). This cryptosystem not only retains the advantages of EMD but also reduces the number of public keys so as to enhance the resistance to the amplitude-phase retrieval algorithm (APRA). In the decryption, two high-quality decrypted images can be obtained with their corresponding wavelengths. To the best of our knowledge, this is the first time that wavelength multiplexing is used to achieve high-quality two-image encryption and decryption. Numerical simulation results show the effectiveness and robustness of this new method.

Single-shot three-input phase retrieval for quantitative back focal plane measurement

This paper presents quantitative measurements facilitated with a new optical system that implements a single-shot three-input phase retrieval algorithm. The new system allows simultaneous acquisition of three distinct input patterns, thus eliminating the requirement for mechanical movement and reducing any registration errors and microphonics. We demonstrate the application of the system for measurement and separation of two distinct attenuation measurements of surface waves, namely, absorption and coupling loss. This is achieved by retrieving the phase in the back focal plane and performing a series of virtual optics computations. This overcomes the need to use a complicated series of hardware manipulations with a spatial light modulator. This gives a far more accurate and faster measurement with a simpler optical system. We also demonstrate that phase measurements allow us to implement different measurement methods to acquire the excitation angle for surface plasmons. Depending on the noise statistics different methods have superior performance, so the best method under particular conditions can be selected. Since the measurements are only weakly correlated, they may also be combined for improved noise performance. The results presented here offer a template for a wider class of measurements in the back focal plane including ellipsometry.

Security-enhanced multiple-image encryption based on modified iterative phase retrieval algorithm with structured phase mask in Fresnel domain

A security-enhanced multiple-image encryption method is proposed based on modified iterative phase retrieval algorithm with structured phase mask in Fresnel domain in this paper, where each plaintext is encoded to only one independent phase-only mask with the help of the used structured phase mask, and a group of phase-only masks retrieved by multiple plaintexts are integrated into the ciphertext with the noise-like distribution by using the phase mask multiplexing technique. In the proposed method, the Fresnel diffraction distances and the wavelength of light as well as the optical parameters of the structured phase mask such as focal length, topological charge number are all used as digital keys to expand the key space. Most importantly, the sensitivities of the digital keys are substantially improved by mapping the digital keys with the initial values of chaotic system thanks to the nonlinear, pseudo-random, high sensitivity of initial value of special phase generated by chaotic system, contributing to an extremely high security of multiple-image encryption method. Moreover, computational simulations are performed to demonstrate the feasibility of the proposed security-enhanced multiple-image encryption method, and the simulation results confirm that the proposed method exhibits high effectiveness and strong robustness.

An In-House Cone-Beam Tomographic Reconstruction Package for Laboratory X-ray Phase-Contrast Imaging

Phase-contrast, and in general, multi-modal, X-ray micro-tomography is proven to be very useful for low-density, low-attention samples enabling much better contrast than its attenuation-based pendant. Therefore, it is increasingly applied in bio- and life sciences primarily dealing with such samples. Although there is a plethora of literature regarding phase-retrieval algorithms, access to implementations of those algorithms is relatively limited and very few packages combining phase-retrieval methods with the full tomographic reconstruction pipeline are available. This is especially the case for laboratory-based phase-contrast imaging typically featuring cone-beam geometry. We present here an in-house cone-beam tomographic reconstruction package for laboratory X-ray phase-contrast imaging. It covers different phase-contrast techniques and phase retrieval methods. The paper explains their implementation and integration in the filtered back projection chain. Their functionality and efficiency will be demonstrated through applications on a few dedicated samples.

Imaging Through Random Scatterer with Spatial Coherence Structure Measurement

Optical coherence is becoming an efficient degree of freedom for light field manipulations and applications. In this work, we show that the image information hidden a distance behind a random scattering medium is encoded in the complex spatial coherence structure of a partially coherent light beam that generates after the random scatterer. We validate in experiment that the image information can be well recovered with the spatial coherence measurement and the aid of the iterative phase retrieval algorithm in the Fresnel domain. We find not only the spatial shape but also the position including the lateral shift and longitudinal distances of the image hidden behind the random scatterer can be reconstructed, which indicates the potential uses in three-dimensional optical imaging through random scattering media.

Measurement of Small-Slope Free-Form Optical Surfaces with the Modified Phase Retrieval.

In this paper, we demonstrate the use of the modified phase retrieval as a method for application in the measurement of small-slope free-form optical surfaces. This technique is a solution for the measurement of small-slope free-form optical surfaces, based on the modified phase retrieval algorithm, whose essence is that only two defocused images are needed to estimate the wave front with an accuracy similar to that of the traditional phase retrieval but with less image capturing and computation time. An experimental arrangement used to measure the small-slope free-form optical surfaces using the modified phase retrieval is described. The results of these experiments demonstrate that the modified phase retrieval method can achieve measurements comparable to those of the standard interferometer.

Robust Phase Retrieval with Complexity-Guidance for Coherent X-Ray Imaging

Reconstruction of a stable and reliable solution from noisy and incomplete Fourier intensity data is a challenging problem for iterative phase retrieval algorithms. The typical methodology employed in the coherent X-ray imaging (CXI) literature involves thousands of iterations of well-known phase retrieval algorithms, e.g., hybrid input-output (HIO) or relaxed averaged alternating reflections (RAAR), that are concluded with a smaller number of error reduction (ER) iterations. Since the single run of this methodology may not provide a reliable solution, hundreds of trial solutions are first obtained by initializing the phase retrieval algorithm with independent random guesses. The resulting trial solutions are then averaged with appropriate phase adjustment, and resolution of the averaged reconstruction is assessed by plotting the phase retrieval transfer function (PRTF). In this work, we examine this commonly used RAAR-ER methodology from the perspective of the complexity parameter introduced by us in recent years. It is observed that the single run of the RAAR-ER algorithm provides a solution with undesirable grainy artifacts that persist to some extent even after averaging the multiple trial solutions. The grainy features are spurious in the sense that they are smaller in size compared to the resolution predicted by the PRTF curve. This inconsistency can be addressed by a novel methodology that we refer to as complexity-guided RAAR (CG-RAAR). The methodology is demonstrated with simulations and experimental data sets from the CXIDB database. In addition to providing consistent solution, CG-RAAR is also observed to require reduced number of independent trials for averaging.

Phase retrieval algorithms: principles, developments and applications (invited)

Phase retrieval algorithms: principles, developments and applications (invited)

Three-dimensional nano-coherent diffraction imaging technology based on high order harmonic X-ray sources

Coherent diffractive imaging (CDI) using ultra-short wavelength light source has become an three-dimensional(3D) nanoimaging technique. In CDI, a target sample is first illuminated by a coherent EUV and soft X-ray light, then the diffraction pattern is recorded by using a charge coupled device (CCD), and finally the image of the sample is obtained based on the pattern by using a phase retrieval algorithm. Of the many currently available coherent EUV and soft X-ray light sources, the high-order harmonic generation (HHG) is the simplest in structure, the lowest in cost, and most compact in size. Therefore, it has become the most promising light source for CDI. Through years of development, HHG based CDI technique(HHG-CDI) has become an outstanding 3D nano-imaging technique with the advantages of no aberration, no damage, and no contact either, and it also possesses the extra-capabilities of probing the dynamics, chemical composition and quantum information in various semiconductor and quantum devices. We believe that the HHG-CDI will soon become a generic nano-imaging tool that can complement or even replace the matured nanoimaging techniques, such as atomic force, near field, X-ray, electron, or scanning tunneling microscopes.

Sparse signal recovery from phaseless measurements via hard thresholding pursuit

In this paper, we consider the sparse phase retrieval problem, recovering an $s$-sparse signal $\bm{x}^{\natural}\in\mathbb{R}^n$ from $m$ phaseless samples $y_i=|\langle\bm{x}^{\natural},\bm{a}_i\rangle|$ for $i=1,\ldots,m$. Existing sparse phase retrieval algorithms are usually first-order and hence converge at most linearly. Inspired by the hard thresholding pursuit (HTP) algorithm in compressed sensing, we propose an efficient second-order algorithm for sparse phase retrieval. Our proposed algorithm is theoretically guaranteed to give an exact sparse signal recovery in finite (in particular, at most $O(\log m + \log(\|\bm{x}^{\natural}\|_2/|x_{\min}^{\natural}|))$) steps, when $\{\bm{a}_i\}_{i=1}^{m}$ are i.i.d. standard Gaussian random vector with $m\sim O(s\log(n/s))$ and the initialization is in a neighborhood of the underlying sparse signal. Together with a spectral initialization, our algorithm is guaranteed to have an exact recovery from $O(s^2\log n)$ samples. Since the computational cost per iteration of our proposed algorithm is the same order as popular first-order algorithms, our algorithm is extremely efficient. Experimental results show that our algorithm can be several times faster than existing sparse phase retrieval algorithms.

ISAR Imaging Algorithm for Maneuvering Targets Based on Fourier Ptychographic Microscopy

Aiming at the problems of low resolution and poor noise suppression for maneuvering targets with traditional inverse synthetic aperture radar (ISAR) imaging algorithms, we propose a motion compensation-driven high-resolution imaging algorithm based on Fourier ptychographic microscopy (FPM) technology. FPM is a phase retrieval technology that uses the concept of synthetic aperture to obtain the high-resolution image of the target. The process of FPM imaging is composed of the enhancement of the known target information in the spatial domain and some fixed restricted alternating iteration in the frequency domain. This article analyzes the mathematical model of maneuvering target echo and then studies the rationality of applying the FPM phase retrieval algorithm to ISAR maneuvering target imaging. The proposed algorithm mainly includes two steps. First, a series of low-resolution ISAR images are generated by sampling echo data through an aperture function. Second, the FPM phase retrieval technology is applied to splice and reconstruct the spectrum of the low-resolution images to achieve the focused ISAR image. The effectiveness of the proposed algorithm is verified by using multiple groups of simulation data and real radar data.

Off-axis digital holographic decarrier phase recovery algorithm combined with linear regression

Benefitting from the high measurement efficiency, off-axis digital holography (DH) has become a most powerful DH technique for fast and high-accuracy measurement. Owing to the carrier frequency, the real image can be isolated easily in the Fourier spectrum of one off-axis hologram, so that the Fourier transform algorithm (FTA) is the most widely used algorithm for off-axis DH to realize the phase retrieval. In the FTA, one of the most important tasks is to figure out the accurate peak position of the real image and then shift the real image to the center of spectrum to remove the carrier. However, owing to the digitalization of the hologram, the peak position of the real spectrum is always not located at an integral pixel position in the practical applications, resulting in carrier residuals, thereby lowering the retrieval quality. Much work on accurately determining the peak position has been conducted to suppress the carrier residuals, such as by using the spectrum centroid method and zero padding. However, those estimation algorithms can achieve only satisfied accuracy in some situations. Then, spatial carrier phase shift (SCPS) is utilized to expand the utilization of space-bandwidth and avoid the spectrum leakage caused by band-pass filtering. The SCPS decomposes one off-axis hologram into several sub-holograms, in which the carrier induces the phase shifts between sub-holograms. Many on-axis phase retrieval algorithms are combined with SCPS to retrieve the phase from one off-axis hologram. However, the retrieved phase is usually composed of the sample phase and the carrier, so the accurate carrier information is also required to remove the carrier and obtain the correct reconstructed phase. In this paper, an accurate phase retrieval with carrier removal from single off-axis hologram by using the linear regression is proposed to achieve the simultaneous phase retrieval and carrier removal. In this method, four phase-shifted sub-holograms are extracted first from one off-axis hologram by SCPS. Since the phase shift between sub-holograms is linearly proportional to the carrier, the linear regression can be combined with least-square method to retrieve the phase and carrier simultaneously. Both the simulation and experimental results show that the proposed method can determine the carrier accurately and obtain correct phase without carrier. We believe that this proposed method can be applied to practical measurement.

Simultaneous intensity projection for projector response computation with application to an active radiometric calibration

One of the most important sources of errors in digital fringe projection (DFP) systems is the nonlinearity in the response of the projector when it uses the three-step phase retrieval algorithm. Thus, it is necessary to increase the accuracy without affecting the efficiency. In this sense, the radiometric rectification methods are used. In this paper, an active radiometric rectification method for digital fringe projection is proposed. This proposal consists in two improvements of traditional active techniques: first, parallel intensity projection is used to obtain the projector response which requires only four dot patterns; and second, a method is provided for the calculation of the inverse polynomial that guarantees symmetry with respect to the response of the projector. Experimental results, in a digital fringe projection system, show that the root-mean-square phase error improves 6.3 times using this proposal.

Compressed optical image encryption in the diffractive-imaging-based scheme by input plane and output plane random sampling

The successful recovery of the plaintext in the simplified diffractive-imaging-based encryption (S-DIBE) scheme needs to record one intact axial intensity map as the ciphertext. By aid of compressive sensing, we propose here a new image encryption approach, referred to as compressed DIBE (C-DIBE), which allows further compression of the intensity map. The plaintext is sampled before being sent to DIBE. Afterwards, the intensity map recorded by the CCD camera is also processed by such sampling operation to generate the ciphertext. For decryption, we first obtain the sparse plaintext using the proposed phase retrieval algorithm, and then reobtain the primary plaintext from it via compressive sensing. Numerical results show that a proper proportion of the intensity map (e.g. 50%) is enough to totally recover a grayscale image. We achieve multiple-image encryption by space multiplexing without enlarging the size of the ciphertext. The robustness of C-DIBE against brute-force attack evidently outperforms S-DIBE due to the extended key space. Numerical simulation has been presented to confirm the proposal.

Poisson Phase Retrieval in Very Low-count Regimes.

This paper discusses phase retrieval algorithms for maximum likelihood (ML) estimation from measurements following independent Poisson distributions in very low-count regimes, e.g., 0.25 photon per pixel. To maximize the log-likelihood of the Poisson ML model, we propose a modified Wirtinger flow (WF) algorithm using a step size based on the observed Fisher information. This approach eliminates all parameter tuning except the number of iterations. We also propose a novel curvature for majorize-minimize (MM) algorithms with a quadratic majorizer. We show theoretically that our proposed curvature is sharper than the curvature derived from the supremum of the second derivative of the Poisson ML cost function. We compare the proposed algorithms (WF, MM) with existing optimization methods, including WF using other step-size schemes, quasi-Newton methods such as LBFGS and alternating direction method of multipliers (ADMM) algorithms, under a variety of experimental settings. Simulation experiments with a random Gaussian matrix, a canonical DFT matrix, a masked DFT matrix and an empirical transmission matrix demonstrate the following. 1) As expected, algorithms based on the Poisson ML model consistently produce higher quality reconstructions than algorithms derived from Gaussian noise ML models when applied to low-count data. Furthermore, incorporating regularizers, such as corner-rounded anisotropic total variation (TV) that exploit the assumed properties of the latent image, can further improve the reconstruction quality. 2) For unregularized cases, our proposed WF algorithm with Fisher information for step size converges faster (in terms of cost function and PSNR vs. time) than other WF methods, e.g., WF with empirical step size, backtracking line search, and optimal step size for the Gaussian noise model; it also converges faster than the LBFGS quasi-Newton method. 3) In regularized cases, our proposed WF algorithm converges faster than WF with backtracking line search, LBFGS, MM and ADMM.

PGPAL: A Monotonic Iterative Algorithm for Phase-Retrieval Under the Presence of Poisson-Gaussian Noise

In this paper, we propose an iterative algorithm named PGPAL for the phase-retrieval problem under the scenario when the measurements follow Poisson plus Gaussian (PG) distribution, which is a more realistic model in applications like astronomy, microscopy, medical imaging, and remote sensing. The proposed algorithm is based on majorization-minimization (MM) framework, in which a simple surrogate function that tightly upperbounds the challenging underlying maximum-likelihood (ML) objective is constructed and minimized iteratively. The proposed algorithm monotonically decreases the ML objective and the algorithm is guaranteed to converge to a stationary point of the ML objective. Numerical simulations are performed for the one-dimensional phase-retrieval case, and the performance of the proposed method is compared with the recently proposed Poisson phase-retrieval algorithm, which assumes the measurement model to be only Poisson distributed. The performance of the proposed MM-based algorithm in terms of the normalized root mean square error (NRMSE) is found to be much better in comparison to the method for Poisson phase-retrieval.