Phase Retrieval Method Research Articles

An effective assessment method for absolute phase retrieval in digital fringe projection profilometry

In this paper, we propose an effective method for assessing absolute phase retrieval quality in digital fringe projection profilometry (DFPP). The method provides a quantitative analysis of the phase retrieval quality with a new integrated quality metric. The phase map of the measured object is pre-defined on the imaging plane and called the inverse ground truth. To obtain the measurement result of the inverse ground truth, object-adapted patterns are projected onto the object and the phase map of the object is retrieved by a phase retrieval technique. Subsequently, the retrieved phase map of the object is compared with the inverse ground truth to assess the phase retrieval quality. The structural similarity between the measurement result and the inverse ground truth is quantified by the proposed quality metric in terms of mean, contrast and structure comparisons. Experiments are carried out to evaluate the effectiveness of the proposed method from two aspects: comparing the accuracies of different phase retrieval techniques and evaluating the influence of different setup parameters on the accuracy. Experimental results show that the analysis by the proposed quality metric is well consistent with the existing findings for the phase retrieval quality in DFPP.

Assessment on the deviation from an ideal Gaussian beam for a real laser beam by Fresnel–Huygens phase-retrieval method

The deviation of a real laser beam from an ideal Gaussian beam can be ascribed either to the non-Gaussianity of the intensity profile or to the non-planarity of the wave front at the beam waist. We demonstrate a method to differentiate these two causes. The contribution of the phase distortion at the beam waist to the deviation from an ideal Gaussian beam can be assessed from the change in modal spectrum when the wave front at the beam waist is replaced by a flat plane. The crucial information about the phase-profile function of the optical field at the beam waist for mode decomposition was calculated from two intensity profiles for different cross-sections around the beam waist using a convergent iterative algorithm based on the Fresnel–Huygens principle. Using a HeNe laser with its beam close to an ideal Gaussian beam for test, we found that both effects have comparable contributions to the deviation from an ideal Gaussian beam.

Adaptive weighted total variation regularized phase retrieval in differential phase-contrast imaging

Phase retrieval with unidirectional differential phase-contrast image requires integration with noisy data, which is an illposed inverse problem. Conventional direct integration method would result in severe streak artifacts. Total variation (TV) regularization-based method would reduce the streak artifacts, but the edges parallel with phase-contrast sensitivity direction are likely to be over smoothed. We propose an improved weighted TV regularization phase retrieval method by introducing a weighting factor to the conventional TV term. When applied to simulation and experimental data, this method shows an advantage of preserving the sharpness of the edges while preserving the ability of reducing streak artifacts compared with conventional TV-regularization method.

Constrained phase retrieval: when alternating projection meets regularization

Simplicity and few finely tuned parameters are the main advantages of alternating projection (AP) methods, a fundamental class of phase retrieval (PR) methods in the optical imaging field. However, AP methods often suffer from low-quality imaging when few diffraction patterns are recorded. Regularized PR methods avoid this deficiency by using some proper regularization models, but many finely tuned parameters are needed. In this work, we propose a novel unified framework called constrained PR (ConPR), which brings the AP method and the regularization approach together. The proposed ConPR framework not only can recover high-quality images from few diffraction patterns, but also does not need fine-tuning of the parameters. Our proposed generalized constrained PR optimization model consists of a relation function term, a regularization term, and a measurement constraint. The measurement constraint ensures that the recovered image matches with the measurement, and the regularization term can impose some desirable properties on the recovered image. The relation function promotes the approximation of the two underlying variables. The sparsity under the block-matching and 3D filtering frame is incorporated into the proposed ConPR framework. The problem formulation consists of an image updating sub-problem and a constrained optimization sub-problem. The epigraph set of the data fidelity function is defined, and the constrained optimization sub-problem is solved via the epigraph concept. Diffraction imaging from one noisy coded diffraction pattern demonstrates the effectiveness of the proposed algorithm.

A fast-converging iterative method based on weighted feedback for multi-distance phase retrieval

Multiple distance phase retrieval methods hold great promise for imaging and measurement due to their less expensive and compact setup. As one of their implementations, the amplitude-phase retrieval algorithm (APR) can achieve stable and high-accuracy reconstruction. However, it suffers from the slow convergence and the stagnant issue. Here we propose an iterative modality named as weighted feedback to solve this problem. With the plug-ins of single and double feedback, two augmented approaches, i.e. the APRSF and APRDF algorithms, are demonstrated to increase the convergence speed with a factor of two and three in experiments. Furthermore, the APRDF algorithm can extend the multiple distance phase retrieval to the partially coherent illumination and enhance the imaging contrast of both amplitude and phase, which actually relaxes the light source requirement. Thus the weighted feedback enables a fast-converging and high-contrast imaging scheme for the iterative phase retrieval.

Direct determination of one-dimensional interphase structures using normalized crystal truncation rod analysis

A one-dimensional non-iterative direct method was employed for normalized crystal truncation rod analysis. The non-iterative approach, utilizing the Kramers–Kronig relation, avoids the ambiguities due to an improper initial model or incomplete convergence in the conventional iterative methods. The validity and limitations of the present method are demonstrated through both numerical simulations and experiments with Pt(111) in a 0.1 M CsF aqueous solution. The present method is compared with conventional iterative phase-retrieval methods.

Single shot multi-wavelength phase retrieval with coherent modulation imaging.

A single shot multi-wavelength phase retrieval method is proposed by combining common coherent modulation imaging (CMI) and a low rank mixed-state algorithm together. A radiation beam consisting of multi-wavelength is illuminated on the sample to be observed, and the exiting field is incident on a random phase plate to form speckle patterns, which is the incoherent superposition of diffraction patterns of each wavelength. The exiting complex amplitude of the sample including both the modulus and phase of each wavelength can be reconstructed simultaneously from the recorded diffraction intensity using a low rank mixed-state algorithm. The feasibility of this proposed method was verified with visible light experimentally. This proposed method not only makes CMI realizable with partially coherent illumination but also can extend its application to various traditionally unrelated fields, where several wavelengths should be considered simultaneously.

Phase retrieval via randomized Kaczmarz: theoretical guarantees

Abstract We consider the problem of phase retrieval, i.e. that of solving systems of quadratic equations. A simple variant of the randomized Kaczmarz method was recently proposed for phase retrieval, and it was shown numerically to have a computational edge over state-of-the-art Wirtinger flow methods. In this paper, we provide the first theoretical guarantee for the convergence of the randomized Kaczmarz method for phase retrieval. We show that it is sufficient to have as many Gaussian measurements as the dimension, up to a constant factor. Along the way, we introduce a sufficient condition on measurement sets for which the randomized Kaczmarz method is guaranteed to work. We show that Gaussian sampling vectors satisfy this property with high probability; this is proved using a chaining argument coupled with bounds on Vapnik–Chervonenkis (VC) dimension and metric entropy.

Advanced Fourier transform analysis method for phase retrieval from a single-shot spatial carrier fringe pattern

An advanced Fourier transform analysis (AFTA) method is proposed in this study for phase retrieval from a single-shot spatial carrier fringe pattern (SCFP). It firstly extracts four phase-shifted fringe patterns (FP) from the SCFP with one pixel malposition, and then performs Fourier transform to each of them for their Fourier spectrums. The innovation of AFTA is making a subtraction in frequency domain directly to calculate the spectrum difference, which can generally eliminate the direct current component, thus it could mitigate the spectrum leakage problem and the edge error of traditional Fourier transform analysis (FTA) method significantly. After that, by making an inverse Fourier transform to the spectrum difference and a simple average operation, the phase information could be reconstructed conveniently. The time cost is approximately twice of FTA, which is still highly efficient and time-saving for dynamic or real-time measurement. AFTA is adequately validated as a promising method in fringe analysis by various simulations and comparative experiments, with emphasis of this study on its performance in terms of accuracy, noise-tolerance, carrier frequency, time cost, etc.

Non-iterative method for phase retrieval and coherence characterization by focus variation using a fixed star-shaped mask

A novel non-iterative phase retrieval method is proposed and demonstrated with a proof-of-principle experiment. The method uses a fixed specially designed mask and through-focus intensity measurements. It is demonstrated that this method is robust to spatial partial coherence in the illumination, making it suitable for coherent diffractive imaging using spatially partially coherent light, as well as for coherence characterization.

Phase retrieval using through-focus images in Lorentz transmission electron microscopy.

In this study, we report on phase retrieval by the maximum likelihood method with conjugate gradient method (CG-MAL) using through-focus images in Lorentz transmission electron microscopy (LTEM). The method was evaluated using 32 simulated and experimentally obtained through-focus images of magnetic bubbles; these images were collected under a defocus range from approximately -3 mm to 3 mm. Consequently, we obtained the magnetic domain structures of the magnetic bubbles in both the simulation and the LTEM experiment. Furthermore, the CG-MAL method showed better convergence behavior than other iterative phase retrieval methods. Therefore, the method can also be widely and effectively applied to the observation of magnetic domain structures other than magnetic bubbles when highly defocused through-focus images are used.

Sparse recovery of undersampled intensity patterns for coherent diffraction imaging at high X-ray energies

Coherent X-ray photons with energies higher than 50 keV offer new possibilities for imaging nanoscale lattice distortions in bulk crystalline materials using Bragg peak phase retrieval methods. However, the compression of reciprocal space at high energies typically results in poorly resolved fringes on an area detector, rendering the diffraction data unsuitable for the three-dimensional reconstruction of compact crystals. To address this problem, we propose a method by which to recover fine fringe detail in the scattered intensity. This recovery is achieved in two steps: multiple undersampled measurements are made by in-plane sub-pixel motion of the area detector, then this data set is passed to a sparsity-based numerical solver that recovers fringe detail suitable for standard Bragg coherent diffraction imaging (BCDI) reconstruction methods of compact single crystals. The key insight of this paper is that sparsity in a BCDI data set can be enforced by recognising that the signal in the detector, though poorly resolved, is band-limited. This requires fewer in-plane detector translations for complete signal recovery, while adhering to information theory limits. We use simulated BCDI data sets to demonstrate the approach, outline our sparse recovery strategy, and comment on future opportunities.

A systematic analysis for the quantitative comparison of phase retrieval methods based on alternating projections

We present a numerical and experimental scheme for the systematic analysis and comparison of phase retrieval techniques based on alternating projection numerical methods. This comparison allows us to evaluate the most common and recently introduced phase retrieval methods. The proposed scheme gives a quantitative comparison that helps to elucidate the differences between them and develop proper technical implementations of phase retrieval. The comparison is made by means of a numerical and experimental scheme that allows us to evaluate phase retrieval experiments. In this work, the drawbacks of using arbitrary random initial seeds to support the phase retrieval numerical algorithms are also analyzed and discussed. Moreover, we show the convenience of using a rough object phase estimation, which is obtained by means of a simple holographic technique, as the initial seed. This seed dramatically reduces the computational load of the algorithms by decreasing the successive iterations from hundreds to less than twenty. The experimental object under study is a random phase object within a micro-channel. As a proof of concept, this micro-channel combined with a millimeter size semicircular hole, which provides a reference wave, conforms a primitive sensor. The performance of the algorithms is not only measured by the usual convergence error, but also by means of a quality index that requires a direct comparison against the generally unknown original phase object. Thus, in order to evaluate the experimental performance of the phase retrieval techniques, we implement an interferometric optical setup that allows us to compare the results obtained by both techniques. The experiment proposed is a valuable tool for quantitative experimental evaluation of phase retrieval techniques in the optical domain.

Model-based multi-fringe interferometry using Zernike polynomials

In this paper, a general phase retrieval method is proposed, which is based on one single interferogram with a small amount of fringes (either tilt or power). Zernike polynomials are used to characterize the phase to be measured; the phase distribution is reconstructed by a non-linear least squares method. Experiments show that the proposed method can obtain satisfactory results compared to the standard phase-shifting interferometry technique. Additionally, the retrace errors of proposed method can be neglected because of the few fringes; it does not need any auxiliary phase shifting facilities (low cost) and it is easy to implement without the process of phase unwrapping.

Phase modulated high density collinear holographic data storage system with phase-retrieval reference beam locking and orthogonal reference encoding.

A novel phase modulation method for holographic data storage with phase-retrieval reference beam locking is proposed and incorporated into an amplitude-encoding collinear holographic storage system. Unlike the conventional phase retrieval method, the proposed method locks the data page and the corresponding phase-retrieval interference beam together at the same location with a sequential recording process, which eliminates piezoelectric elements, phase shift arrays and extra interference beams, making the system more compact and phase retrieval easier. To evaluate our proposed phase modulation method, we recorded and then recovered data pages with multilevel phase modulation using two spatial light modulators experimentally. For 4-level, 8-level, and 16-level phase modulation, we achieved the bit error rate (BER) of 0.3%, 1.5% and 6.6% respectively. To further improve data storage density, an orthogonal reference encoding multiplexing method at the same position of medium is also proposed and validated experimentally. We increased the code rate of pure 3/16 amplitude encoding method from 0.5 up to 1.0 and 1.5 using 4-level and 8-level phase modulation respectively.

Substructure determination using phase-retrieval techniques.

Thus far, the application of phase-retrieval methods in crystallography has mainly been aimed at variants of charge flipping or structure-factor flipping. In this work, the relaxed averaged alternating reflections (RAAR) algorithm is applied to determine anomalously scattering substructures from single-wavelength anomalous diffraction (SAD) data of macromolecules. The algorithm has been implemented in a new program, PRASA, and has been shown to significantly outperform charge flipping in determining anomalously scattering substructures on a test sample of 169 SAD data sets with resolutions up to 3.88 Å.

Low-dose quantitative phase contrast medical CT

X-ray computed tomography (CT) is a powerful and routinely used clinical diagnostic technique, which is well tolerated by patients, and which provides high-resolution images and volumetric information about the body. However, two important limitations still affect this examination procedure: (1) its low sensitivity with respect to soft tissues, and (2) the hazards associated with x-ray exposure. Conventional radiology is based on the detection of the different photon absorption properties that characterize biological tissues, and thus the obtainable image contrast from soft and/or similar tissues is intrinsically limited. In this scenario, x-ray phase contrast imaging (XPCI) has been extensively tested and proven to overcome some of the main issues surrounding standard x-ray imaging. In addition to the absorption signal, XPCI relies on detecting the phase shifts induced by an object. Interestingly, as the order of magnitude of the phase contrast is higher than that of absorption, XPCI can, in principle, offer higher sensitivity at lower radiation doses. However, other technical aspects may counterbalance this gain, and an optimized setup and image processing solutions need to be implemented. The work presented here describes the strategies and developments we have realized, with the aim of controlling the radiation dose for the highly sensitive and quantitative XPCI-CT. Different algorithms for the phase retrieval and CT reconstruction of the XPCI data are presented. The CT algorithms we have implemented, namely the equally sloped tomography and the dictionary learning method, allow the image quality to be preserved while reducing the number of angular projections required by a factor of five. The results applied to breast imaging report accurate reconstructions at clinically compatible doses of the 3D distribution of the refractive properties of full human organs obtained by using three different phase retrieval methods. The described methodologies and the presented results have been validated by a team of clinical radiologists and represent an important step in the exploitation of XPCI-CT for in vivo and possible clinical applications.

Extracting attosecond delays from spectrally overlapping interferograms

Attosecond interferometry is becoming an increasingly popular technique for measuring the dynamics of photoionization in real time. Whereas early measurements focused on atomic systems with very simple photoelectron spectra, the technique is now being applied to more complex systems including isolated molecules and solids. The increase in complexity translates into an augmented spectral congestion, unavoidably resulting in spectral overlap in attosecond interferograms. Here, we discuss currently used methods for phase retrieval and introduce two new approaches for determining attosecond photoemission delays from spectrally overlapping photoelectron spectra. We show that the previously used technique, consisting in the spectral integration of the areas of interest, does in general not provide reliable results. Our methods resolve this problem, thereby opening the technique of attosecond interferometry to complex systems and fully exploiting its specific advantages in terms of spectral resolution compared to attosecond streaking.

Complex amplitude reconstruction by iterative amplitude-phase retrieval algorithm with reference

Multi-image iterative phase retrieval methods have been successfully applied in plenty of research fields due to their simple but efficient implementation. However, there is a mismatch between the measurement of the first long imaging distance and the sequential interval. In this paper, an amplitude-phase retrieval algorithm with reference is put forward without additional measurements or priori knowledge. It gets rid of measuring the first imaging distance. With a designed update formula, it significantly raises the convergence speed and the reconstruction fidelity, especially in phase retrieval. Its superiority over the original amplitude-phase retrieval (APR) method is validated by numerical analysis and experiments. Furthermore, it provides a conceptual design of a compact holographic image sensor, which can achieve numerical refocusing easily.

Talbot-Lau x-ray deflectometry phase-retrieval methods for electron density diagnostics in high-energy density experiments.

Talbot-Lau x-ray interferometry uses incoherent x-ray sources to measure refraction index changes in matter. These measurements can provide accurate electron density mapping through phase retrieval. An adaptation of the interferometer has been developed in order to meet the specific requirements of high-energy density experiments. This adaptation is known as a moiré deflectometer, which allows for single-shot capabilities in the form of interferometric fringe patterns. The moiré x-ray deflectometry technique requires a set of object and reference images in order to provide electron density maps, which can be costly in the high-energy density environment. In particular, synthetic reference phase images obtained ex situ through a phase-scan procedure, can provide a feasible solution. To test this procedure, an object phase map was retrieved from a single-shot moiré image obtained from a plasma-produced x-ray source. A reference phase map was then obtained from phase-stepping measurements using a continuous x-ray tube source in a small laboratory setting. The two phase maps were used to retrieve an electron density map. A comparison of the moiré and phase-stepping phase-retrieval methods was performed to evaluate single-exposure plasma electron density mapping for high-energy density and other transient plasma experiments. It was found that a combination of phase-retrieval methods can deliver accurate refraction angle mapping. Once x-ray backlighter quality is optimized, the ex situ method is expected to deliver electron density mapping with improved resolution. The steps necessary for improved diagnostic performance are discussed.