Accuracy Of Phase Retrieval Research Articles

Using convex optimization of autocorrelation with constrained support and windowing for improved phase retrieval accuracy.

In imaging modalities recording diffraction data, such as the imaging of viruses at X-ray free electron laser facilities, the original image can be reconstructed assuming known phases. When phases are unknown, oversampling and a constraint on the support region in the original object can be used to solve a non-convex optimization problem using iterative alternating-projection methods. Such schemes are ill-suited for finding the optimum solution for sparse data, since the recorded pattern does not correspond exactly to the original wave function. Different iteration starting points can give rise to different solutions. We construct a convex optimization problem, where the only local optimum is also the global optimum. This is achieved using a modified support constraint and a maximum-likelihood treatment of the recorded data as a sample from the underlying wave function. This relaxed problem is solved in order to provide a new set of most probable "healed" signal intensities, without sparseness and missing data. For these new intensities, it should be possible to satisfy the support constraint and intensity constraint exactly, without conflicts between them. By making both constraints satisfiable, traditional phase retrieval with superior results is made possible. On simulated data, we demonstrate the benefits of our approach visually, and quantify the improvement in terms of the crystallographic R factor for the recovered scalar amplitudes relative to true simulations from .405 to .097, as well as the mean-squared error in the reconstructed image from .233 to .139. We also compare our approach, with regards to theory and simulation results, to other approaches for healing as well as noise-tolerant phase retrieval. These tests indicate that the COACS pre-processing allows for best-in-class results.

Dual-channel simultaneous spatial and temporal polarization phase-shifting interferometry.

Without the limitations of fringe number and fringe shape, a dual-channel simultaneous spatial and temporal polarization (DC-SSTP) phase-shifting interferometry system is proposed to achieve rapid and accurate phase retrieval through only one-time phase-shifting procedure with unknown phase shifts. First, an arbitrary phase shifts is simultaneously introduced into two channels of DC-SSTP system by a spatial light modulator (SLM). Second, by performing the subtraction operation between each pair of phase-shifting interferograms captured in the same channel, the background deduction of interferogram can be achieved easily, so the accurate phase can be retrieved rapidly. Especially, it is found even if the fringe number in interferogram is less than one, the proposed DC-SSTP method still reveals high accuracy of phase retrieval. Both the simulation and experimental results demonstrate the outstanding performance of proposed DC-SSTP method in phase measurement.

High-accuracy wavefront sensing by phase diversity technique with bisymmetric defocuses diversity phase

We investigate a specific diversity phase for phase diversity (PD) phase retrieval, which possesses higher accuracy than common PD, especially for large-scale and high-frequency wavefront sensing. The commonly used PD algorithm employs the image intensities of the focused plane and one defocused plane to build the error metric. Unlike the commonly used PD, we explore a bisymmetric defocuses diversity phase, which employs the image intensities of two symmetrical defocused planes to build the error metric. This kind of diversity phase, named PD-BD (bisymmetric defocuses phase diversity), is analysed with the Cramer-Rao lower bound (CRLB). Statistically, PD-BD shows smaller CRLBs than the commonly used PD, which indicates stronger capacity of phase retrieval. Numerical simulations also verify that PD-BD has higher accuracy of phase retrieval than the commonly used PD when dealing with large-scale and high-frequency wavefront aberrations. To further affirm that PD-BD possesses higher accuracy of wavefront sensing than PD, we also perform a simple verification experiment.

Spatial dual-orthogonal (SDO) phase-shifting algorithm by pre-recomposing the interference fringe.

In the case that the phase distribution of interferogram is nonuniform and the background/modulation amplitude change rapidly, the current self-calibration algorithms with better performance like principal components analysis (PCA) and advanced iterative algorithm (AIA) cannot work well. In this study, from three or more phase-shifting interferograms with unknown phase-shifts, we propose a spatial dual-orthogonal (SDO) phase-shifting algorithm with high accuracy through using the spatial orthogonal property of interference fringe, in which a new sequence of fringe patterns with uniform phase distribution can be constructed by pre-recomposing original interferograms to determine their corresponding optimum combination coefficients, which are directly related with the phase shifts. Both simulation and experimental results show that using the proposed SDO algorithm, we can achieve accurate phase from the phase-shifting interferograms with nonuniform phase distribution, non-constant background and arbitrary phase shifts. Specially, it is found that the accuracy of phase retrieval with the proposed SDO algorithm is insensitive to the variation of fringe pattern, and this will supply a guarantee for high accuracy phase measurement and application.

Enhanced two-frequency phase-shifting method based on generalized phase-shifting algorithm

In this paper, we develop a novel dual-frequency pattern which not only releases the restriction of conventional two-frequency phase-shifting algorithm that at least six fringe patterns are needed, but reduces the noise impact by decreasing the frequency ratio between the high- and low-components. To decrease the number of necessary patterns to five, a novel composite dual-frequency pattern scheme combining together a high- with a low-frequency pattern is employed. To make the algorithm less sensitivity to noise, the low-frequency component is with more than one period fringes, which is relatively prone to recover the continuous result by simple spatial phase unwrapping approach. Besides, the restriction of special phase shifts between two-frequency components in conventional methods is released by the generalized phase-shifting algorithm. Simulations and experiments are conducted to demonstrate that in addition to maintaining the minimum number of patterns, the proposed method reveals higher accuracy of phase retrieval.

An advanced phase retrieval algorithm in N-step phase-shifting interferometry with unknown phase shifts

In phase-shifting interferometry with unknown phase shifts, a normalization and orthogonalization phase-shifting algorithm (NOPSA) is proposed to achieve phase retrieval. The background of interferogram is eliminated through using the orthogonality of complex sinusoidal function; and the influence of phase shifts deviation on accuracy of phase retrieval is avoided through both normalization and orthogonalization processing. Compared with the current algorithms with unknown phase shifts, the proposed algorithm reveals significantly faster computation speed, higher accuracy, better stability and non-sensitivity of phase shifts deviation.

Performance analysis of quantitative phase retrieval method in Zernike phase contrast X-ray microscopy*Supported by the State Key Project for Fundamental Research (2012CB825801), National Natural Science Foundation of China (11475170, 11205157 and 11179004) and Anhui Provincial Natural Science Foundation (1508085MA20).

Since the invention of Zernike phase contrast method in 1930, it has been widely used in optical microscopy and more recently in X-ray microscopy. Considering the image contrast is a mixture of absorption and phase information, we recently have proposed and demonstrated a method for quantitative phase retrieval in Zernike phase contrast X-ray microscopy. In this contribution, we analyze the performance of this method at different photon energies. Intensity images of PMMA samples are simulated at 2.5 keV and 6.2 keV, respectively, and phase retrieval is performed using the proposed method. The results demonstrate that the proposed phase retrieval method is applicable over a wide energy range. For weakly absorbing features, the optimal photon energy is 2.5 keV, from the point of view of image contrast and accuracy of phase retrieval. On the other hand, in the case of strong absorption objects, a higher photon energy is preferred to reduce the error of phase retrieval. These results can be used as guidelines to perform quantitative phase retrieval in Zernike phase contrast X-ray microscopy with the proposed method.

Overcoming the limitation of phase retrieval using Gerchberg-Saxton-like algorithm in optical fiber time-stretch systems.

We investigate the fundamental limitation of the full-field retrieval of optical pulses based on a time-equivalent Gerchberg-Saxton (GS)-like algorithm, in which the Fourier transformation of the temporal signal is performed by the group velocity dispersion (GVD) of optical fibers. The insufficient knowledge of the input pulse intensity fundamentally limited by the bandwidth of the photodetector influences the accuracy of the retrieved phase using the algorithm. To this end, we propose a modified GS algorithm, which involves multi-step intensity-only measurements, to mitigate this limitation. By studying the influence of spectral phase and amplitude modulations on the retrieval error, we show that the modified GS algorithm demonstrates a significant improvement in phase retrieval accuracy. More importantly, we anticipate the present study will provide valuable insight in understanding how the spectral variation generally influences the recovery accuracy in time-stretch-based GS algorithm, and thus identify its potential applications, particularly for those requiring ultrafast measurements.

Iterative projection approach for phase retrieval of semi-sparse wave field

Abstract In the paper, we consider the problem of two-dimensional (2D) phase retrieval, which recovers a 2D complex-valued wave field from magnitudes of both wave field and its Fourier transform. Due to the absence of the phase measurements, prior information on wave field is needed in order to recover phase, which is feasible when the phases of the wave field are sparse. In this paper, we improve the phase retrieval accuracy by incorporating phase sparse constraint of wave field. As a sequel to previous iterative projection approaches, iterative projection approaches with phase sparse constraint are realized based on ‘soft thresholding’. It has superior performances in terms of convergence, residual error, noise stability, and suitability in large-scale phase retrieval problems. Numerical experiments illustrate that the proposed approach outperforms existing iterative projection approaches.

Non-linear regularized phase retrieval for unidirectional X-ray differential phase contrast radiography

Phase retrieval from unidirectional radiographic differential phase contrast images requires integration of noisy data. A method is presented, which aims to suppress stripe artifacts arising from direct image integration. It is purely algorithmic and therefore, compared to alternative approaches, neither additional alignment nor an increased scan time is required. We report on the theory of this method and present results using numerical as well as experimental data. The method shows significant improvements on the phase retrieval accuracy and enhances contrast in the phase image. Due to its general applicability, the proposed method provides a valuable tool for various 2D imaging applications using differential data.

Transport of intensity phase imaging from multiple intensities measured in unequally-spaced planes

A method based on the transport of intensity equation (TIE) for phase retrieval is presented, which can retrieve the optical phase from intensity measurements in multiple unequally-spaced planes in the near-field region. In this method, the intensity derivative in the TIE is represented by a linear combination of intensity measurements, and the coefficient of the combination can be expressed by explicitly analytical form related to the defocused distances. The proposed formula is a generalization of the TIE with high order intensity derivatives. The numerical experiments demonstrate that the proposed method can improve the accuracy of phase retrieval with higher-order intensity derivatives and is more convenient for practical application.

Wavefield characterization of nearly diffraction-limited focused hard x-ray beam with size less than 10 nm

In situ wavefront compensation is a promising method to realize a focus size of only a few nanometers for x-ray beams. However, precise compensation requires evaluation of the wavefront with an accuracy much shorter than the wavelength. Here, we characterized a one-dimensionally focused beam with a width of 7 nm at 20 keV using a multilayer mirror. We demonstrate that the wavefront can be determined precisely from multiple intensity profiles measured around the beamwaist. We compare the phase profiles recovered from intensity profiles measured under the same mirror condition but with three different aperture sizes and find that the accuracy of phase retrieval is as small as λ∕12.

Transport of Intensity imaging with higher order derivatives

We demonstrate a method for improving the accuracy of phase retrieval based on the Transport of Intensity equation by using intensity measurements at multiple planes to estimate and remove the artifacts due to higher order axial derivatives. We suggest two similar methods of higher order correction, and demonstrate their ability for accurate phase retrieval well beyond the 'linear' range of defocus that TIE imaging traditionally requires. Computation is fast and efficient, and sensitivity to noise is reduced by using many images.

Window size selection in windowed Fourier transform for phase retrieval

Windowed Fourier transform (WFT) has been proved to be one of the most effective algorithms for extracting phase from various optical fringe patterns. The accuracy of phase retrieval using WFT depends on the preset window size, but how to determine an appropriate window size is still a problem. In this paper, a mathematical relationship between the optimal window size and the phase derivative is deduced and several advices on window size selection in practical applications are proposed. The correctness of the proposed window size selection criterions is demonstrated using both numeral simulations and real experiments. Comparison of phase values retrieved by conventional approaches and WFT shows that the WFT with an optimal window size can give more precise results.

An improved windowed Fourier transform for fringe demodulation

A modified algorithm of windowed Fourier transform (WFT) for phase retrieval from electronic speckle-shearing fringe patterns with carriers is proposed. The algorithm is based on the introduction of a fast Fourier transform (FFT) in WFT to reduce computation time for fringe demodulation. Since boundary effects in FFT will influence the accuracy of phase retrieval, the Gerchberg method is employed to extrapolate the fringe pattern at the boundaries to reduce boundary effects. A theoretical analysis of the algorithm is presented. Both simulated and experimental results show that the proposed method has reduced the computation time significantly compared with the convolution method of WFT without sacrificing measurement accuracy.

Tilt-shift error detection in phase-shifting interferometry

Temporal phase shifting in automatic interferogram analysis offers very high accuracy of phase retrieval providing that several experimental conditions are met. The paper is focused on the calibration error of unequal phase changes across the interferogram field, i.e., tilt-shift error. For its detection the lattice-site representation of phase shift angles is proposed. The error can be readily discerned using (N+1) algorithms with the last frame overlapping the first one. Four and five frame algorithms are considered. The influence of experimental parameters on the error detection sensitivity is discussed. Numerical studies are complemented by experimental results.

Remote sensing of cloud properties using MODIS airborne simulator imagery during SUCCESS: 2. Cloud thermodynamic phase

Methods to infer cloud thermodynamic phase (ice or water) are investigated using multispectral imagery. An infrared (IR) trispectral algorithm using the 8.52‐, 11‐, and 12‐μm bands [Ackerman et al., 1990; Strabala et al., 1994] forms the basis of this work and will be applied to data from the Earth Observing System (EOS) Moderate‐Resolution Imaging Spectroradiometer (MODIS) instrument. Since the algorithm uses IR bands, it can be applied to either daytime or nighttime data and is not sensitive to the presence of cloud shadows. A case study analysis is performed with a MODIS airborne simulator (MAS) scene collected during the Subsonic Aircraft Contrail and Cloud Effects Special Study (SUCCESS) on April 21, 1996, at 2000 UTC. The scene under scrutiny is quite complex, containing low‐level broken water clouds, cirrus cells, cloud shadows, sub visual cirrus, and contrails. The IR trispectral algorithm is found to be less accurate in (1) regions where more than one cloud phase type occurs and (2) regions of thin cirrus overlying a lower‐level cloud layer. To improve the phase retrieval accuracy in these areas of difficulty, additional bands located at 0.65, 1.63, and 1.90 μm are incorporated. Radiative transfer (RT) calculations are performed to simulate the MAS bands using the cirrus and water cloud models detailed by Baum et al. [this issue]. The RT calculations are performed for single‐layer cirrus‐ and water‐phase clouds as well as for the case of thin cirrus overlying a lower‐level water droplet cloud. Both modeled results and application of the theory to a case study suggest that the cloud thermodynamic phase retrieval accuracy can be improved by inclusion of the visible and near infrared bands.

Shop-floor laser interferometer

In order to reduce the influence of nonlinear factors such as gamma nonlinearity and color crosstalk on the phase retrieval accuracy with color fringe projection profilometry (CFPP) in rapid measurement, a single-shot high-precision backpropagation (BP) neural network based CFPP is proposed. Firstly, neural networks are trained from samples through a training process. The neural network inputs are the RGB components of composite color fringe patterns. The numerator and denominator of the arctangent function are used as network outputs. The monochromatic fringe patterns are adopted. The numerator and denominator of the arctangent function are calculated by the 15-step phase-shift method, which serves as the network training target. Applying this neural network to the prediction of the standard phase can reduce the influence of nonlinear factors on the phase accuracy. Then, the absolute phase was unwrapped by using geometric constraints. Finally, the 3D reconstruction with high precision can be realized by using only a single-shot color fringe. The theoretical model of the proposed method and the process of neural network training are introduced. The results of 3D reconstruction are compared with other algorithms in the same condition. The experimental results demonstrate that the RMS of the proposed algorithm can reach 0.0107 mm, compared with 0.0121 mm of the traditional 3-step phase-shift method. The proposed method is applied to the online quality inspection of relay components, and the results verify the high accuracy and high efficiency of measurement.