Phase Retrieval Method Research Articles

Euclidean matrix norm algorithm for simultaneous dual-wavelength phase-shifting interferometry with blind phase shifts

In this study, a simultaneous dual-wavelength phase-shifting interferometry system using two reference arms was built, and a simple phase retrieval method for the system with blind phase shifts was proposed using five phase-shifted interferograms. First, the unknown phase shifts were controlled independently to separate the information of single wavelength, which can be calculated directly by using the Euclidean matrix norm algorithm. Second, a wrapped phase of a single wavelength was obtained using a three-step phase-shifting algorithm. Finally, a continuous phase of a synthetic wavelength was obtained. The simulation and experimental results indicated the feasibility and greater accuracy of the proposed method. Moreover, the strict requirement of the phase shifter was eliminated because of the use of arbitrary and unknown phase shifts, thereby making experimental operation easier.

Сomparison of on-line and off-line Fried parameter estimation methods

We address the problem of high-accuracy estimation of the Fried parameter r0 by comparing two approaches based on estimation of statistical properties of intensity-based wavefront measurements with parameter-fitting to the theoretically predicted values. In the first approach, the phase of the aberration-degraded field is restored from the measurements to obtain the statistical estimate for the structure function. Due to the iterative nature of the most phase retrieval methods, this approach requires significant computational time and thus cannot provide results in real time. In the second approach, the structure function of the sub-aperture wavefront slopes is directly calculated and related to the turbulence parameters in real time. We describe the equations used to obtain the estimate of the Fried parameter by both methods and check their accuracy with numerical simulations.

Investigation of the tolerance of the phase retrieval algorithm to missing information at the center of a detector in the case of coherent scattering from an ordered structure

The hybrid input-output algorithm is a phase retrieval method that provides solution for the phase problem of coherent X-ray diffraction imaging of micro- and nano-objects from the diffraction pattern alone without using any focusing optics. In this paper, we have studied a tolerance of this algorithm to missing information at the center of the diffraction pattern, which is a frequent problem in problems of small-angle scattering. We considered the particular problem of the stability of the algorithm in the case of scattering from an ordered structure and provided a qualitative and quantitative description of the degradation of image reconstruction with an increase in the amount of missing information.

Suppression of zero-order term and twin-image in in-line digital holography by a single hologram

Suppressing zero-order term and twin-image is a crucial technique in in-line digital holography, which can obtain more accurate reconstructed images. A robust and effective algorithm to suppress zero-order term and twin-image in in-line digital holography is proposed in this paper, which combines a phase retrieval method and spatial spectrum filtering in a single hologram. In order to more accurately distinguish the effective information and noise information (zero-order term and twin-images), we propose a synthetic threshold condition of frequency domain and spatial domain in the iterative process. Several experiments have been carried out to validate the effectiveness of the proposed algorithm. The results demonstrate that the proposed algorithm not only has better visual effects, but also has a better accuracy.

Improved unwrapped phase retrieval method of a fringe projection profilometry system based on fewer phase-coding patterns.

In this paper, based on two additional phase-coding patterns, an improved phase demodulation method is proposed. First, six equally spaced coding phases in the interval [$ - \pi $-π, $\pi $π] are embedded in different periods of the coded fringes following a certain sequence. Subsequently, since a group of phase orders can be uniquely determined by the four adjacent coding phases, the phase-order map of the object can be generated. To ensure the accuracy of decoding results, the interference coding numbers should be corrected in advance. In the meantime, the connected regions exhibiting the same orders are classified and then labeled for simplifying the decoding process. The simulation results verify the feasibility of the proposed method. By two groups of 3D imaging experiments, the applicability of this method to multiple objects and discontinuous objects is confirmed.

Phase-contrast tomography from a series of single-shot measurements using a deterministic phase-retrieval method in X-ray imaging

An X-ray tomographic imaging method is proposed that is capable of reconstructing the complex refractive index of an object from a data set that involves only a single-shot image obtained at each tomographic view angle. This method is based on a deterministic phase retrieval method from a single diffraction intensity by use of an aperture array, and so the real and imaginary components, which are independent of each other, of the complex refractive index distribution can be reconstructed without object’s assumptions about negligible absorption or phase-attenuation duality, which have been frequently required in the other single-shot phase-retrieval methods for tomography. The validity of the method is shown in computer-simulated examples of three-dimensional object reconstruction.

Phase-retrieval method for measuring small contact angles of pentane on water.

Pentane drops on a water surface are predicted to have contact angles of the order of 1 degree or less in the phase of frustrated complete wetting. We have developed an optical method of measuring such small contact angles, applicable to cases where the refractive indices of the substrate and adsorbate are very similar and the fluid dynamics do not allow delay between image acquisitions, by using phase retrieval to map the surface profile of the drops. It is empirically shown that, with our method, a difference of nanometer order can be achieved for the phase-retrieved dimensions relative to their expected value. Results agree with numerical predictions by Weiss and Widom [Physica A292, 137 (2001)PHYADX0378-437110.1016/S0378-4371(00)00619-1].

A Gaussian mixture method for specific differential phase retrieval at X-band frequency

Abstract. The specific differential phase Kdp is one of the most important polarimetric radar variables, but the variance σ2(Kdp), regarding the errors in the calculation of the range derivative of the differential phase shift Φdp, is not well characterized due to the lack of a data generation model. This paper presents a probabilistic method based on the Gaussian mixture model for Kdp estimation at X-band frequency. The Gaussian mixture method can not only estimate the expected values of Kdp by differentiating the expected values of Φdp, but also obtain σ2(Kdp) from the product of the square of the first derivative of Kdp and the variance of Φdp. Additionally, the ambiguous phase and backscattering differential phase shift are corrected via the mixture model. The method is qualitatively evaluated with a convective event of a bow echo observed by the X-band dual-polarization radar in the University of Missouri. It is concluded that Kdp estimates are highly consistent with the gradients of Φdp in the leading edge of the bow echo, and large σ2(Kdp) occurs with high variation of Kdp. Furthermore, the performance is quantitatively assessed by 2-year radar–gauge data, and the results are compared to linear regression model. It is clear that Kdp-based rain amounts have good agreement with the rain gauge data, while the Gaussian mixture method gives improvements over the linear regression model, particularly for far ranges.

Inverse elastic scattering problems with phaseless far field data

This paper is concerned with uniqueness, phase retrieval and shape reconstruction methods for inverse elastic scattering problems with phaseless far field data. The phaseless far field data is closely related to the outward energy flux, which is easily measured in practice. Systematically, we study two basic models, i.e. inverse scattering of plane waves by rigid bodies and inverse scattering of sources with compact support. For both models, we show that the phaseless far field data is invariant under translation of the underlying scattering objects, which implies that the location of the objects can not be uniquely recovered by the data. To solve this problem, we consider simultaneously the incident point sources with one fixed source point and at most three scattering strengths. With this technique, we establish some uniqueness results for source scattering problem with multi-frequency phaseless far field data. Furthermore, a fast and stable phase retrieval approach is proposed based on a simple geometric result which provides a stable reconstruction of a point in the plane from three distances to given points. Difficulties arise for inverse scattering by rigid bodies due to the additional unknown far field pattern of the point sources. To overcome this difficulty, we introduce an artificial rigid body into the system and show that the underlying rigid bodies can be uniquely determined by the corresponding phaseless far field data at a fixed frequency. Noting that the far field pattern of the scattered field corresponding to point sources is very small if the source point is far away from the scatterers, we propose an appropriate phase retrieval method for obstacle scattering problems, without using the artificial rigid body. Finally, we propose several sampling methods for shape reconstruction with phaseless far field data directly. For inverse obstacle scattering problems, two different direct sampling methods are proposed with data at a fixed frequency. For inverse source scattering problems, we introduce two direct sampling methods for source supports with sparse multi-frequency data. The phase retrieval techniques are also combined with the classical sampling methods for the shape reconstructions. Extended numerical examples in two dimensions are conducted with noisy data, and the results further verify the effectiveness and robustness of the proposed phase retrieval techniques and sampling methods.

Generation of hollow beams by using phase filtering with multi-distance phase retrieval

We propose a method to generate beams according to transmission limitation condition by using the multi-distance phase retrieval algorithm. Propagation properties of hollow beams were determined by several intensity patterns along the travel axis. Predicted beams were obtained using a two-step phase filtering from a Gaussian beam with fundamental mode. An iterative phase retrieval method was employed for calculating the phase distribution in the filtering mask. This beam generation approach matched special propagation characteristics of the beams in practical applications.

Optical phase retrieval using four rotated versions of a single binary amplitude modulating mask

In recent years, phase retrieval methods recovering the phase of an object from coded diffraction patterns have gained popularity. A numerical phase retrieval method called PhaseLift that recovers the phase of an object from a very limited number of coded diffraction patterns was recently proposed. Performance of PhaseLift has been analyzed for different types and the number of masks modulating an object. We present a unique application of PhaseLift that uses four rotations of a single mask, modulating only the amplitude of an object. In simulations, a phase screen with the root-mean-square (RMS) value 0.294 μm was used as the test object. The RMS value of the retrieved phase screen after smoothing was 0.257 μm. In experiments, the RMS value of a wavefront measured with a Shack–Hartmann wavefront sensor was 0.094 while that of the retrieved wavefront after smoothing was 0.054 μm. While PhaseLift is able to recover a wavefront using this kind of modulation, a serious limitation to applicability of this method is its high computational cost and time.

Adaptive Subsurface 3-D Imaging Based on Peak Phase-Retrieval and Complex-Valued Self-Organizing Map

We propose an adaptive subsurface 3-D visualization system based on a complex-valued self-organizing map (CSOM). Conventionally buried things can be detected in the so-called B-scan images obtained by a ground penetrating radar. In contrast, our proposed method is able not only to detect their presence but also to classify the targets by the self-organizing dynamics in the CSOM. Instead of utilizing only the amplitude information in the time domain, we use both the amplitude and the phase information to obtain the scattering coefficients of scatterers by use of the phase retrieval method.

Hybrid method for accurate phase retrieval based on higher order transport of intensity equation and multiplane iteration

Because phase contains specific information of the sample, especially transparent specimens such as biological cells and tissues, it is of utmost significance to accurately extract the phase. Although the phase can be directly solved from multiplane intensities through the prevalent transport of intensity equation (TIE) method, too many approximations in the solving process ultimately reduce its recovery accuracy. In addition, an iterative idea as a classic but potential tool allows quantitatively reconstructing phase while it is prone to trap in local stagnation due to nonconvexity. In order to restore the phase quantitatively and precisely, we propose a hybrid method combining the initial phase value via higher order TIE for system error reduction, and a final recovery solution using multiplane iteration for phase accuracy enhancement. Simulation and experimental results show that the proposed method not only has good noise suppression ability but also high phase recovery accuracy, and thus can be used in fields such as biological observation and medical diagnosis in the future.

Accelerated single-beam multiple-intensity reconstruction using unordered propagations.

In conventional multiple-plane phase retrieval method, the wave propagations that proceed in the same ordered sequence could slow down convergence or lead to stagnation. In this Letter, a novel, to the best of our knowledge, algorithmic technique to accelerate phase retrieval using an unordered sequence of propagations is demonstrated experimentally. The main advantage of the technique is the significant increase in the change in amplitude, a key and essential element for a successful iterative phase retrieval. For N planes, the number of possible ways to implement the sequence of propagations is increased by a factor of N!(N-1)!, providing diversity for the change in the amplitude. Compared to the conventional algorithm, the technique performed 2 times faster and reduced the number of needed intensity patterns for the test object wave used. The technique may be adopted in the other multiple intensity-based phase retrieval methods.

An improved phase-coding method for absolute phase retrieval based on the path-following algorithm

Although the traditional phase-coding method for absolute phase retrieval possesses strong robustness, the method leads to unwrapping artifacts as a mass of codewords required. To solve this problem, a novel improved method is proposed in this paper. The incorrect points in the recovered phase via the traditional method are located using a computational framework, and the path-following algorithm is adopted as an aid to correct the unwrapping artifacts. The effectiveness of the proposed method is experimentally verified by an established measurement system. The reconstructed geometry of standard sphere has sufficient accuracy and the absolute phase retrieval of a complex star object has high quality.

Phase retrieval for Bragg coherent diffraction imaging at high x-ray energies

Coherent X-ray beams with energies $\geq 50$ keV can potentially enable three-dimensional imaging of atomic lattice distortion fields within individual crystallites in bulk polycrystalline materials through Bragg coherent diffraction imaging (BCDI). However, the undersampling of the diffraction signal due to Fourier space compression at high X-ray energies renders conventional phase retrieval algorithms unsuitable for three-dimensional reconstruction. To address this problem we utilize a phase retrieval method with a Fourier constraint specifically tailored for undersampled diffraction data measured with coarse-pitched detector pixels that bin the underlying signal. With our approach, we show that it is possible to reconstruct three-dimensional strained crystallites from an undersampled Bragg diffraction data set subject to pixel-area integration without having to physically upsample the diffraction signal. Using simulations and experimental results, we demonstrate that explicit modeling of Fourier space compression during phase retrieval provides a viable means by which to invert high-energy BCDI data, which is otherwise intractable.

Imaging Performance of Polarization-Insensitive Metalenses

Metasurfaces have recently emerged as a promising technology to realize flat optical components with customized functionalities. In particular, their application to lenses in various imaging systems is of significant interest. However, a systematic and complete study of the focusing and imaging behavior of metalenses has not yet been conducted. In this work we analyze not only the on-axis focusing performance, but also the field-dependent wavefront aberrations via a phase-retrieval optimization method. We find that, particularly for high-NA metalenses, the field-dependent geometrical aberrations like coma are dominant at the design wavelength, while for longer and shorter operation wavelengths, the effective numerical aperture is decreased and mainly spherical aberrations are dominant. Additionally, we investigate the spectral and angular bandwidth of a polarization-insensitive metalens by analyzing the metalens efficiencies as a function of numerical aperture, field angle, and wavelength. We then compare...

A new phase retrieval algorithm with pure generalized three-step phase-shifting under matrix norm processing

ABSTRACTPhase-shifting interferometry (PSI) is one of the most effective techniques in imaging a phase specimen, in which the phase retrieval is a basic and significant process. A new phase retrieval method based on the matrix norm algorithm in PSI is proposed in this paper. In this algorithm, the value of phase shift can be determined by three different matrix norms of the intensity difference between two phase-shifted interferograms, and then the phase can be retrieved. Neither the iterative calculation nor the extra measurements of other parameters are necessary on account of this algorithm which only requires three phase-shifted interferograms. The feasibility and accuracy of this algorithm are demonstrated by the simulated results. Experimentally, the generalized phase shift can be realized by a simple device which adjusting the angle of glass accurately. It is found that this algorithm has a good application prospect in the field of dynamic imaging.

Identification of the dynamics of time-varying phase aberrations from time histories of the point-spread function.

To optimally compensate for time-varying phase aberrations with adaptive optics, a model of the dynamics of the aberrations is required to predict the phase aberration at the next time step. We model the time-varying behavior of a phase aberration, expressed in Zernike modes, by assuming that the temporal dynamics of the Zernike coefficients can be described by a vector-valued autoregressive (VAR) model. We propose an iterative method based on a convex heuristic for a rank-constrained optimization problem, to jointly estimate the parameters of the VAR model and the Zernike coefficients from a time series of measurements of the point-spread function (PSF) of the optical system. By assuming the phase aberration is small, the relation between aberration and PSF measurements can be approximated by a quadratic function. As such, our method is a blind identification method for linear dynamics in a stochastic Wiener system with a quadratic nonlinearity at the output and a phase retrieval method that uses a time-evolution-model constraint and a single image at every time step.

Fast three-dimensional phase retrieval in propagation-based X-ray tomography.

The following article describes a method for 3D reconstruction of multi-material objects based on propagation-based X-ray phase-contrast tomography (PB-CT) with phase retrieval using the homogeneous form of the transport of intensity equation (TIE-Hom). Unlike conventional PB-CT algorithms that perform phase retrieval of individual projections, the described post-reconstruction phase-retrieval method is applied in 3D to a localized region of the CT-reconstructed volume. This work demonstrates, via numerical simulations, the accuracy and noise characteristics of the method under a variety of experimental conditions, comparing it with both conventional absorption tomography and 2D TIE-Hom phase retrieval applied to projection images. The results indicate that the 3D post-reconstruction method generally achieves a modest improvement in noise suppression over existing PB-CT methods. It is also shown that potentially large computational gains over projection-based phase retrieval for multi-material samples are possible. In particular, constraining phase retrieval to a localized 3D region of interest reduces the overall computational cost and eliminates the need for multiple CT reconstructions and global 2D phase retrieval operations for each material within the sample.