Phase Retrieval Method Research Articles

ISAR Autofocus Imaging Algorithm for Maneuvering Targets Based on Phase Retrieval and Keystone Transform

In the current scenario of high-range resolution radar and noncooperative target, the rotational motion parameters of the target are unknown and migration through resolution cells (MTRC) is apparent in the obtained inverse synthetic aperture radar (ISAR)images, in both slant-range and cross-range directions. In the case of the high-speed maneuvering target with a small value of rotation, the phase retrieval algorithm can be applied to compensate for the translational motion to form an autofocusing image. However, when the target has a relatively large rotation angle during the coherent integration time, phase retrieval method cannot get an acceptable image for viewing and analysis as the location of the scatterer will not be true due to the Doppler shift imposed by the target’s rotational motion. In this paper, a novel ISAR imaging method for maneuvering targets based on phase retrieval and keystone transform is proposed, which can effectively solve the above problems. First, the keystone transform is used to solve the MTRC effects caused by the rotation component. Next, phase errors caused by the remaining translational motion will be removed by employing phase retrieval algorithm, allowing the scatterers are always kept in their range cells. Finally, the Doppler frequency shifts of scatterers will be time invariant in the phase of the received signal. Furthermore, this approach does not need to estimate the motion parameters of the target, which simplifies the processing steps. The simulated results demonstrate the validity of this method.

Fourier ptychographic microscopy reconstruction with multiscale deep residual network.

Fourier ptychographic microscopy (FPM) is a newly developed microscopic technique for large field of view, high-resolution and quantitative phase imaging by combining the techniques from ptychographic imaging, aperture synthesizing and phase retrieval. In FPM, an LED array is utilized to illuminate the specimen from different angles and the corresponding intensity images are synthesized to reconstruct a high-resolution complex field. As a flexible and low-cost approach to achieve high-resolution, wide-field and quantitative phase imaging, FPM is of enormous potential in biomedical applications such as hematology and pathology. Conventionally, the FPM reconstruction problem is solved by using a phase retrieval method, termed Alternate Projection. By iteratively updating the Fourier spectrum with low-resolution-intensity images, the result converges to a high-resolution complex field. Here we propose a new FPM reconstruction framework with deep learning methods and design a multiscale, deep residual neural network for FPM reconstruction. We employ the widely used open-source deep learning library PyTorch to train and test our model and carefully choose the hyperparameters of our model. To train and analyze our model, we build a large-scale simulation dataset with an FPM imaging model and an actual dataset captured with an FPM system. The simulation dataset and actual dataset are separated as training datasets and test datasets, respectively. Our model is trained with the simulation training dataset and fine tuned with the fine-tune dataset, which contains actual training data. Both our model and the conventional method are tested on the simulation test dataset and the actual test dataset to evaluate the performances. We also show the reconstruction result of another neural network-based method for comparison. The experiments demonstrate that our model achieves better reconstruction results and consumes much less time than conventional methods. The results also point out that our model is more robust under system aberrations such as noise and blurring (fewer intensity images) compared with conventional methods.

Method for spectral phase retrieval of single attosecond pulses utilizing the autocorrelation of photoelectron streaking spectra

We propose an algorithm for retrieving the spectral phase of an isolated attosecond pulse based on the photoelectron spectra generated in atoms by the attosecond pulse in the presence of a time-delayed infrared laser. Instead of employing the whole set of the streaking spectra as in the previous phase retrieval algorithms, we calculate the autocorrelation (AC) of the streaking trace, and use it to extract the spectral phase. We illustrate that this method can be used to extract narrow- as well as broadband attosecond pulses. The method converges much faster and is more accurate. We also define two parameters, $V$ and $A$. By comparing the AC pattern and the $V$ and $A$ parameters from the experiment and from the retrieved pulse, this method provides a metric to evaluate the accuracy of the retrieved pulse.

Full-field ultrafast oscilloscope based on temporal imaging.

As the ultrafast laser has been developed, the measurement of ultra-speed transient and dynamic processes has gained much attention. An ultrafast oscilloscope based on the optical stretch method is promising to address this problem, but the measurement is restricted to the temporal profile. This means that the temporal phase is lost. In this work, we propose a full-field ultrafast oscilloscope using two temporal phase retrieval methods: the temporal annealing Gerchberg-Saxton (TAGS) algorithm and temporal ptychography. These could provide complete information, including temporal profile and phase, of high-rate repetitive transient pulses. The functions of an ultrafast oscilloscope with 230 GHz bandwidth and the two phase retrieval methods are verified by simulation and experimental results. This full-field ultrafast oscilloscope promises more applications in phase encoding, phase-contrast imaging, and sensing in the time domain.

Impact of a single distance phase retrieval algorithm on spatial resolution in X-ray inline phase sensitive imaging.

A single-projection based phase retrieval method based on the phase attenuation duality principle (PAD) was used to compare the spatial resolution of the acquired phase sensitive and PAD processed phase retrieved images. An inline phase sensitive prototype was used to acquire the phase sensitive images. The prototype incorporates a micro-focus x-ray source and a flat panel detector with a 50 μm pixel pitch. A phantom composed of a 2 cm thick 50-50 adipose-glandular mimicking slab sandwiched with a 0.82 cm thick slanted PMMA sharp edge was used. Phase sensitive image of the phantom was acquired at 120 kV, 3.35 mAs with a 16 μm tube focal spot size under a geometric magnification (M) of 2.5. The PAD based method was applied to the acquired phase sensitive image for the retrieval of phase values. With necessary data processing, modulation transfer function (MTF) curves were determined for the estimation and comparison of the spatial resolution. The PAD processed phase retrieved values of the phantom were in good agreement with the theoretically calculated values. Phase sensitive images showed higher spatial resolution at all spatial frequencies compared to the phase retrieved images. It was noted that the high-frequency signal components in the retrieved image were suppressed that resulted in lower MTF values. When compared to the phase sensitive image, the cutoff resolution (10% MTF) for phase retrieved image dropped 32% from 15.6 lp/mm (32μm) to 10.6 lp/mm (47μm). The resolution offered by this phase sensitive prototype is radiographically enough to detect breast cancer.

A novel phase retrieval method in ghost diffraction

We propose a new kind of ghost diffraction scheme to find the phase distribution of the object. By inserting reference screens into the reference path, we can directly obtain the phase of the Fourier-transformed spectrum of the object via the four-step phase-shifting method so that both the amplitude and the phase of the object can be easily retrieved. Realistic simulations have been performed to demonstrate experimental feasibility.

Technical Note: Single-shot phase retrieval method for synchrotron-based high-energy x-ray grating interferometry.

X-ray grating interferometry (XGI) provides substantially increased contrast over conventional absorption-based imaging methods and therefore shows great potential for future biomedical applications. In this work, we propose a single-shot phase retrieval method for synchrotron-based high-energy x-ray grating interferometry. Contrary to existing retrieval methods, the presented novel approach enables direct retrieval of the object's phase map quantitatively from a single projection image, thus significantly simplifying the experimental procedure and reducing data acquisition times. The phase retrieval method is analytically derived, based on the phase-attenuation duality of soft tissues when being imaged with high-energy x rays. The sensitivity of the retrieved phase map, quantified by the standard deviation, is evaluated as a function of the photon number. Numerical experiments are performed to validate the proposed method and provide some quantitative insight. The numerical results show that the method can provide high-quality phase images, where the well-known streak artifacts are significantly suppressed. Moreover, the retrieved phase maps confirm that the method is highly stable with respect to statistical noise. Thanks to simplified experimental procedure and reduced acquisition time and dose deposition to the sample, we believe that this new method can find its potential in biomedical imaging and invivo studies. Future work will focus on the adaptation of the method to polychromatic x ray from tube source and to computed tomography.

Optimized Denoising Method for Fourier Ptychographic Microscopy Based on Wirtinger Flow

Fourier ptychographic microscopy (FPM) is a recently developed technique for high-resolution complex imaging with a wide field-of-view. FPM combines captured low-resolution images corresponding to angle-varying illuminations in the Fourier domain with phase retrieval and synthetic aperture algorithms; consequently, it can surpass the space-bandwidth product limit of an objective lens. In typical FPM systems, the reconstructed result is sensitive to various types of system and environmental noise; thus, the quality can be significantly degraded. In this paper, based on the Wirtinger flow gradient descent framework, we propose an optimized phase retrieval method, which searches for the optimization results with iteratively changing weights and suppresses noise by utilizing captured noise and its corresponding constrained item. A correction term is further introduced into the new algorithm. This method cannot only adapt to varying sampling conditions but also provide an accurate optimum solution. We validate our method with both simulated and experimental images. The reconstructed results all illustrate the effectiveness and robustness of our method and demonstrate that the proposed method can significantly depress noise and improve the reconstruction performance of high-resolution complex images.

Non-locally coded Fourier-transform ghost imaging.

A non-locally coded Fourier-transform ghost imaging (FGI) scheme and relevant coded phase retrieval method have been proposed to improve the image quality in ghost imaging. By inserting masks in the reference beam, the sample in the test beam is non-locally modulated, and coded Fourier-transform diffraction patterns of the sample are obtained via intensity correlation calculations between the two beams. Encoding and decoding procedures are incorporated in the phase retrieval process based on traditional hybrid input-output algorithm. Simulation and experiment results show that the spatial information of samples is successfully recovered from the coded diffraction patterns obtained using three masks, and the image quality is improved remarkably. If promoting this approach to x-ray FGI systems, it may realize high-resolution x-ray microscopy without increasing the sample's radiation damage.

High-speed 3D shape measurement using the optimized composite fringe patterns and stereo-assisted structured light system.

In this paper, we propose a high-speed 3D shape measurement technique based on the optimized composite fringe patterns and stereo-assisted structured light system. Stereo phase unwrapping, as a new-fashioned method for absolute phase retrieval based on the multi-view geometric constraints, can eliminate the phase ambiguities and obtain a continuous phase map without projecting any additional patterns. However, in order to ensure the stability of phase unwrapping, the period of fringe is generally around 20, which limits the accuracy of 3D measurement. To solve this problem, we develop an optimized method for designing the composite pattern, in which the speckle pattern is embedded into the conventional 4-step phase-shifting fringe patterns without compromising the fringe modulation, and thus the phase measurement accuracy. We also present a simple and effective evaluation criterion for the correlation quality of the designed speckle pattern in order to improve the matching accuracy significantly. When the embedded speckle pattern is demodulated, the periodic ambiguities in the wrapped phase can be eliminated by combining the adaptive window image correlation with geometry constraint. Finally, some mismatched regions are further corrected based on the proposed regional diffusion compensation technique (RDC). These proposed techniques constitute a complete computational framework that allows to effectively recover an accurate, unambiguous, and distortion-free 3D point cloud with only 4 projected patterns. Experimental results verify that our method can achieve high-speed, high-accuracy, robust 3D shape measurement with dense (64-period) fringe patterns at 5000 frames per second.

Phase retrieval with complexity guidance.

Iterative phase retrieval methods based on the Gerchberg-Saxton (GS) or Fienup algorithm typically show stagnation artifacts even after a large number of iterations. We introduce a complexity parameter ζ that can be computed directly from the Fourier magnitude data and provides a measure of fluctuations in the desired phase retrieval solution. It is observed that when initiated with a constant or a uniformly random phase map, the complexity of the Fienup solution containing stagnation artifacts stabilizes at a numerical value that is higher than ζ. We propose a modified Fienup algorithm that uses a controlled sparsity-enhancing step such that in every iteration the complexity of the resulting guess solution is explicitly made close to ζ. This approach, which we refer to as complexity-guided phase retrieval, is seen to provide an artifact-free phase retrieval solution within a few hundred iterations. Numerical illustrations are provided for both amplitude as well as phase objects with and without Poisson noise introduced in the Fourier intensity data. The complexity-guidance concept may potentially be combined with a variety of phase retrieval algorithms and can enable several practical applications.

A Generalization of Wirtinger Flow for Exact Interferometric Inversion

Interferometric inversion involves recovery of a signal from cross-correlations of its linear transformations. A close relative of interferometric inversion is the generalized phase retrieval problem, which consists of recovering a signal from the auto-correlations of its linear transformations. Recently, significant advancements have been made in phase retrieval methods despite the ill-posed, and non-convex nature of the problem. One such method is Wirtinger Flow (WF), a non-convex optimization framework that provides high probability guarantees of exact recovery under certain measurement models, such as coded diffraction patterns, and Gaussian sampling vectors. In this paper, we develop a generalization of WF for interferometric inversion, which we refer to as Generalized Wirtinger Flow (GWF). GWF theory extends the probabilistic exact recovery results of WF to arbitrary measurement models characterized in the equivalent lifted problem, hence covers a larger class of measurement models. Our framework unifies the theory of low rank matrix recovery (LRMR) and the non-convex optimization approach of WF, thereby establishes theoretical advantages of the non-convex approach over LRMR. We identify a new sufficient condition on the lifted forward model that directly implies exact recovery conditions of standard WF. This condition is less stringent than those of LRMR, which is the state of the art approach for exact interferometric inversion. We establish our sufficient condition for the cross-correlations of linear measurements collected by complex Gaussian sampling vectors, and show that the exact recovery conditions of standard WF imply our sufficient condition. As a result, the regularity condition of WF becomes redundant in solving the interferometric inversion problem. Finally, we demonstrate the effectiveness of GWF numerically in a deterministic multi-static radar imaging scenario.

Raster Grid Pathology and the Cure

Blind ptychography is a phase retrieval method using multiple coded diffraction patterns from different, overlapping parts of the unknown extended object illuminated with an unknown window function. The window function is also known as the probe in the optics literature. As such blind ptychography is an inverse problem of simultaneous recovery of the object and the window function given the intensities of the windowed Fourier transform and has a multiscale set-up in which the probe has an intermediate scale between the pixel scale and the macroscale of the extended object. The uniqueness problem for blind ptychography is analyzed rigorously for the raster scan (of a constant step size $\tau$) and its perturbations. The block phases are shown to form an arithmetic progression and the complete characterization of the raster scan ambiguities is given, including, first, the periodic raster grid pathology of degrees of freedom proportional to $\tau^{2}$, and, second, a nonperiodic, arithmetically progressing phase shift from block to block. Finally, irregularly perturbed raster scans are shown to remove all ambiguities other than the inherent ambiguities of the scaling factor and the affine phase factor under general requirements, including roughly the minimum overlap ratio 50%.

Inverse Acoustic Scattering with Phaseless Far Field Data: Uniqueness, Phase Retrieval, and Direct Sampling Methods

The well-known translation invariance property of the phaseless far field patterns with incident plane waves make it is impossible to reconstruct the location of the underlying scatterers even for multiple incident directions and frequencies. To overcome this difficulty, we take the superpositions of plane waves and point sources as the incident fields. The point sources have fixed source location with at most three different scattering strengths. By adding a reference sound-soft obstacle into the scattering system, we show some uniqueness results with phaseless far field data. Of possibly even more interest are the phase retrieval technique and direct sampling methods, without using the reference obstacle. With the help of the point sources, we propose a simple and stable phase retrieval algorithm and design some novel direct sampling methods using the phaseless data directly. The novel phase retrieval algorithm can then be combined with the classical sampling methods for scatterer reconstruction. Finally, we present a variant of numerical examples in two dimensions to verify the effectiveness and robustness of the proposed algorithms.

Inverse Electromagnetic Source Scattering Problems with MultiFrequency Sparse Phased and Phaseless Far Field Data

This paper is concerned with uniqueness, phase retrieval, and shape reconstruction methods for solving inverse electromagnetic source scattering problems with multifrequency sparse phased or phaseless far field data. With the phased data, we show that the smallest strip containing the source support with the observation direction as the normal can be uniquely determined by the multifrequency far field pattern at a single observation direction. A phased direct sampling method is also proposed to reconstruct the strip. The phaseless far field data is closely related to the outward energy flux, which can be measured more easily in practice. We show that the phaseless far field data is invariant under the translation of the sources, which implies that the location of the sources cannot be uniquely recovered by the data. To solve this problem, we consider simultaneously the scattering of magnetic dipoles with one fixed source point and at most three scattering strengths. With this technique, a fast and stable phase retrieval approach is proposed based on a simple geometrical result which provides a stable reconstruction of a point in the plane from three distances to the given points. A novel phaseless direct sampling method is also proposed to reconstruct the strip. The phase retrieval approach can also be combined with the phased direct sampling method to reconstruct the strip. Finally, to obtain a source support, we just need to superimpose the indicators with respect to the sparse observation directions. Extended numerical examples in three dimensions are conducted with noisy data, and the results further verify the effectiveness and robustness of the proposed phase retrieval approach and direct sampling methods.

Multidirectional wavelet-based fringe analysis for surface diagnosis by optical profilometry

Scientists, engineers, and manufacturers have a critical need for better techniques of diagnosis and quality control. The optical metrology uses the practical light and computer sciences, in order to perform simulation, design, computing and inspection for many scientific and industrial applications, such as optics, mechanics, aeronautics, electronics and many others. The fringe pattern analysis is a method to perform some operations on the optical images, in order to get an interferometric phase map and then to extract some useful information from it. In this paper, an exciting improvements of local fringe demodulation algorithm are presented, which are based on a new multidirectional wavelet. The numerical and experimental works show interesting gains compared to the other standard algorithms. Our approach executes as fast as the popular phase retrieval methods, but with substantially improved accuracy in demodulating noisy fringes. All of this comes without any pre-processing by filtering models.

Phaseless inverse source scattering problem: Phase retrieval, uniqueness and direct sampling methods

Similar to the obstacle or medium scattering problems, an important property of the phaseless far field patterns for source scattering problems is the translation invariance. Thus it is impossible to reconstruct the location of the underlying sources. Furthermore, the phaseless far field pattern is also invariant if the source is multiplied by any complex number with modulus one. Therefore, the source can not be uniquely determined, even the multi-frequency phaseless far field patterns are considered. By adding a reference point source into the model, we propose a simple and stable phase retrieval method and establish several uniqueness results with phaseless far field data. We proceed to introduce a novel direct sampling method for shape and location reconstruction of the source by using broadband sparse phaseless data directly. We also propose a combination method with the novel phase retrieval algorithm and the classical direct sampling methods with phased data. Numerical examples in two dimensions are also presented to demonstrate their feasibility and effectiveness.

Under-Sampled Phase Retrieval of Single Interference Fringe Based on Hilbert Transform

Phase retrieval from single interference fringe is important and effective method in obtaining the real phase distribution. The original phase can be retrieved by the line integral of its gradient expressed as sine and cosine components, which were gained by the Hilbert transform twice from a single interference fringe pattern. However, this method fails when the phase transformation of the interference fringe is too fast. In this paper, a novel method to recover the continuous phase of the whole field is proposed to solve the above problems. The shear interference technique is introduced into the phase retrieval method to build an exponential 2-D complex light field of natural base for the phase slope obtained by the Hilbert transform. Then, the expressions of phase slopes in x- and y-directions are constructed as a discrete Poisson equation. Therefore, the calculation of phase retrieval is equivalent to solve the discrete Poisson equation mathematically. Finally, the real phase is gotten by the weighted discrete cosine transform (WDCT) of the discrete Poisson equation. The simulation results verify the validity of this method and show that the proposed method can achieve the phase retrieval of the phase discontinuity in x- and y-directions, which leads to the under-sampled problem. It can restore the whole field phase distribution rapidly and accurately. Moreover, this method is applied to phase retrieval of interferometric synthetic aperture radar (InSAR) with the under-sampled problem in this paper. The experimental results show that this method can recover the phase of InSAR with the under-sampled problems caused by terrain abrupt change and so on. Compared with other commonly used methods, it achieved satisfactory results. This method provides a new idea for solving the under-sampled problem in the phase retrieval from a single-frame interference fringe.

Optimal pulse width modulation technique combined with stair phase-coding method for absolute phase retrieval with projector defocusing

The accurate three-dimensional shape measurement of a complex surface is significantly important in industrial testing. However, three-dimensional profilometry by conventional sinusoidal fringe projection using a phase-shifting algorithm performs suboptimally because of the nonlinear intensity response of projectors. To overcome this problem, the present paper proposes using a combined optimizing sinusoidal pulse width modulation (SPWM) technique and stair phase-coding approach to obtain the unwrapped phase. Properly optimizing SPWM with a small degree of a defocusing projector to generate sinusoidal fringe patterns can overcome the undesired harmonics and nonlinear gamma effect. Two groups of four-step phase-shifting fringe patterns are used. One group of the stripes contains four sinusoidal patterns generated by the SPWM technique, which is used to determine the wrapped phase. The other group of stripes contains four sinusoidal patterns with the codeword embedded into the stair phase, whose stair changes are perfectly aligned with the 2π discontinuities of the sinusoidal fringe phase, which is used to determine the fringe order for the phase unwrapping. Moreover, for the defocusing projection, because the frequency of the phase-coding fringe increases, the fringe order decision becomes less reliable. Thus, a self-correction phase unwrapping method is applied for phase retrieval. Experiments were conducted to verify the performance of the proposed method.

Absolute Phase Retrieval Using One Coded Pattern and Geometric Constraints of Fringe Projection System

Fringe projection technologies have been widely used for three-dimensional (3D) shape measurement. One of the critical issues is absolute phase recovery, especially for measuring multiple isolated objects. This paper proposes a method for absolute phase retrieval using only one coded pattern. A total of four patterns including one coded pattern and three phase-shift patterns are projected, captured, and processed. The wrapped phase, as well as average intensity and intensity modulation, are calculated from three phase-shift patterns. A code word encrypted into the coded pattern can be calculated using the average intensity and intensity modulation. Based on geometric constraints of fringe projection system, the minimum fringe order map can be created, upon which the fringe order can be calculated from the code word. Compared with the conventional method, the measurement depth range is significantly improved. Finally, the wrapped phase can be unwrapped for absolute phase map. Since only four patterns are required, the proposed method is suitable for real-time measurement. Simulations and experiments have been conducted, and their results have verified the proposed method.