Phase Retrieval Method Research Articles

PhlatCam: Designed Phase-Mask Based Thin Lensless Camera.

We demonstrate a versatile thin lensless camera with a designed phase-mask placed at sub-2 mm from an imaging CMOS sensor. Using wave optics and phase retrieval methods, we present a general-purpose framework to create phase-masks that achieve desired sharp point-spread-functions (PSFs) for desired camera thicknesses. From a single 2D encoded measurement, we show the reconstruction of high-resolution 2D images, computational refocusing, and 3D imaging. This ability is made possible by our proposed high-performance contour-based PSF. The heuristic contour-based PSF is designed using concepts in signal processing to achieve maximal information transfer to a bit-depth limited sensor. Due to the efficient coding, we can use fast linear methods for high-quality image reconstructions and switch to iterative nonlinear methods for higher fidelity reconstructions and 3D imaging.

Digital volume correlation technique for characterizing subsurface deformation behavior of a laminated composite

In the present study, the digital volume correlation (DVC) technique is used to study the deformation behavior occurring inside a carbon fibre-reinforced epoxy composite. While a tensile test on a symmetric angle-ply laminated composite specimen is being performed, X-ray images are recorded using synchrotron radiation. The radiographs are processed to reconstruct 3D μCT images. The reconstruction process utilizes a filtered back-projection (FBP) algorithm combined with a phase retrieval method to enhance the contrast of the CT images. Phase contrast is crucial since the natural microstructural pattern is analogous to a speckle pattern for digital image correlation (DIC) analysis. A slight misalignment of the specimen, typically negligible at the macroscopic scale, is corrected in the CT images prior to the DVC analysis. The accuracy of the DVC results is improved by implementing a sub-voxel registration approach to achieve a resolution less than one pixel. The DVC results are compared with finite element analysis (FEA) results. Although the overall results agree well with the FEA results, few subvolumes are prone to fail to find their correct pairs within DVC volumes. Raw 2D tomograms are individually examined and compared to show that a high correlation coefficient value does not always guarantee a correct match. Detailed discussions about the misleading correlations are provided and thereby the validations and limitations of the DVC (and DIC) technique are addressed.

Coherent diffraction imaging based on iterative fractional Fourier transform

This paper presents a phase retrieval method which is based on a fractional Fourier transform with estimating on-line the order of the transform during the iterative projections between space domain and space-frequency domain. This method overcomes the limitations of the same-kind methods which require a prior knowledge of the accurate order of fractional Fourier transform. Simulations and experiments demonstrated that the proposed method converges faster and achieves less error.

A new phase retrieval method using sequential phase modulations

In this paper, a new phase retrieval method using sequential phase modulations is proposed. Behind the unknown object, adding sequential phase modulations will change the diffraction intensities received by sensor. Through increasing the number of diffraction intensities patterns, the difficulty of retrieving the unknown object is decreased. To better select these modulation phases, the complexity parameter is defined to evaluate the complexity of unknown object. When the complexity parameter of unknown object is larger, it contains more spectrum information on different frequency bands and will be harder to retrieve. The complexity of unknown object should be contained between the maximum and minimum complexities of modulation phases. In this way, the information of each frequency band on the unknown object can be effectively retrieved. Meanwhile, the distribution of modulation phases should be continuous to avoid introducing high frequency noise. In addition, there is no limit on what kind of modulation phase distribution to choose. The effectiveness and fast convergence of this new method has been proved.

Imaging around corners in the mid-infrared using speckle correlations.

Speckle correlation imaging offers the ability to see objects through diffusive materials and around corners. Imaging self-illuminating thermal objects in non-line-of-sight scenarios is of particular interest. Here, using bispectrum and phase retrieval methods, we demonstrate speckle correlation imaging of mid-infrared objects through diffusers and around corners at resolutions near the diffraction limit. The images agree well with those recorded by conventional cameras with line-of-sight to the same objects.

Single-shot sequential projection phase retrieval and 3D localization from chromatic aberration.

A phase retrieval method based on sequential projection and chromatic aberration is reported. Construction of this method includes a red, green and blue (RGB) LED, an objective and a color camera. Owing to the chromatic aberration characteristics of the objective, three color images obtained by the color camera correspond to three equivalent propagation planes. Equivalent relative distances among these planes can be obtained by defining and iteratively minimizing the convergence index. Then, sequential projection strategy is used for phase retrieval in each image plane. Based on the recovered phase information and angular spectrum propagation principle, digital refocusing and 3D localization can be achieved for each subregion of the sample. Finally, the feasibility of our method is demonstrated by simulations and experiments.

Accurate phase retrieval algorithm based on normalized vector of three interferograms

A phase retrieval method from three interferograms based on normalized vector is proposed to achieve high accuracy and less time computation. It needn’t pre-filtering for the interferograms, which reduces its complexity and shortens the processing time. The difference maps are used to filter out the background intensity after compensating the fluctuation. Then, the normalized vector is used to obtain tested phase distribution with high precision. It is suitable for simple and complex fringes. Moreover, the phase can be correctly extracted when the fluctuation of background intensity and modulation amplitude is existed. The reliability of the algorithm is verified with simulated and experimental data.

Enhanced Phase Retrieval Method Based on Random Phase Modulation

The phase retrieval method based on random phase modulation can wipe out any ambiguity and stagnation problem in reconstruction. However, the two existing reconstruction algorithms for the random phase modulation method are suffering from problems. The serial algorithm from the spread-spectrum phase retrieval method can realize rapid convergence but has poor noise immunity. Although there is a parallel framework that can suppress noise, the convergence speed is slow. Here, we propose a random phase modulation phase retrieval method based on a serial–parallel cascaded reconstruction framework to simultaneously achieve quality imaging and rapid convergence. The proposed serial–parallel cascaded method uses the phased result from the serial algorithm to serve as the initialization of the subsequent parallel process. Simulations and experiments demonstrate that the superiorities of both serial and parallel algorithms are fetched by the proposed serial–parallel cascaded method. In the end, we analyze the effect of iteration numbers from the serial process on the reconstruction performance to find the optimal allocation scope of iteration numbers.

Improved Fourier transformation based method for accurate phase and amplitude retrieval in spectral interferometry

Although Fourier transformation (FT) is the most commonly used method to retrieve the phase and amplitude information of a spectral interferogram, its accuracy is usually degraded due to effects such as fringe truncation, uneven distribution of fringes, noise and fringe density, etc. In this work, we propose an improved FT based method to process the spectral interferogram. By using a Gaussian envelope and a self-designed flat-top cap window function, significant improvements in accuracy and robustness can be achieved compared to the conventional FT method. Besides, by analyzing the correlation between the fringe density and the data processing error, the optimal configuration in an actual spectral interferometry experiment can be screened out. At last, we have applied the proposed method on the actual spectral interferograms obtained in our laser-induced shock wave loading experiment for effectiveness demonstration, and the results show a dramatical reduction of the measurement error compared to the traditional FT method.

Rapid and precise phase retrieval from two-frame tilt-shift based on Lissajous ellipse fitting and ellipse standardization.

A rapid and precise phase-retrieval method based on Lissajous ellipse fitting and ellipse standardization is demonstrated. It only requires two interferograms without pre-filtering, which reduces its complexity and shortens the processing time. The elliptic coefficients obtained by ellipse fitting are used for ellipse standardization. After compensating phase-shift errors by ellipse standardization, the phase distribution is extracted with high precision. It is suitable for fluctuation, noise, tilt-shift, simple and complex fringes. This method is effective for the number of fringes less than 1. The reliability of the method is verified by simulations and experiments, indicating high accuracy and less time consumption.

Constrained Non-Linear Phase Retrieval for Single Distance Xray Phase Contrast Tomography

X-ray phase contrast tomography (XPCT) is widely used for 3D imaging of objects with weak contrast in X-ray absorption index but strong contrast in refractive index decrement. To reconstruct an object imaged using XPCT, phase retrieval algorithms are first used to estimate the X-ray phase projections, which is the 2D projection of the refractive index decrement, at each view. Phase retrieval is followed by refractive index decrement reconstruction from the phase projections using an algorithm such as filtered back projection (FBP). In practice, phase retrieval is most commonly solved by approximating it as a linear inverse problem. However, this linear approximation often results in artifacts and blurring when the conditions for the approximation are violated. In this paper, we formulate phase retrieval as a non-linear inverse problem, where we solve for the transmission function, which is the negative exponential of the projections, from XPCT measurements. We use a constraint to enforce proportionality between phase and absorption projections. We do not use constraints such as large Fresnel number, slowly varying phase, or Born/Rytov approximations. Our approach also does not require any regularization parameter tuning since there is no explicit sparsity enforcing regularization function. We validate the performance of our non-linear phase retrieval (NLPR) method using both simulated and real synchrotron datasets. We compare NLPR with a popular linear phase retrieval (LPR) approach and show that NLPR achieves sharper reconstructions with higher quantitative accuracy.

Is high sensitivity always desirable for a grating-based differential phase contrast imaging system?

In grating-based x-ray differential phase contrast (DPC) imaging, the measured signal amplitude of the phase shift induced by an image object is proportional to the so-called system sensitivity. Therefore, to achieve a better signal-to-noise (SNR) for improved imaging performance, it is generally believed that one should increase the system sensitivity by reducing the period of the analyzer grating or increasing the distance between the phase grating and analyzer grating. The purpose of this work is to theoretically and experimentally demonstrate that there is an optimal system sensitivity to attain the highest SNR for a given task provided that the standard phase-stepping acquisition and phase retrieval methods are used. When system sensitivity goes beyond this optimal value, SNR decreases and the imaging performance deteriorates. Due to the fundamental fact that the measured phase signal is a cyclic variable, the phase wrapping effect is inevitable in DPC imaging when the system sensitivity increases. The phase wrapping effect appears in both signal and noise measurements. The effect in the signal measurement is manifested in the so-called signal statistical bias and such effect often impacts the accuracy of the measurement. The phase wrapping effect also appears in the noise variance measurement and impacts the precision of the measurement. A thorough theoretical analysis was performed in this work to demonstrate the quantitative impacts of phase wrapping on both signal bias and noise variance and thus on the actual system SNR. The joint effect of phase wrapping in both the signal bias and noise variance yields an optimal system sensitivity to achieve the highest SNR. Both extensive numerical simulation studies and experimental studies were performed to validate the theoretical analysis. Both theoretical analysis and experimental studies show that the SNR of the DPC signal is not always proportional to the sensitivity due to the cyclic nature of the signal and the phase wrapping effect. For a given refraction angle and exposure level, there exists an optimal sensitivity factor that maximizes the SNR, beyond which, increasing the sensitivity will decrease the SNR. Increase of system sensitivity does not always improve x-ray DPC imaging performance provided that the standard phase-stepping acquisition and phase retrieval methods are used.

Analysis of non-iterative phase retrieval based on machine learning

In this paper, we analyze a machine-learning-based non-iterative phase retrieval method. Phase retrieval and its applications have been attractive research topics in optics and photonics, for example, in biomedical imaging, astronomical imaging, and so on. Most conventional phase retrieval methods have used iterative processes to recover phase information; however, the calculation speed and convergence with these methods are serious issues in real-time monitoring applications. Machine-learning-based methods are promising for addressing these issues. Here, we numerically compare conventional methods and a machine-learning-based method in which a convolutional neural network is employed. Simulations with several conditions show that the machine-learning-based method realizes fast and robust phase recovery compared with the conventional methods. We also numerically demonstrate machine-learning-based phase retrieval from noisy measurements with a noisy training data set for improving the noise robustness. The machine-learning-based approach used in this study may increase the impact of phase retrieval, which is useful in various fields, where phase retrieval has been used as a fundamental tool.

Phase retrieval using axial diffraction patterns and a ptychographic iterative engine.

We propose a phase retrieval method using axial diffraction patterns under planar and spherical wave illuminations. The proposed method uses a ptychographic iterative engine (PIE) for the phase retrieval algorithm. The proposed approach uses multiple diffraction patterns. Thus, adjusting the alignment of each diffraction pattern is mandatory, and we propose a method to adjust the alignment. In addition, a random selection of the measured diffraction patterns is used to further accelerate the convergence of the PIE-based optimization. To confirm the effectiveness of the proposed method, we compare the conventional and proposed methods using a simulation and optical experiments.

Frequency expanded non-interferometric phase retrieval for holographic data storage

A novel frequency expanded non-interferometric phase retrieval method for holographic data storage is proposed. The limitation of twice Nyquist frequency for phase retrieval with high fidelity is broken through. Only 1 times Nyquist size frequency of the hologram is recorded, reconstructed and detected in our method which is the smallest range for complex multi-level phase retrieval in the current holographic data storage report. Required high-order frequencies for quick phase retrieval are generated artificially by expanding 1 times Nyquist frequency to high-order frequencies with utilizations of frequency periodicity. Therefore, our proposed method can increase storage density due to reducing the recording consumption of media and decreasing the recording area by 4 times compared with twice Nyquist frequency recording.

Partially Coherent Compressive Phase Retrieval for Millimeter-Wave Massive MIMO Channel Estimation

Channel estimation is challenging for millimeter-wave (mmWave) massive MIMO systems. Leveraging the spatial sparsity of mmWave channels, compressive sensing (CS) based channel estimation been extensively studied. However, the mmWave hardware imperfections may introduce random phase distortions to the received pilots, which makes the conventional CS methods fail to estimate the channel. The compressive phase retrieval (CPR) method can be employed to solve this challenging problem. In this paper, we exploit the partial coherence in hybrid mmWave systems, i.e., the pilots sent from different radio frequency (RF) chains share the same phase distortion in the same time frame, while the phase distortions are different across different time frames. Based on this property, we propose an on-grid partially coherent CPR (PC-CPR) algorithm for mmWave channel estimation in the presence of severe phase distortions. Unlike the existing coherent channel estimation schemes that require perfect phase information or the noncoherent channel estimation schemes that ignore the phases of measurements, the proposed on-grid PC-CPR algorithm exploits the partial coherence property to estimate the sparse angle-domain channel vector. Furthermore, to solve the resolution limitation of the on-grid PC-CPR algorithm, we propose an off-grid PC-CPR algorithm that directly estimates the parameters of channel paths. The proposed partially coherent channel estimation framework subsumes the existing coherent and noncoherent channel estimation methods as special cases. Simulation results show that under the presence of random phase distortions, the proposed PC-CPR algorithms outperform noncoherent channel estimation methods with higher reliability and lower pilot overhead by leveraging the partial coherence.

Interferometric Phase Retrieval for Multimode InSAR via Sparse Recovery

Modern spaceborne synthetic aperture radar (SAR) features a capacity of multiple imaging modes. It comes with Earth-observation data archives consisting of SAR images acquired in various modes with different resolutions and coverage. In this context, in addition to using single-mode images for SAR interferometry (InSAR), exploiting images acquired in different imaging modes for InSAR can provide extra interferograms and, thus, favors the retrieval of interferometric information. The interferometric processing of multimode image pairs requires special considerations due to significant variations in the Doppler spectra. Conventionally, the InSAR technique only uses the spectral band common in both master and slave images, and the remaining band is discarded before interferogram formation. Therefore, conventional processing cannot make full use of the observed data, and the interferogram quality is limited by the common band spectra. In this article, by exploiting the conventionally discarded spectrum, we present a new interferometric phase retrieval method for multimode InSAR data to improve interferogram quality. To this end, first, we propose a linear model to characterize the interferometric phase of a multimode image pair based on image spectral relation. Second, we adopt a sparse recovery method to inverse the linear model for the retrieval of the interferometric phase. Finally, we present real-data experiments on TerraSAR-X staring spotlight to sliding spotlight interferometry and Sentinel-1 strip map to Terrain Observation by Progressive Scan (TOPS) interferometry to test the proposed method. The experiment results show that the proposed method can provide interferograms with reduced phase noise and defocusing effect for multimode InSAR.

PhaseEqual: Convex Phase Retrieval via Alternating Direction Method of Multipliers

The problem of estimating an unknown complex signal from its magnitude-only measurements is a classical problem known as phase retrieval. This problem arises naturally because in some applications, it is difficult or costly to measure the phase of the measurements. In this paper, we propose a new convex formulation, referred to as PhaseEqual, for solving the phase retrieval problem. Similar to PhaseMax T. Goldstein and C. Studer, 2018 by Goldstein and Studer (IEE TIT, 2018), PhaseEqual works in the natural signal space and thus is computationally efficient. We then extend PhaseEqual to the sparse signal case, with the proposed convex formulation termed as compressed PhaseEqual. Different from existing convex compressed phase retrieval methods, the proposed compressed PhaseEqual formulation does not involve any regularization parameter and thus is free of the parameter tuning issue which is always tricky in practice. Order-wise recovery conditions for PhaseEqual and its sparse version (i.e. compressed PhaseEqual) are analyzed. Our theoretical results show that PhaseEqual (resp. compressed PhaseEqual) achieves perfect recovery with $\mathcal {O}(n)$ (resp. $\mathcal {O}(k\text{log}\frac{n}{k})$ ) magnitude measurements, provided that a well-correlated reference vector is available, where $k$ and $n$ denote the number of nonzero entries in the complex sparse signal and the dimension of the signal, respectively. Simulation results are provided to illustrate the effectiveness of the proposed methods.

One-shot phase-recovery using a cellphone RGB camera on a Jamin-Lebedeff microscope.

Jamin-Lebedeff (JL) polarization interference microscopy is a classical method for determining the change in the optical path of transparent tissues. Whilst a differential interference contrast (DIC) microscopy interferes an image with itself shifted by half a point spread function, the shear between the object and reference image in a JL-microscope is about half the field of view. The optical path difference (OPD) between the sample and reference region (assumed to be empty) is encoded into a color by white-light interference. From a color-table, the Michel-Levy chart, the OPD can be deduced. In cytology JL-imaging can be used as a way to determine the OPD which closely corresponds to the dry mass per area of cells in a single image. Like in other interference microscopy methods (e.g. holography), we present a phase retrieval method relying on single-shot measurements only, thus allowing real-time quantitative phase measurements. This is achieved by adding several customized 3D-printed parts (e.g. rotational polarization-filter holders) and a modern cellphone with an RGB-camera to the Jamin-Lebedeff setup, thus bringing an old microscope back to life. The algorithm is calibrated using a reference image of a known phase object (e.g. optical fiber). A gradient-descent based inverse problem generates an inverse look-up-table (LUT) which is used to convert the measured RGB signal of a phase-sample into an OPD. To account for possible ambiguities in the phase-map or phase-unwrapping artifacts we introduce a total-variation based regularization. We present results from fixed and living biological samples as well as reference samples for comparison.

Enhanced multiple-plane phase retrieval using adaptive support.

In the single-plane phase retrieval method, the use of a fixed object support is not efficient and could lead to inaccurate reconstructions. While there have been adaptive support strategies for the single-plane method, numerical processing is slow because such strategies are based in the space domain. Here, an adaptive support strategy based in the Fourier domain in conjunction with the multiple-plane phase retrieval method is presented and demonstrated through simulations and experiments. Optimizations of Fourier filter size and mask threshold parameters result in 3× faster convergence compared to the conventional multiple-plane method for the test object waves used. The proposed strategy offers fast, automated determination of the object support and affords the use of fewer intensity patterns.