Transfer orthogonal sparsifying transform learning for phase retrieval

Phase retrieval for sub-Gaussian measurements

In the process of signal and image acquisition and transmission, there is not only the additive white Gaussian noise interference, but also the non-Gaussian noise interference. The performance of a standard reconstruction algorithm, known as compressed sensing, can be significantly reduced if it is used to recover the signal affected by non-Gaussian noise. In this paper, LpLq-ADMM algorithm $(p \in(0,2), q \in(0,1))$ based on alternating direction method of multipliers is proposed to improve the reconstruction accuracy and the robustness of signal and image restoration in impulsive SαS (symmetric α-stable) noise. Lp-norm is used as the loss function term to enhance the data restoration ability in SαS noise and Lq-norm is used as the generalized non-convex penalty term to guarantee the sparsity of the signal. In order to efficiently solve the cost function model with minimum non-convex and nonsmooth, reduce the computational complexity of the objective function and improve the processing ability of high-dimensional data, in this paper, the alternating direction method of multipliers (ADMM) is used to solve the cost function model. The appropriate p and q values are selected via numerical simulation experiments to achieve the optimal reconstruction accuracy and robustness of sparse signals in SαS noise. The reconstruction probability of sparse signals and image recovery performance are analyzed under the optimal p and q values. The simulation results show that LpLq-ADMM algorithm has great noise suppression ability to SαS noise, and its Peak Signal to Noise Ratio (PSNR) and signal recovery probability is higher than other recovery algorithms.

The ill-posed problem of phase retrieval in optics, using one or more intensity measurements, has a multitude of applications using electromagnetic or matter waves. Many phase retrieval algorithms are computed on pixel arrays using discrete Fourier transforms due to their high computational efficiency. However, the mathematics underpinning these algorithms is typically formulated using continuous mathematics, which can result in a loss of spatial resolution in the reconstructed images. Herein we investigate how phase retrieval algorithms for propagation-based phase-contrast X-ray imaging can be rederived using discrete mathematics and result in more precise retrieval for single- and multi-material objects and for spectral image decomposition. We validate this theory through experimental measurements of spatial resolution using computed tomography (CT) reconstructions of plastic phantoms and biological tissues, using detectors with a range of imaging system point spread functions (PSFs). We demonstrate that if the PSF substantially suppresses high spatial frequencies, the potential improvement from utilising the discrete derivation is limited. However, with detectors characterised by a single pixel PSF (e.g. direct, photon-counting X-ray detectors), a significant improvement in spatial resolution can be obtained, demonstrated here at up to 17%.

Reconstructing the interesting complex image from multiple diffraction patterns is the goal of the ptychography. Previous ptychographic algorithms often suffer from low reconstruction quality under the low overlap ratios. To address this issue, we proposed a novel ptychographic phase retrieval (PR) algorithm of exploiting the sparsity of the image in dual-tree complex wavelet domain. The Fourier magnitude measurements are utilized to construct a data fidelity term, and the sparse representation model of the image over the dual-tree complex wavelet transform is utilized for the sparse induced regularization term. The data fidelity term and the proposed regularization term are combined to formulate a ptychographic PR optimization problem. Alternating direction method of multipliers (ADMM) and gradient descent algorithm are utilized for solving the corresponding optimization problem. Compared with previous algorithms, the experimental results indicate that the proposed algorithm can obtain reconstructed images with high quality even at low overlap ratios.

Phase retrieval is a promising method for optical system and surface metrology that makes use of intensity measurements of diffraction patterns. An iterative algorithm is used to solve the inverse problem to find the phase of the field producing the measured intensity distributions. For practical reasons, such as the reduction of coherent artifacts or to improve the signal-to-noise ratio of the measured data, it is often desirable to measure intensity distributions using broadband illumination. It is possible to perform phase retrieval with broadband data by incorporating a broadband model of the system into the phase retrieval algorithm. To do this, the system is modeled at several discrete wavelengths and the results from each are summed incoherently to produce a broadband result. This significantly increases the computational load. We show here that when aberrations are small, accurate estimates of the OPD distribution, on the level of λ/1000 RMS error, can be achieved using data with bandwidth up to about 10% as the input to a phase retrieval algorithm that assumes monochromatic data.

Iterative algorithms for phase retrieval from intensity data are compared to gradient search methods. Both the problem of phase retrieval from two intensity measurements (in electron microscopy or wave front sensing) and the problem of phase retrieval from a single intensity measurement plus a non-negativity constraint (in astronomy) are considered, with emphasis on the latter. It is shown that both the error-reduction algorithm for the problem of a single intensity measurement and the Gerchberg-Saxton algorithm for the problem of two intensity measurements converge. The error-reduction algorithm is also shown to be closely related to the steepest-descent method. Other algorithms, including the input-output algorithm and the conjugate-gradient method, are shown to converge in practice much faster than the error-reduction algorithm. Examples are shown.

We present a simple and robust technique for measuring the nonlinear refractive index. The principle is based on an iterative phase retrieval algorithm with a pump-probe system. Different strong phase modulations are intentionally introduced into the probe beam, and corresponding diffraction intensity patterns are recorded. The recordings are used in the phase retrieval algorithm to reconstruct the pump-induced phase on the probe beam. The nonlinear refractive index is then extracted from the reconstructed phase. The reconstruction method offers a straightforward procedure and a simple lensless setup. Simulations validate the proposed method. The effects of different characteristics of the pump and probe beams on the quality of reconstructions are investigated. The obtained results demonstrate that the reconstructions are accurate even for the probe beams with complex-valued fields and non-Gaussian pump beams; it removes the requirement for smooth fields of the pump and probe beams. The validity of the method in noisy conditions is also shown.

Fourier phase recovery techniques focus on how to reconstruct object information from phaseless measurement. Generally, such model-based phase recovery algorithms are difficult to obtain high-quality reconstructions in the presence of noise interference. Hence, we proposed a phase retrieval algorithm with deep denoiser networks. Firstly, an optimization model is constructed for the phase retrieval problem, then the alternating direction method of multipliers method is used to solve optimization problem iteratively. Besides, a well-trained deep neural network act as plug-and-play denoiser to participate the process of algorithm. Our method combines the model information of traditional phase retrieval algorithm and the fitting ability of the deep neural network, experiments show that it can achieve higher reconstruction result in the face of noisy image, and the generalization ability is also improved compared to end-to-end method.

There are several important remote sensing applications where the development of Ground Penetrating Synthetic Aperture Radar (GPENSAR) is the logical approach, e.g., searching for buried military facilities, minefield mapping, survey of underground pipelines. Penetration of sufficient soil depth for useful results require a SAR to operate at VHF/UHF frequencies, e.g., 200 - 300 MHz. At these frequencies a satellite SAR will encounter substantial distortion in the double passage of the SAR signal through the ionosphere. One of the ionospheric distortions is equivalent the phase aberrations caused in imaging through the turbulent atmosphere, and the problem of phase retrieval for the GPENSAR becomes a necessity. For GPENSR there are imaging concepts that exploit dual polarization radiation of the SAR pulse. The phase retrieval problem then becomes one of compensation for the phase aberrations induced in each of the polarization components returned to the satellite receiver. We discuss the use of the two polarizations to cancel the ionospheric phase aberrations. Unfortunately, the resulting signal has only relative phase of the two polarizations. We discuss an algorithm for the retrieval of the absolute phase. The algorithm is based on an optimization approach. Although phase retrieval by optimization is difficult because of local minima, the retrieval of absolute phase in the dual polarization case is substantially less difficult, because the two polarizations constrain the solution sufficiently to eliminate many local minima.

This paper investigates the application of unsupervised learning methods for computed tomography reconstruction. To motivate our work, we review several existing priors, namely the truncated Gaussian prior, the prior, the total variation prior, and the deep image prior (DIP). We find that DIP outperforms the other three priors in terms of representational capability and visual performance. However, the performance of DIP deteriorates when the number of iterations exceeds a certain threshold due to overfitting. To address this issue, we propose a novel method (MCDIP‐ADMM) based on multi‐code deep image prior (MCDIP) and plug‐and‐play alternative direction method of multipliers (ADMM). Specifically, MCDIP utilizes multiple latent codes to generate a series of feature maps at an intermediate layer within a generator model. These maps are then composed with trainable weights, representing the complete image prior. Experimental results demonstrate the superior performance of the proposed MCDIP‐ADMM compared to three existing competitors. In the case of parallel beam projection with Gaussian noise, MCDIP‐ADMM achieves an average improvement of 4.3 dB over DIP, 1.7 dB over ADMM DIP‐weighted total variation (WTV) and 1.2 dB over PnP‐DIP in terms of peak‐signal‐to‐noise ratio (PSNR). Similarly, for fan‐beam projection with Poisson noise, MCDIP‐ADMM achieves an average improvement of 3.09 dB over DIP, 1.86 dB over ADMM DIP‐WTV and 0.84 dB over PnP‐DIP in terms of PSNR.

The problem of phase retrieval is revisited and studied from a fresh perspective. In particular, we establish a connection between the phase retrieval problem and the sensor network localization problem, which allows us to utilize the vast theoretical and algorithmic literature on the latter to tackle the former. Leveraging this connection, we develop a two-stage algorithm for phase retrieval that can provably recover the desired signal. In both sparse and dense settings, our proposed algorithm improves upon prior approaches simultaneously in the number of required measurements for recovery and the reconstruction time. We present numerical results to corroborate our theory and to demonstrate the efficiency of the proposed algorithm. As a side result, we propose a new form of phase retrieval problem and connect it to the complex rigidity theory proposed by Gortler and Thurston (in: Connelly R, Ivic Weiss A, Whiteley W (eds) Rigidity and symmetry, Springer, New York, pp 131–154, 2014).

Phase retrieval is an inverse problem which consists in recovering an unknown signal from a set of absolute squared projections. Recently, gradient descent algorithms have been developed to solve this problem. However, their optimization cost functions are non-convex and non-smooth. To address the non-smoothness of the cost function, some of these methods use truncation thresholds to calculate a truncated step gradient direction. But, the truncation requires designing parameters to obtain a desired performance in the phase recovery, which drastically modifies the search direction update, increasing the sampling complexity. Therefore, this paper develops the Phase Retrieval Smoothing Conjugate Gradient method (PR-SCG) which uses a smoothing function to retrieve the signal. PR-SCG is based on the smooth-ing projected gradient method which is useful for non-convex optimization problems. PR-SCG uses a nonlinear conjugate gradient of the smoothing function as the search direction to accelerate the convergence. Furthermore, the incremental Stochastic Smoothing Phase Retrieval algorithm (SSPR) is developed. SSPR involves a single equation per iteration which results in a simple, scalable, and fast approach useful when the size of the signal is large. Also, it is shown that SSPR converges linearly to the true signal, up to a global unimodular constant. Additionally, the proposed methods do not require truncation parameters. Simulation results are provided to validate the efficiency of PR-SCG and SSPR compared to existing phase retrieval algorithms. It is shown that PR-SCG and SSPR are able to reduce the number of measurements and iterations to recover the phase, compared with recently developed algorithms.

Multiplicative noise and blur removal problems have attracted much attention in recent years. In this paper, we propose an efficient minimization method to recover images from input blurred and multiplicative noisy images. In the proposed algorithm, we make use of the logarithm to transform blurring and multiplicative noise problems into additive image degradation problems, and then employ l 1-norm to measure in the data-fitting term and the total variation to measure the regularization term. The alternating direction method of multipliers (ADMM) is used to solve the corresponding minimization problem. In order to guarantee the convergence of the ADMM algorithm, we approximate the associated nonconvex domain of the minimization problem by a convex domain. Experimental results are given to demonstrate that the proposed algorithm performs better than the other existing methods in terms of speed and peak signal noise ratio.

Signal recovery from the magnitude of the Fourier transform, or equivalently, from the autocorrelation, is a classical problem known as phase retrieval. Due to the absence of phase information, some form of additional information is required in order to be able to uniquely identify the underlying signal. In this work, we consider the problem of phase retrieval using masks. Due to our interest in developing robust algorithms with theoretical guarantees, we explore a convex optimization-based framework. In this work, we show that two specific masks (each mask provides 2n Fourier magnitude measurements) or five specific masks (each mask provides n Fourier magnitude measurements) are sufficient for a convex relaxation of the phase retrieval problem to provably recover almost all signals (up to global phase). We also show that the recovery is stable in the presence of measurement noise. This is a significant improvement over the existing results, which require O(log2 n) random masks (each mask provides n Fourier magnitude measurements) in order to guarantee unique recovery (up to global phase). Numerical experiments complement our theoretical analysis and show interesting trends, which we hope to explain in a future publication.

Intensity correlation imaging(ICI) method can obtain high resolution image with ground-based low precision mirrors, in the imaging process, phase retrieval algorithm should be used to reconstituted the object’s image. But the algorithm now used(such as hybrid input-output algorithm) is sensitive to noise and easy to stagnate. However the signal-to-noise ratio of intensity interferometry is low especially in imaging astronomical objects. In this paper, we build the mathematical model of phase retrieval and simplified it into a constrained optimization problem of a multi-dimensional function. New error function was designed by noise distribution and prior information using regularization method. The simulation results show that the regularization method can improve the performance of phase retrieval algorithm and get better image especially in low SNR condition