Phase Retrieval Algorithm Research Articles

Imaging of color targets through scattering media based on mixed speckle pattern separation

Color imaging of objects hidden behind scattering media is significant to many imaging scenarios. Although various color imaging methods have been proposed, most of them need to get prior or reference information. To avoid this, an imaging method of reconstructing color objects through scattering media is proposed in this paper, which is based on the irrelevance between speckles of different wavelengths and speckle correlation imaging. The speckles of different wavelengths are separated from the detected broadband speckles by the improved FastICA algorithm. Then the separated speckles are respectively reconstructed by the phase retrieval algorithm. This method is also applicable to reconstructing color objects surrounded by scattering media. The proposed method explores a direction for color imaging through scattering media based on spectral unmixing algorithms that separate speckles of different wavelengths from broadband speckles and has the potential value to be further developed to realize multispectral anti-scattering imaging.

Single-pixel coherent diffractive imaging based on super-pixel holography

Quantitative phase imaging (QPI) is a powerful approach to study dynamics associated with both thickness and refractive index fluctuations. In this work, we propose a coherent diffractive QPI scheme based on single-pixel imaging structure and coherent modulation imaging. In this scheme, digital micro-mirror device with high refresh rate is employed for structured illumination. Synchronously, a single point detector is used to perform mode selected measurement of diffraction intensity. Based on the illumination structure and point signals, phase imaging is realized using a reweighted amplitude flow phase retrieval algorithm. Non-interference modality omits the reference arm, which simplifies the apparatus and improves the system robustness. This proposed technique is demonstrated by QPI of both digital binary and grayscale objects. The three-dimensional profile measurement of a plano-convex lens shows the possibility of phase retrieval for real objects. Accordingly our technique will promote the application of single-pixel phase imaging to biomedical imaging, x-ray diffraction imaging and three-dimensional profilometry.

Known-plaintext attack-based analysis of double random phase encoding using multiple known plaintext-ciphertext pairs.

Double random phase encoding (DRPE) is vulnerable to known-plaintext attacks (KPAs) based on phase retrieval algorithms. We previously analyzed DRPE resistance to KPA cryptanalysis with multiple known plaintext-ciphertext image pairs and obtained secret keys at a higher probability rate than when performing KPA cryptanalysis using one known image pair. However, the robustness of KPA in the presence of noise or occlusion in DRPE is unclear. We analyzed KPA properties in relation to DRPE when white Gaussian noise was gradually added to the Fourier amplitude or phase of a known ciphertext complex amplitude image. Additionally, we analyzed KPA properties when the Fourier phase of the known ciphertext image was gradually occluded by zero-valued pixels. The results showed that KPAs performed using multiple known plaintext-ciphertext image pairs were largely affected by noise added to the Fourier phase and thus are not always a strong tool for DRPE cryptanalysis.

Binocular phase retrieval based on the transport of intensity equation under multiwavelength illumination

In view of the limitation that the transport of intensity equation method can only be used at a single wavelength, we propose a binocular phase retrieval algorithm based on multiwavelength illumination. First, we solve for the corresponding single-phase results at different wavelengths and then calculate the surface height of the object by combining the obtained synthetic wavelengths and synthetic phases, in order to reconstruct the initial phase results at different wavelengths. To solve the problem of amplification of traditional noise and surface profile noise in the phase diagram in the phase synthesis step, we introduce a step-by-step noise reduction method that can reduce the noise in the initial phase results to the level for a single wavelength and finally obtain the corresponding high-precision phase results at different wavelengths. The intensity images collected by the proposed dual-microscope system verify the correctness and effectiveness of the algorithm.

Phase retrieval for block sparsity based on adaptive coupled variational Bayesian learning

Phase retrieval (PR) of block-sparse signals is a new branch of sparse PR that causes rising research, which focusses with methods owing a high successful rate. However, the recovery performances of existing methods for block sparsity are usually unfit for large-scale problems with unacceptable compute complexity. We derive an algorithm for PR of block sparsity via variational Bayesian learning with expectation maximisation to mitigate this drawback. In the proposed algorithm, the block-sparse structure is modelled by the hierarchical constructional priors with a novel adaptive coupled pattern, which provides a strong relationship between the neighbour blocks. Simulations indicate that the proposed algorithm outperforms the existing methods in success rate, noise-robustness, and signal detection rate in large-scale cases with acceptable computation complexity.

Improved two-photon photopolymerisation and optical trapping with aberration-corrected structured light

We demonstrate the effectiveness of phase only aberration corrections of structured light and their application to versatile optical trapping setups. We calculate phase corrections before (ex-situ) and after (in-situ) a high numerical aperture microscope objective using a spatial light modulator (SLM), and investigate how these corrections can be used to improve the efficiency and resolution of micro-structures fabricated through two-photon-photopolymerisation (2PP). We apply a phase retrieval algorithm to correct for distortions in a femtosecond laser that enables the fabrication of 3D structures using as many as 50 simultaneous foci. The inclusion of aberration correction in the fabrication process shows improved confinement of optically trapped particles and more efficient polymerisation while minimising intensity variations at individual foci, which potentially damage the structure during fabrication. We find that phase corrections allow for consistent voxel sizes, increased sharpness, and an expanded effective printing range when using an SLM, while also allowing for closer proximity of individual trap foci, minimising interference effects that hinder fabrication resolution.

Accurate unsupervised estimation of aberrations in digital holographic microscopy for improved quantitative reconstruction.

In the context of digital in-line holographic microscopy, we describe an unsupervised methodology to estimate the aberrations of an optical microscopy system from a single hologram. The method is based on the Inverse Problems Approach reconstructions of holograms of spherical objects. The forward model is based on a Lorenz-Mie model distorted by optical aberrations described by Zernike polynomials. This methodology is thus able to characterize most varying aberrations in the field of view in order to take them into account to improve the reconstruction of any sample. We show that this approach increases the repeatability and quantitativity of the reconstructions in both simulations and experimental data. We use the Cramér-Rao lower bounds to study the accuracy of the reconstructions. Finally, we demonstrate the efficiency of this aberration calibration with image reconstructions using a phase retrieval algorithm as well as a regularized inverse problems algorithm.

Speckle spatial correlations aiding optical transmission matrix retrieval: the smoothed Gerchberg–Saxton single-iteration algorithm

The estimation of the transmission matrix of a disordered medium is a challenging problem in disordered photonics. Usually, its reconstruction relies on a complex inversion that aims at connecting a fully controlled input to the deterministic interference of the light field scrambled by the device. At the moment, iterative phase retrieval protocols provide the fastest reconstructing frameworks, converging in a few tens of iterations. Exploiting the knowledge of speckle correlations, we construct a new phase retrieval algorithm that reduces the computational cost to a single iteration. Besides being faster, our method is practical because it accepts fewer measurements than state-of-the-art protocols. Thanks to reducing computation time by one order of magnitude, our result can be a step forward toward real-time optical imaging that exploits disordered devices.

Hybrid brightfield and darkfield transport of intensity approach for high-throughput quantitative phase microscopy

Transport of intensity equation (TIE) is a well-established non-interferometric phase retrieval approach that enables quantitative phase imaging (QPI) by simply measuring intensity images at multiple axially displaced planes. The advantage of a TIE-based QPI system is its compatibility with partially coherent illumination, which provides speckle-free imaging with resolution beyond the coherent diffraction limit. However, TIE is generally implemented with a brightfield (BF) configuration, and the maximum achievable imaging resolution is still limited to the incoherent diffraction limit (twice the coherent diffraction limit). It is desirable that TIE-related approaches can surpass this limit and achieve high-throughput [high-resolution and wide field of view (FOV)] QPI. We propose a hybrid BF and darkfield transport of intensity (HBDTI) approach for high-throughput quantitative phase microscopy. Two through-focus intensity stacks corresponding to BF and darkfield illuminations are acquired through a low-numerical-aperture (NA) objective lens. The high-resolution and large-FOV complex amplitude (both quantitative absorption and phase distributions) can then be synthesized based on an iterative phase retrieval algorithm taking the coherence model decomposition into account. The effectiveness of the proposed method is experimentally verified by the retrieval of the USAF resolution target and different types of biological cells. The experimental results demonstrate that the half-width imaging resolution can be improved from 1230 nm to 488 nm with 2.5 × expansion across a 4 × FOV of 7.19 mm2, corresponding to a 6.25 × increase in space-bandwidth product from ∼5 to ∼30.2 megapixels. In contrast to conventional TIE-based QPI methods where only BF illumination is used, the synthetic aperture process of HBDTI further incorporates darkfield illuminations to expand the accessible object frequency, thereby significantly extending the maximum available resolution from 2NA to ∼5NA with a ∼5 × promotion of the coherent diffraction limit. Given its capability for high-throughput QPI, the proposed HBDTI approach is expected to be adopted in biomedical fields, such as personalized genomics and cancer diagnostics.

Ghost diffraction: A spatial statistical approach

We report a new method to reconstruct a two-dimensional object in Ghost diffraction scheme using a spatial statistical optics approach. This is implemented by using a static diffuser rather than a pseudo thermal light source generated with a rotating diffuser as used in conventional ghost imaging systems. The experimental implementation makes use of spatial ergodicity and spatial stationarity for spatially distributed random fields by following an analogy between time average and space average. The ensemble average is replaced by space average under the condition of spatial stationarity and ergodicity. A strategy to realize the Ghost diffraction scheme through spatial intensity correlation with a phase retrieval algorithm permits reconstruction of the transparency.

Fast algorithms for nonlinear and constrained phase retrieval in near-field X-ray holography based on Tikhonov regularization.

Based on phase retrieval, lensless coherent imaging and in particular holography offers quantitative phase and amplitude images. This is of particular importance for spectral ranges where suitable lenses are challenging, such as for hard x-rays. Here, we propose a phase retrieval approach for inline x-ray holography based on Tikhonov regularization applied to the full nonlinear forward model of image formation. The approach can be seen as a nonlinear generalization of the well-established contrast transfer function (CTF) reconstruction method. While similar methods have been proposed before, the current work achieves nonlinear, constrained phase retrieval at competitive computation times. We thus enable high-throughput imaging of optically strong objects beyond the scope of CTF. Using different examples of inline holograms obtained from illumination by a x-ray waveguide-source, we demonstrate superior image quality even for samples which do not obey the assumption of a weakly varying phase. Since the presented approach does not rely on linearization, we expect it to be well suited also for other probes such as visible light or electrons, which often exhibit strong phase interaction.

Coordinate-transformation iteration algorithm for phase modulation of arbitrary axial light distribution.

Axial light distribution modulation is widely applied in optical tweezers, hard-brittle material cutting, multilayer laser direct writing, etc. To generate arbitrary axial light distribution, the coordinate-transformation iteration (CTI) algorithm is presented. The CTI algorithm unifies equations in low and high numerical aperture (NA) scenarios, using the same iterative algorithm to produce phase computer-generated holograms. In a low NA scenario, twin-foci, flattop, and sin2 distributions have been achieved. In high NA scenarios, multirings, multifoci, and needle-like distributions have been realized in simulation with specific polarized incident beams. Since the CTI algorithm is inherently an efficient one-dimensional phase retrieval algorithm that is not limited by NA, this method has the potential to become a well-received solution for axial light distribution modulation.

Single frame color fringe projection profilometry based on empirical mode decomposition imbalance correction algorithm for fast three-dimensional reconstruction

Phase-shifting profilometry (PSP) is considered to be the most accurate technique for phase retrieval with fringe projection profilometry (FPP) systems. However, PSP requires that multiple phase-shifted fringe patterns be acquired, usually sequentially, which limits PSP to static or quasistatic imaging. We introduce color FPP (CFPP), which provides three-dimensional (3D) imaging using a single acquisition, for the application of real-time profiling. A single frame acquisition provides all three phase-shifted fringe patterns needed for the PSP phase retrieval algorithm. However, some problems must be overcome in CFPP, namely, color coupling and imbalance. Following up on the study, we propose a fixed coefficient decoupling method based on point-to-point correction and then introduce the empirical mode decomposition algorithm to decompose a color fringe pattern into three high-frequency components and three low-frequency components for imbalance correction, which improves measurement accuracy and realizes a single frame measurement for complex and discontinuous objects. After correction by the proposed method, the shape of the tested object can be recovered by one a single frame color fringe pattern, which meets the need for dynamic measurement, and the accuracy is not limited by the movement speed of object. The simulations and experimental results show that the proposed method can significantly reduce the influence of color coupling and imbalance in CFPP, and thus realize dynamic and high-precision 3D reconstruction.

Accelerating Reconstruction of Reflective Fourier Ptychographic Microscopy by Employing a Global Optimal Search Algorithm in a Graphics Processing Unit

Reflective-type Fourier ptychographic microscopy, an outstanding solution for surface inspection, employs a phase retrieval algorithm to reconstruct a high-resolution image with a wide field-of-view. This technique is time-consuming because it requires processing a large number of low-resolution images coming from the dark-field illuminator and bright-field illuminator. To decrease the computation time, we propose a new Fourier ptychography recovery method running in parallel in a graphics processing unit. We also utilized an adaptive step-size incremental gradient descent searching algorithm. The approach was validated by measuring a USAF reflective resolution target and by inspecting the surface of a smartphone integrated circuit. The processing time was remarkably reduced to 88.29 s, about 11.1 times less than the state-of-the-art method, while obtaining about 250 nm half-pitch resolution and about 1 × 1 mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> field-of-view.

Phase retrieval with dynamic linear combination in multiple intensity measurements

In the present methods of multiple intensity measurements, the recorded different intensity patterns are supposed to have the identical quality and the contribution of different images to the result is the same. However, intensity patterns are of different quality, since the raw patterns are affected by environment noise and focusing distance in the process of data acquisition. The equal weight scheme in data processing is not optimum in the multiple intensity measurements. A novel, to the best of our knowledge, phase retrieval algorithm with dynamic linear combination is proposed to improve resolution and robustness. In the algorithm, each pattern is assigned different weight according to its quality during reconstruction. The image quality evaluation function is applied to determine the weight coefficient. A higher-resolution image is reconstructed by the linear combination of all images, which is superior to the conventional algorithm.

Jitter-Robust Phase Retrieval Wavefront Sensing Algorithms.

Phase retrieval wavefront sensing methods are now of importance for imaging quality maintenance of space telescopes. However, their accuracy is susceptible to line-of-sight jitter due to the micro-vibration of the platform, which changes the intensity distribution of the image. The effect of the jitter shows some stochastic properties and it is hard to present an analytic solution to this problem. This paper establishes a framework for jitter-robust image-based wavefront sensing algorithm, which utilizes two-dimensional Gaussian convolution to describe the effect of jitter on an image. On this basis, two classes of jitter-robust phase retrieval algorithms are proposed, which can be categorized into iterative-transform algorithms and parametric algorithms, respectively. Further discussions are presented for the cases where the magnitude of jitter is unknown to us. Detailed simulations and a real experiment are performed to demonstrate the effectiveness and practicality of the proposed approaches. This work improves the accuracy and practicality of the phase retrieval wavefront sensing methods in the space condition with non-ignorable micro-vibration.

Phase retrieval for signals with block sparsity using BOMP: Algorithms and recovery guarantees

Phase retrieval (PR) from the magnitude-only measurements is a challenging problem that has attracted researchers in many scientific fields regarding the loss of phase information. However, PR for a block-sparse signal with non-zero elements present in several blocks has only a few appropriate methods that are not desirable in large-scale problems. This paper focuses on improving the adaptive capacity of PR of block sparsity for large-scale cases. Two practical greedy PR algorithms are proposed. The first is based on block orthogonal matching pursuit (BOMP) called BOMP-coordinate descent (CD). The other one invokes a tree search strategy of multipath matching pursuit (MMP), termed BMMP-CD, to promote the chance to the optimum. Theoretical conditions of BOMP-CD and BMMP-CD to select the correct index show the advantages of the proposed algorithms. Simulations show that the BOMP-CD is faster, while the recovery probability of BMMP-CD surpasses the compared counterparts.

Phase retrieval based on a total-variation-regularized Poisson model for X-ray ptychographic imaging of low-contrast objects

Hard X-ray ptychography has become an indispensable tool for observing the microscopic structure of a thick specimen. It measures diffraction patterns by scanning an X-ray beam and visualizes the complex-valued refractive index of the specimen by a computational reconstruction called phase retrieval. The quality of imaging is dependent on the used phase-retrieval algorithm, especially when the intensity of the diffraction patterns in the high-spatial-frequency range is low and/or when the spatial overlap of the illumination area is small. In this paper, a phase-retrieval algorithm, AMPAM, based on the Poisson model and total variation (TV) is proposed. It applies alternating minimization using primal-dual splitting and gradient-descent algorithms to compute the result without matrix inversion. The imaging capability of the proposed algorithm from low-dose and/or sparsely scanned data was investigated by numerical simulations. The proposed algorithm was compared with ADPr, which is the state-of-the-art algorithm based on the TV-regularized Poisson model. The results indicated that AMPAM can provide good-quality images with a computational cost 7–11 times less than ADPr. In addition, ink toner and macroporous silica particles were imaged at SPring-8 BL24XU to confirm the applicability of the algorithm to actual measurements.

Unified Initial Preprocessing for Phaseless Characterization of Quiet Zone in Millimeter-Wave Compact Antenna Test Range

This letter proposes a unified initial preprocessing technique to address phaseless characterization of a compact antenna test range (CATR) system within millimeter-wave range. With the aid of phase retrieval algorithm, the missing phase information of quiet zone (QZ) can be reconstructed by only two amplitude-only measurements. Here, the initial complex information consists of two parts: back propagation of measurements in QZ as initial amplitude and Zernike-Fourier expansion as initial phase. To further validate the superiority and robustness of the proposed approach, we conduct extensive simulations over the state-of-the-art phase retrieval algorithms under different operating frequencies and compare it to different initial preprocessing methods. In addition to be readily available in practical applications, the proposed technique is also verified under experimental conditions. In a nutshell, this letter paves the way toward the applications of phaseless characterization in CATR system.

Iterative phase retrieval with a sensor mask.

As an important inverse imaging problem in diffraction optics, Fourier phase retrieval aims at estimating the latent image of the target object only from the magnitude of its Fourier measurement. Although in real applications alternating methods are widely-used for Fourier phase retrieval considering the constraints in the object and Fourier domains, they need a lot of initial guesses and iterations to achieve reasonable results. In this paper, we show that a proper sensor mask directly attached to the Fourier magnitude can improve the efficiency of the iterative phase retrieval algorithms, such as alternating direction method of multipliers (ADMM). Furthermore, we refer to the learning-based method to determine the sensor mask according to the Fourier measurement, and unrolled ADMM is used for phase retrieval. Numerical results show that our method outperforms other existing methods for the Fourier phase retrieval problem.