Phase Diversity Algorithm Research Articles

Determination of the SO/PHI-HRT wavefront degradation using multiple defocused images

Context. The Polarimetric and Helioseismic Imager on board the Solar Orbiter mission (SO/PHI) offers refocusing capabilities to cope with the strongly varying thermal environment of the optical system along the spacecraft’s elliptical orbit. The series of images recorded during in-flight focus calibrations can be employed for phase diversity analyses. Aims. In this work we infer the wavefront degradation caused by the thermo-optical effects in the High Resolution Telescope (HRT) from images taken during the fine and coarse focus scans performed in the commissioning phase of the instrument. The difference between these two series of images are mainly related to the employed defocused step (smaller for the fine scans) and the signal-to-noise ratio (higher for the coarse scans). We use the retrieved wavefronts to reconstruct the original scene observed during the calibration of the instrument. Methods. We applied a generalized phase diversity algorithm that allowed us to use several images taken with different amounts of defocus to sense the wavefront degradation caused by the instrument. The algorithm also uses information from both the inferred wavefront and the series of images to restore the solar scene. Results. We find that most of the retrieved Zernike coefficients tend to converge to the same value when increasing the number of images employed for PD for both the fine and the coarse focusing scans. The restored scenes also show signs of convergence, and the merit function is minimized more as K increases. Apart from a defocus, the inferred wavefronts are consistent for the two datasets (λ/10 − λ/11). For the fine scan images, the quiet-sun contrast improves from 4.5% for the original focused image up to about 10%. For the coarse scan images, the contrast of the restored scene is as high as 11%.

Adaptive Optics Compensation for Orbital Angular Momentum Optical Wireless Communications

Adaptive optics (AO) can efficiently compensate for turbulence-induced distortion in orbital angular momentum (OAM)-based optical wireless communication (OWC) systems. In this paper, we design a modified phase diversity algorithm (MPDA)-based wavefront sensor to enhance the reconstruction accuracy of distorted OAM wavefront information. Aiming to further strike a compelling trade-off between AO system complexity and compensation accuracy, we first construct a novel AO system that applies a quickly and electronically controlled focus-tunable lens (FTL). It decontaminates distorted OAM signaling beams while having a low systemic complexity and superior convergence performance. Furthermore, we propose the 3-modified phase diversity algorithm (3-MPDA) AO scheme relying upon a Fourier intensity and two defocused intensities as the prior information, which beneficially balances the compensation effect and the number of defocused intensities and exhibits good noise robustness against charge-coupled device (CCD) detectors. In summary, this paper provides new insight for designing AO schemes with high compensation performance in communication links.

Cophasing multiple aperture telescopes with linearized analytic phase diversity

Focal plane wavefront sensing is an appealing technique to cophase multiple aperture telescopes. Phase diversity (PD), operable with any aperture configuration or source extension, generally suffers from high computing load. We introduce, characterize, and experimentally validate the linearized analytical phase diversity (LAPD) algorithm, based on a fast linearized PD algorithm with a capture range comparable to classic PD. We demonstrate that a typical performance of λ / 75 RMS wavefront error can be reached.

Phase diversity method based on an improved particle swarm algorithm used in co-phasing error detection.

The phase diversity (PD) algorithm is an image-based co-phasing error detection method for segmented mirror synthetic aperture optical systems. Particle swarm optimization (PSO) is suitable for PD for its fast convergence speed and simple structure. However, with the increase of the numbers of subapertures, optimization of cost function formed by the PD algorithm becomes a high-dimension non-linear optimization problem, which would lead PSO to result in a premature solution. Regarding the problem above, this paper presented a modified PSO in the PD algorithm to co-phase the segmented primary mirror, which overcame drawbacks of the traditional PSO, such as premature convergence and deactivation of particles, and enhanced the dig ability of the algorithm. Vast simulation results show that the modified PSO in the PD algorithm can effectively restore the segmented primary mirror, reducing the peak-to-valley (PV) value to 0.0012λ and the root mean square error to 0.0007λ, and raising the Strehl ratio to over 0.999. A comparison was also conducted to show that the modified PSO has advantages over two existing algorithms in higher accuracy and faster convergence speed.

On-sky verification of Fast and Furious focal-plane wavefront sensing: Moving forward toward controlling the island effect at Subaru/SCExAO

Context. High-contrast imaging (HCI) observations of exoplanets can be limited by the island effect (IE). The IE occurs when the main wavefront sensor (WFS) cannot measure sharp phase discontinuities across the telescope’s secondary mirror support structures (also known as spiders). On the current generation of telescopes, the IE becomes a severe problem when the ground wind speed is below a few meters per second. During these conditions, the air that is in close contact with the spiders cools down and is not blown away. This can create a sharp optical path length difference between light passing on opposite sides of the spiders. Such an IE aberration is not measured by the WFS and is therefore left uncorrected. This is referred to as the low-wind effect (LWE). The LWE severely distorts the point spread function (PSF), significantly lowering the Strehl ratio and degrading the contrast. Aims. In this article, we aim to show that the focal-plane wavefront sensing (FPWFS) algorithm, Fast and Furious (F&F), can be used to measure and correct the IE/LWE. The F&F algorithm is a sequential phase diversity algorithm and a software-only solution to FPWFS that only requires access to images of non-coronagraphic PSFs and control of the deformable mirror. Methods. We deployed the algorithm on the SCExAO HCI instrument at the Subaru Telescope using the internal near-infrared camera in H-band. We tested with the internal source to verify that F&F can correct a wide variety of LWE phase screens. Subsequently, F&F was deployed on-sky to test its performance with the full end-to-end system and atmospheric turbulence. The performance of the algorithm was evaluated by two metrics based on the PSF quality: (1) the Strehl ratio approximation (SRA), and (2) variance of the normalized first Airy ring (VAR). The VAR measures the distortion of the first Airy ring, and is used to quantify PSF improvements that do not or barely affect the PSF core (e.g., during challenging atmospheric conditions). Results. The internal source results show that F&F can correct a wide range of LWE phase screens. Random LWE phase screens with a peak-to-valley wavefront error between 0.4 μm and 2 μm were all corrected to a SRA > 90% and an VAR ⪅ 0.05. Furthermore, the on-sky results show that F&F is able to improve the PSF quality during very challenging atmospheric conditions (1.3–1.4″seeing at 500 nm). Closed-loop tests show that F&F is able to improve the VAR from 0.27–0.03 and therefore significantly improve the symmetry of the PSF. Simultaneous observations of the PSF in the optical (λ = 750 nm, Δλ = 50 nm) show that during these tests we were correcting aberrations common to the optical and NIR paths within SCExAO. We could not conclusively determine if we were correcting the LWE and/or (quasi-)static aberrations upstream of SCExAO. Conclusions. The F&F algorithm is a promising focal-plane wavefront sensing technique that has now been successfully tested on-sky. Going forward, the algorithm is suitable for incorporation into observing modes, which will enable PSFs of higher quality and stability during science observations.

Extending Capture Range for Piston Error in Segmented Primary Mirror Telescopes Based on Wavelet Support Vector Machine With Improved Particle Swarm Optimization

In the co-phasing techniques applied to the segmented telescope, a Shack-Hartmann wavefront sensor cannot accurately detect the piston error of the segment. Although the phase diversity (PD) algorithm can detect the piston error of each segment, it fails to reconstruct the wavefront quickly, and its dynamic range is small. Other technologies, such as prisms or micro-lens arrays, will significantly increase the complexity and construction cost of the optical system. Moreover, they may also introduce non-common path errors. In this study, we propose an approach to address this challenging problem via curvature sensing. This method uses multi-wavelength to eliminate the influence of $2\pi $ ambiguity and improve the capture range of co-phasing detection. However, curvature sensing is easily influenced by atmospheric seeing. We propose a wavelet support vector machine optimized via particle swarm optimization (PSO-WSVM) method to deal with this problem, and to improve the application scope of curvature sensing. We reshape SVM with a wavelet kernel function, and improve the PSO algorithm. We train the SVM to build a prediction model to distinguish the piston error range of each pair of adjacent segments and surpass $2\pi $ ambiguity. First, we obtain defocused images by means of the convolution technique. Second, we propose a prediction model based on SVM. We select the correlation coefficient between the sampling signal and the template signal at different wavelengths as the input vector, and we choose a wavelet basis function as the kernel function of SVM. Third, we improve the PSO algorithm with the exponential decreasing inertia weight (EDIW) to tune the parameters of SVM. Finally, we perform a simulation experiment on a real optical system model based on the Keck telescope. The results indicate that the performance of this method is better than that of other state-of-the-art SVM-based classifiers, and it works rapidly during the observation.

Phase diversity algorithm with high noise robust based on deep denoising convolutional neural network.

The wave-front phase expanded on the Zernike polynomials is estimated from a pair of images by the use of a maximum-likelihood approach, the in-focus image and the defocus image, which contaminated by noise, will greatly reduce the solution accuracy of the phase diversity (PD) algorithm. In the study, we introduce the deep denoising convolutional neural networks (DnCNNs) into the image preprocessing of PD to denoise the in-focus image and defocus the image containing gaussian white noise to improve the robustness of PD to noise. The simulation results show that the composite PD algorithm with DnCNNs is better than the traditional PD algorithm in both RMSE of phase estimation and SSIM, and the mean of the RMSE of the phase estimation of the improved PD algorithm is reduced by 78.48%, 82.35%, 71.09% and 73.67% compared with the mean of the RMSE of the phase estimation of the traditional PD algorithm. The well-trained DnCNNs runs fast, which does not increase the running time of traditional PD algorithms, and the compound approach may be widely used in various domains, such as the measurements of intrinsic aberrations in optical systems and compensations for atmospheric turbulence.

Large-scale piston error detection technology for segmented optical mirrors via convolutional neural networks.

In the cophasing of the segmented optical mirrors, the Shack-Hartmann wavefront sensor is not sensitive to the submirror piston error and the large range piston errors beyond the cophasing detection range of phase diversity algorithm. It is necessary to introduce specific sensors (e.g.,microlenses or prisms), but they greatly increase the complexity and manufacturing cost of the optical system. In this Letter, we introduce the convolutional neural network (CNN) to distinguish the piston error range of each submirror. To get rid of the dependence of the CNN dataset on the imaging target, we construct the feature vector by the in-focal and defocused images. The method surpasses the fundamental limit of the detection range by using different wavelengths. Finally, the results of the simulation experiment indicate that the method is effective.

Variable step size adaptive cuckoo search optimization algorithm for phase diversity.

The phase diversity (PD) algorithm will eventually be converted into a large-scale nonlinear numerical optimization problem, so the selection of numerical optimization algorithm will directly determine the accuracy and speed of the algorithm settlement. In this paper, we introduce the cuckoo search optimization algorithm, which has the advantages of simple model, few parameters, and easy implementation, to the phase diversity algorithm. By improving the step size control factor in the original cuckoo search algorithm, we can make it have faster optimization speed for PD. In the simulation experiments, we further proved and gave a simple explanation in theory that in the case of large-scale wavefront sensing, compared to the traditional particle swarm algorithm, this improved algorithm has higher accuracy and faster convergence speed. Finally, we set up a simple experimental system and proved the effectiveness of the improved cuckoo search algorithm for PD.

Laboratory verification of Fast & Furious phase diversity: Towards controlling the low wind effect in the SPHERE instrument

Context. The low wind effect (LWE) refers to a characteristic set of quasi-static wavefront aberrations seen consistently by the SPHERE instrument when dome-level wind speeds drop below 3 ms−1. The LWE produces bright low-order speckles in the stellar point-spread function (PSF), which severely limit the contrast performance of SPHERE under otherwise optimal observing conditions. Aims. In this paper we propose the Fast & Furious (F&F) phase diversity algorithm as a viable software-only solution for real-time LWE compensation, which would utilise image sequences from the SPHERE differential tip-tilt sensor (DTTS) and apply corrections via reference slope offsets on the AO system’s Shack-Hartmann wavefront sensor. Methods. We evaluated the closed-loop performance of F&F on the MITHIC high-contrast test-bench, under conditions emulating LWE-affected DTTS images. These results were contrasted with predictive simulations for a variety of convergence tests, in order to assess the expected performance of an on-sky implementation of F&F in SPHERE. Results. The algorithm was found to be capable of returning LWE-affected images to Strehl ratios of greater than 90% within five iterations, for all appropriate laboratory test cases. These results are highly representative of predictive simulations, and demonstrate stability of the algorithm against a wide range of factors including low image signal-to-noise ratio (S/N), small image field of view, and amplitude errors. It was also found in simulation that closed-loop stability can be preserved down to image S/N as low as five while still improving overall wavefront quality, allowing for reliable operation even on faint targets. Conclusions. The Fast & Furious algorithm is an extremely promising solution for real-time compensation of the LWE, which can operate simultaneously with science observations and may be implemented in SPHERE without requiring additional hardware. The robustness and relatively large effective dynamic range of F&F also make it suitable for general wavefront optimisation applications, including the co-phasing of segmented ELT-class telescopes.

Fast correction approach for wavefront sensorless adaptive optics based on a linear phase diversity technique.

Wavefront sensorless (WFSless) adaptive optics (AO) systems have been widely studied in recent years. To reach optimum results, such systems require an efficient correction method. This paper presents a fast wavefront correction approach for a WFSless AO system mainly based on the linear phase diversity (PD) technique. The fast closed-loop control algorithm is set up based on the linear relationship between the drive voltage of the deformable mirror (DM) and the far-field images of the system, which is obtained through the linear PD algorithm combined with the influence function of the DM. A large number of phase screens under different turbulence strengths are simulated to test the performance of the proposed method. The numerical simulation results show that the method has fast convergence rate and strong correction ability, a few correction times can achieve good correction results, and can effectively improve the imaging quality of the system while needing fewer measurements of CCD data.

High-accuracy wavefront sensing by phase diversity technique with bisymmetric defocuses diversity phase

We investigate a specific diversity phase for phase diversity (PD) phase retrieval, which possesses higher accuracy than common PD, especially for large-scale and high-frequency wavefront sensing. The commonly used PD algorithm employs the image intensities of the focused plane and one defocused plane to build the error metric. Unlike the commonly used PD, we explore a bisymmetric defocuses diversity phase, which employs the image intensities of two symmetrical defocused planes to build the error metric. This kind of diversity phase, named PD-BD (bisymmetric defocuses phase diversity), is analysed with the Cramer-Rao lower bound (CRLB). Statistically, PD-BD shows smaller CRLBs than the commonly used PD, which indicates stronger capacity of phase retrieval. Numerical simulations also verify that PD-BD has higher accuracy of phase retrieval than the commonly used PD when dealing with large-scale and high-frequency wavefront aberrations. To further affirm that PD-BD possesses higher accuracy of wavefront sensing than PD, we also perform a simple verification experiment.

Detecting wavefront amplitude and phase using linear phase diversity.

We propose a simplified phase diversity method based on the first-order Taylor expansion of the optical transfer function (OTF) as the light intensity on the pupil plane is not uniform but presents Gaussian distribution. By extending the Zernike polynomial term to the complex field, we regard the distribution of the light intensity as the imaginary part of the phase, which can be solved together with the phase aberration (the real part of the phase). The approximation of the OTF using the first-order Taylor expansion can simplify the relationship between the phase and OTF to linear relation, so that the aberration can be quickly solved. We present an experiment that validates the advantage of our method. The result shows that the modified method has a higher accuracy with less time compared to the traditional phase diversity algorithm, and can acquire the distribution of the light on the pupil plane.

Parallel blind deconvolution reconstruction using phase diversity for three-dimensional imaging

Phase diversity (PD) is the special case of multiframe blind deconvolution algorithm, where an object is observed with a stack intentionally defocused image. The known defocused information between phases is used to estimate the object and aberration function. This paper presents a parallel three-dimensional (3-D) wavefront aberration estimation using blind deconvolution. We used multiplane PD for widefield fluorescence microscopy images of 3-D objects. Parallel multiplane PD yields a speedup factor of 4.33 with respect to the counterpart sequential algorithm for an image size of 256×256. We used the outcome of the PD algorithm to generate 3-D point cloud data from images from one view.

Simplified Phase Diversity algorithm based on a first-order Taylor expansion.

We present a simplified solution to phase diversity when the observed object is a point source. It utilizes an iterative linearization of the point spread function (PSF) at two or more diverse planes by first-order Taylor expansion to reconstruct the initial wavefront. To enhance the influence of the PSF in the defocal plane which is usually very dim compared to that in the focal plane, we build a new model with the Tikhonov regularization function. The new model cannot only increase the computational speed, but also reduce the influence of the noise. By using the PSFs obtained from Zemax, we reconstruct the wavefront of the Hubble Space Telescope (HST) at the edge of the field of view (FOV) when the telescope is in either the nominal state or the misaligned state. We also set up an experiment, which consists of an imaging system and a deformable mirror, to validate the correctness of the presented model. The result shows that the new model can improve the computational speed with high wavefront detection accuracy.

An Efficient Correction Algorithm for Eliminating Image Misalignment Effects on Co-Phasing Measurement Accuracy for Segmented Active Optics Systems.

The misalignment between recorded in-focus and out-of-focus images using the Phase Diversity (PD) algorithm leads to a dramatic decline in wavefront detection accuracy and image recovery quality for segmented active optics systems. This paper demonstrates the theoretical relationship between the image misalignment and tip-tilt terms in Zernike polynomials of the wavefront phase for the first time, and an efficient two-step alignment correction algorithm is proposed to eliminate these misalignment effects. This algorithm processes a spatial 2-D cross-correlation of the misaligned images, revising the offset to 1 or 2 pixels and narrowing the search range for alignment. Then, it eliminates the need for subpixel fine alignment to achieve adaptive correction by adding additional tip-tilt terms to the Optical Transfer Function (OTF) of the out-of-focus channel. The experimental results demonstrate the feasibility and validity of the proposed correction algorithm to improve the measurement accuracy during the co-phasing of segmented mirrors. With this alignment correction, the reconstructed wavefront is more accurate, and the recovered image is of higher quality.

相位差异法探测波前的误差分析及消除方法

相位差异法探测波前的误差分析及消除方法

Co-phasing of the segmented mirror and image retrieval based on phase diversity using a modified algorithm.

The conventional Broyden-Fletcher-Goldfarb-Shanno (BFGS) method used to solve the cost function of a phase diversity (PD) algorithm converges to a global optimum only when the cost function is convex. We present a modified BFGS method, which has fine global convergences for both convex and nonconvex functions, guarantees that the solutions will converge to the global minimum, corresponding to the actual wavefront coefficients, and apply it to minimize the PD cost function to co-phase the segmented active optics system and recover the unknown object under different noise levels. The noise amplification effect on the accuracy of the algorithm is removed by our proposed estimated strategy of the regularization parameter for the PD problem. The vast contrast results demonstrate that the modified method has a much higher accuracy than the conventional BFGS method for the nonconvex condition even under a considerably high noise level.

Redundancy computation analysis and implementation of phase diversity based on GPU

Phase diversity method is not only used as an image restoration technique, but also as a wavefront sensor. However, its computations have been perceived as being too burdensome to achieve its real-time applications on a desktop computer platform. In this paper, the implementation of the phase diversity algorithm based on graphic processing unit (GPU) is presented. The redundancy computations for the pupil function, point spread function, and optical transfer function are analyzed. Two kinds of implementation methods based on GPU are compared: one is the general method which is accomplished by GPU library CUFFT without precision loss (method-1) and the other one performed by our own custom FFT with little damage of precision considering the redundant calculations (method-2). The results show the cost and gradient functions can be speeded up by method-2 in contrast with the method-1 and the overhead of global memory access by kernel fusion can be reduced. For the image of 256 x 256 with the sampling factor of 3, the results reveal that method-2 achieves speedup of 1.83x compared with method-1 when the central 128 x 128 pixels of the point spread function are used.

Phase diversity for calibrating noncommon path aberrations of adaptive optics system under nonideal measurement environment

When the defocus cannot be measured and the wavefront solution set is restricted by a multi-channel, some practical problems exist in the calibration of the noncommon path aberrations of the adaptive optics system. To solve these problems, an evaluation function of phase diversity algorithm is constructed in this paper. We use the method that the estimated aberration and the modulated deformable mirror iterate each other to make up the nonideal measurement environment. Then the ill-pose problem of the solution by phase diversity, produced as relaxing constraints of the diversity defocus on the wavefront solution set, is solved. We have adopted the proposed method to measure the noncommon path aberrations of the adaptive optics system on a 1.23m telescope. Experimental results demonstrate that wavefront solution is more accurate and the whole imaging quality is improved effectively by using the deformable mirror to compensate the aberration measured.