Zernike Polynomials Research Articles

Following the Flux in diffracted fields – An efficient numerical method for tracing the Eikonal function

We report on our recently developed method for tracing lines of flux in three-dimensional diffracted wavefields using an approach we call ’non-linear ray tracing’ but which can be more accurately described as tracing the lines of flux in the context of the Eikonal function. These ’rays’ navigate through the diffracted field guided by the derivative of the phase at a sequence of successive points. Our approach is based on the Angular Spectrum method, a numerical algorithm that accurately and efficiently calculates diffracted fields for numerical apertures <0.7. This is used to generate a three-dimensional grid of complex wavefield samples in the focal region, followed by tracing the flux through this volume. The ray propagates in a straight line between two consecutive planes within the volume; the phase derivative is calculated at each plane to direct the ray on the next step of its journey. We demonstrate the effectiveness of our approach by generating results for focused laser beams with TEM00 and TEM01 laser profiles. We also simulate the effects of optical aberrations, described by Zernike polynomials, on the three-dimensional focused wavefield. Our non-linear ray tracing method provides a powerful and efficient approach for tracing lines of flux through complex three-dimensional wavefields, offering an exciting new tool for understanding these complex systems.

Convolutional neural network optimisation to enhance ESPI fringe visibility

The use of convolutional neuronal networks (CNN) for the treatment of interferometric fringes has been introduced in recent years. In this paper, we optimize and build a CNN model, based U-NET architecture, to maximize its performance processing electronic speckle interferometry fringes (ESPI). The proposed approach is based on quick and light trainings to select the architecture parameters (network depth and kernel sizes) to maximize the performance of the neural network improving the visibility of ESPI images. To measure the performance, the structural similarity index (SSMI) will be the lead indicator, and the need for large datasets to train neural networks, unavailable for ESPI images, forces the use of a simulated ESPI image dataset along the process. This dataset is computed using Zernike polynomials to simulate local surface deformations in the specimen under test and simulated true speckle fields for the reference and object field involved in ESPI techniques.

Multi-Field-of-View Sparse Aperture Imaging Based on Double Zernike Polynomials

Multi-Field-of-View Sparse Aperture Imaging Based on Double Zernike Polynomials

Hyperspectral compressive wavefront sensing

Abstract Presented is a novel way to combine snapshot compressive imaging and lateral shearing interferometry in order to capture the spatio-spectral phase of an ultrashort laser pulse in a single shot. A deep unrolling algorithm is utilized for snapshot compressive imaging reconstruction due to its parameter efficiency and superior speed relative to other methods, potentially allowing for online reconstruction. The algorithm’s regularization term is represented using a neural network with 3D convolutional layers to exploit the spatio-spectral correlations that exist in laser wavefronts. Compressed sensing is not typically applied to modulated signals, but we demonstrate its success here. Furthermore, we train a neural network to predict the wavefronts from a lateral shearing interferogram in terms of Zernike polynomials, which again increases the speed of our technique without sacrificing fidelity. This method is supported with simulation-based results. While applied to the example of lateral shearing interferometry, the methods presented here are generally applicable to a wide range of signals, including Shack–Hartmann-type sensors. The results may be of interest beyond the context of laser wavefront characterization, including within quantitative phase imaging.

Loss Minimized Data Reduction in Single-Cell Tomographic Phase Microscopy Using 3D Zernike Descriptors

Tomographic phase microscopy (TPM) in flow cytometry is one of the most promising computational imaging techniques for the quantitative 3-dimensional (3D) analysis of unstained single cells. Continuous cells’ flow, combined with the stain-free mode, can assure the high-throughput collection of quantitative and informative 3D data. TPM promises to allow rapid cells’ screening by a nondestructive technique and with statistically relevant data. The current leading-edge research aimed at developing TPM systems in flow cytometry has already demonstrated the possibility of acquiring thousands of single-cell tomograms. Nevertheless, a key unsolved problem exists about the efficient storage and easy handling of such a huge amount of 3D data that prevents rapid analysis for cell diagnosis. Here, we show, for the first time, an effective encoding strategy of single-cell tomograms that can completely overcome this critical bottleneck. Essentially, by using the 3D version of Zernike polynomials, we demonstrate that the 3D refractive index distribution of a cell can be straightforwardly encoded in 1D with negligible information loss (<1%), thus greatly streamlining the data handling and storage. The performance analysis of the proposed method has been first assessed on simulated tomographic cell phantom, while the experimental validation has been extensively proofed on tomographic data from experiments with different cell lines. The results achieved here imply an intriguing breakthrough for TPM that promises to unlock computational pipelines for analyzing 3D data that were unattainable until now.

Limitations of Reconstructing Pentacam Rabbit Corneal Tomography by Zernike Polynomials.

The study aims to investigate the likelihood of Zernike polynomial being used for reconstructing rabbit corneal surfaces as scanned by the Pentacam segment tomographer, and hence evaluate the accuracy of corneal power maps calculated from such Zernike fitted surfaces. The study utilised a data set of both eyes of 21 rabbits using a reverse engineering approach for deductive reasoning. Pentacam raw elevation data were fitted to Zernike polynomials of orders 2 to 20. The surface fitting process to Zernike polynomials was carried out using randomly selected 80% of the corneal surface data points, and the root means squared fitting error (RMS) was determined for the other 20% of the surface data following the Pareto principle. The process was carried out for both the anterior and posterior surfaces of the corneal surfaces that were measured via Pentacam scans. Raw elevation data and the fitted corneal surfaces were then used to determine corneal axial and tangential curvature maps. For reconstructed surfaces calculated using the Zernike fitted surfaces, the mean and standard deviation of the error incurred by the fitting were calculated. For power maps computed using the raw elevation data, different levels of discrete cosine transform (DCT) smoothing were employed to infer the smoothing level utilised by the Pentacam device. The RMS error was not significantly improved for Zernike polynomial orders above 12 and 10 when fitting the anterior and posterior surfaces of the cornea, respectively. This was noted by the statistically non-significant increase in accuracy when the order was increased beyond these values. The corneal curvature calculations suggest that a smoothing process is employed in the corneal curvature maps outputted by the Pentacam device; however, the exact smoothing method is unknown. Additionally, the results suggest that fitting corneal surfaces to high-order Zernike polynomials will incur a clinical error in the calculation of axial and tangential corneal curvature of at least 0.16 ± 01 D and 0.36 ± 0.02 D, respectively. Rabbit corneal anterior and posterior surfaces scanned via the Pentacam were optimally fitted to orders 12 and 10 Zernike polynomials. This is essential to get stable values of high-order aberrations that are not affected by Zernike polynomial fittings, such as comas for Intracorneal Ring Segments (ICRS) adjustments or spherical aberration for pre-cataract operations. Smoothing was necessary to replicate the corneal curvature maps outputted by the Pentacam tomographer, and fitting corneal surfaces to Zernike polynomials introduces errors in the calculation of both the axial and tangential corneal curvatures.

Anterior Corneal High-order Aberrations in Fuchs' Endothelial Corneal Dystrophy Classified by Scheimpflug Tomography.

Visual acuity may be limited after endothelial keratoplasty in Fuchs' endothelial corneal dystrophy. This may be due to increased anterior high-order aberrations that persist after endothelial keratoplasty. It is not clear at what stage these aberrations begin. We aimed to evaluate how subclinical corneal edema affects anterior corneal high-order aberrations. We aimed to evaluate anterior corneal high-order aberrations according to the presence of subclinical corneal edema in Fuchs' endothelial corneal dystrophy using Scheimpflug topography. All the patients underwent corneal topography (Sirius Scheimpflug-Placido disc camera; CSO Ophthalmic, Scandicci, Italy). Fifty-two eyes of 31 patients with Fuchs' endothelial corneal dystrophy and 52 eyes of 35 patients without corneal problems were included. In addition, patients with Fuchs' endothelial corneal dystrophy were divided into two groups according to subclinical corneal edema as with Fuchs' endothelial corneal dystrophy-edema and Fuchs' endothelial corneal dystrophy-nonedema. Subclinical corneal edema was defined based on the presence of at least two of three findings of the loss of regular isopachs, displacement of the thinnest point of the cornea, and posterior depression. High-order aberrations were expressed as Zernike polynomials in the 4- and 6-mm optical zone. We found that subclinical corneal edema in Fuchs' endothelial corneal dystrophy caused a larger 4-mm trefoil II and a smaller 6-mm spherical aberration II measurements than the nonedema group. The 6-mm total high-order aberrations, 6-mm coma aberration, and 4- and 6-mm trefoil II aberration measurements were larger, and the 6-mm spherical aberration II was smaller in Fuchs' endothelial corneal dystrophy patients with edema compared with the healthy eyes. The presence of subclinical corneal edema in Fuchs' endothelial corneal dystrophy results in larger 4-mm trefoil II and smaller 6-mm spherical aberration II measurements. More patients with intermediate values are needed to evaluate the effect of these aberrations on visual acuity.

The -isomorphism property for a class of singularly-weighted x-ray transforms

We study a one-parameter family of self-adjoint normal operators for the x-ray transform on the closed Euclidean disk , obtained by considering specific singularly weighted L 2 topologies. We first recover the well-known singular value decompositions in terms of orthogonal disk (or generalized Zernike) polynomials, then prove that each such realization is an isomorphism of . As corollaries: we give some range characterizations; we show how such choices of normal operators can be expressed as functions of two distinguished differential operators. We also show that the isomorphism property also holds on a class of constant-curvature, circularly symmetric simple surfaces. These results allow to design functional contexts where normal operators built out of the x-ray transform are provably invertible, in Fréchet and Hilbert spaces encoding specific boundary behavior.

Single-pixel imaging using discrete Zernike moments.

A novel single-pixel imaging (SPI) technique based on discrete orthogonal Zernike moments is proposed. In this technique, the target object is illuminated by two sets of Zernike basis patterns which satisfy the Zernike polynomials. The Zernike moments of object image are obtained by measuring the reflected light intensities through a single-pixel detector. And the object image is reconstructed by summing the product of Zernike polynomials and detected intensities iteratively. By theoretical and experimental demonstrations, an image with high quality is retrieved under compressive sampling. Moreover, the Zernike illuminating patterns are used for object classification due to the rotation invariant of Zernike moments. By measuring the amplitudes of a few specific Zernike moments through the SPI system, the rotated images with different angles and the same content are classified into the same class on experiment. This classification technique has the advantages of high efficiency and high accuracy due to the high modulation speed and high sensitivity of SPI system.

Zernike mode rescaling extends capabilities of adaptive optics for microscopy

Zernike polynomials are widely used mathematical models of experimentally observed optical aberrations, and they have found widespread use in adaptive optic realizations that are used to correct wavefront aberrations. However, Zernike aberrations lose their orthogonality when used in combination with Gaussian beams and, as a consequence, start to cross-couple between each other, a phenomenon that does not occur for Zernike aberrations in plane waves. Here, we describe how the aberration radius (i.e. the radius of the beam relative to the active aperture of an active optical element) influences this cross-coupling of Zernike aberrations in a way that is distinct from simple truncation or balancing. Furthermore, we show that this effect can actually be harnessed to allow efficient compensation of higher-order aberrations using only low-order Zernike modes. This finding has important practical implications, as it suggests the possibility of using adaptive optics devices with low element numbers to compensate aberrations that would normally require more complex and expensive devices.

Phase retrieval of two random phase-shifting interferograms using Zernike coefficient extraction network.

This paper proposes a deep learning method for phase retrieval from two interferograms. The proposed method converts phase retrieval into the Zernike coefficient extraction problem, which can achieve Zernike coefficient extraction from two interferograms with random phase shifts. After knowing Zernike coefficients, the phase distribution can be retrieved using Zernike polynomials. The pre-filtering and phase unwrapping process are not required using the proposed method. The simulated data are analyzed, and the root mean square (RMS) of phase error reaches 0.01 λ. The effectiveness of the method is verified by the measured data.

Estimation of far-field wavefront error of tilt-to-length distortion coupling in space-based gravitational wave detection

In space-based gravitational wave detection, the estimation of far-field wavefront error of the distorted beam is the precondition for the noise reduction. Zernike polynomials are used to describe the wavefront error of the transmitted distorted beam. The propagation of a laser beam between two telescope apertures is calculated numerically. Far-field wavefront error is estimated with the absolute height of the peak-to-valley phase deviation between the distorted Gaussian beam and a reference distortion-free Gaussian beam. The results show that the pointing jitter is strongly related to the wavefront error. Furthermore, when the jitter decreases 10 times from 100 nrad to 10 nrad, the wavefront error reduces for more than an order of magnitude. In the analysis of multi-parameter minimization, the minimum of wavefront error tends to Z[5,3] Zernike in some parameter ranges. Some Zernikes have a strong correlation with the wavefront error of the received beam. When the aperture diameter increases at Z[5,3] Zernike, the wavefront error is not monotonic and has oscillation. Nevertheless, the wavefront error almost remains constant with the arm length increasing from 10−1 Mkm to 103 Mkm. When the arm length decreases for three orders of magnitude from 10−1 Mkm to 10−4 Mkm, the wavefront error has only an order of magnitude increasing. In the range of 10−4 Mkm to 103 Mkm, the lowest limit of the wavefront error is from 0.5 fm to 0.015 fm at Z[5,3] Zernike and 10 nrad jitter.

Direct retrieval of Zernike-based pupil functions using integrated diffractive deep neural networks

Retrieving the pupil phase of a beam path is a central problem for optical systems across scales, from telescopes, where the phase information allows for aberration correction, to the imaging of near-transparent biological samples in phase contrast microscopy. Current phase retrieval schemes rely on complex digital algorithms that process data acquired from precise wavefront sensors, reconstructing the optical phase information at great expense of computational resources. Here, we present a compact optical-electronic module based on multi-layered diffractive neural networks printed on imaging sensors, capable of directly retrieving Zernike-based pupil phase distributions from an incident point spread function. We demonstrate this concept numerically and experimentally, showing the direct pupil phase retrieval of superpositions of the first 14 Zernike polynomials. The integrability of the diffractive elements with CMOS sensors shows the potential for the direct extraction of the pupil phase information from a detector module without additional digital post-processing.

Estimation of Zernike polynomials for a highly focused electromagnetic field using polarimetric mapping images and neural networks

In this communication, we present a method to estimate the aberrated wavefront at the focal plane of a vectorial diffraction system. In contrast to the phase, the polarization state of optical fields is simply measurable. In this regard, we introduce an alternative approach for determining the aberration of the wavefront using polarimetric information. The method is based on training a convolutional neural network using a large set of polarimetric mapping images obtained by simulating the propagation of aberrated wavefronts through a high-NA microscope objective; then, the coefficients of the Zernike polynomials could be recovered after interrogating the trained network. On the one hand, our approach aims to eliminate the necessity of phase retrieval for wavefront sensing applications, provided the beam used is known. On the other hand, the approach might be applied for calibrating the complex optical system suffering from aberrations. As proof of concept, we use a radially polarized Gaussian-like beam multiplied by a phase term that describes the wavefront aberration. The training dataset is produced by using Zernike polynomials with random coefficients. Two thousand random combinations of polynomial coefficients are simulated. For each one, the Stokes parameters are calculated to introduce a polarimetric mapping image as the input of a neural network model designed and trained for predicting the polynomial coefficients. The accuracy of the neural network model is tested by predicting an unseen dataset (test dataset) with a high success rate.

Effect of crystalline lens on visual quality in pterygium patients before surgery

AbstractPurpose: The aim of our study was to assess the visual quality of pterygium patients with total ocular aberrations, and not only corneal aberrations.Methods: 46 eyes of 37 pterygium patients (mean age 48 ± 15 years) were evaluated in a retrospective study. Inclusion criteria were primary pterygium, without any previous surgical treatment. Before surgery, subjective refraction, corrected distance visual acuity (CDVA), with an EDTRS optotype chart, and Zernike coefficients for a 5 millimetres (mm) pupil diameter measured with an Alcon LadarWave total aberrometer (Fort Worth, Texas, USA) were determined. Pterygium size in mm was measured by an experienced ophthalmologist using a slit beam of light from the limbus to the limit of the pterygium. Statistical analysis was performed using SPSS software version 26.0. The Kruskal‐Wallis test with the Tukey–Kramer criterion was used to examine the influence of the extent of corneal invasion on manifest refraction and Zernike coefficients. One‐way analysis of variance (ANOVA) with the Bonferroni criterion was performed to compare different extent values. A p value less than or equal to 0.05 was considered statistically significant.Results: The sample was divided in four groups, according to the extent of corneal invasion. The upper limits of each category were 1, 2, 3 and 4 mm, with 9, 20, 11 and 6 patients in each group, respectively. Preoperatively spheres more positive were found at high corneal invasion groups, statistically significant between 1 and 3 mm, and between 2 and 3 mm (p = 0.013 and p = 0.011) and cylinder axes were between 20° and 50° in all groups. CDVA was lower in patients with greater corneal invasion. Regarding total ocular aberrations, statistically significant differences between groups were found only in a reduced set of Zernike polynomials but oblique astigmatism (2,‐2) appeared in most of the comparatives. Most of the Zernike coefficients, were out of normal limits, compared to control values, including spherical aberration.Conclusions: Pterygium‐induced aberrations also modify the visual quality of patients when the effect of the crystalline lens aberrations is considered, the conclusions are similar to those derived from corneal aberrations. CDVA was lower preoperatively in patients with greater corneal invasion (4 mm).

Novel Simulation and Analysis of Mie-Scattering Lidar for Detecting Atmospheric Turbulence Based on Non-Kolmogorov Turbulence Power Spectrum Model.

The Mie-scattering lidar can detect atmospheric turbulence intensity by using the return signals of Gaussian beams at different heights. The power spectrum method and Zernike polynomial method are used to simulate the non-Kolmogorov turbulent phase plate, respectively, and the power spectrum method with faster running speed is selected for the subsequent simulation. In order to verify the possibility of detecting atmospheric turbulence by the Mie-scattering lidar, some numerical simulations are carried out. The power spectrum method is used to simulate the propagation of the Gaussian beam from the Mie-scattering lidar in a vertical path. The propagation characteristics of the Gaussian beam using a non-Kolmogorov turbulence model are obtained by analyzing the intensity distribution and spot drift effect. The simulation results show that the scintillation index of simulation is consistent with the theoretical value trend, and the accuracy is very high, indicating that the method of atmospheric turbulence detection using Mie-scattering lidar is effective. The simulation plays a guiding role for the subsequent experimental platform construction and equipment design.

Zernike-like Laguerre-Gaussian orthonormal polynomials for optical field reconstruction.

Analytic closed form expressions for orthonormal polynomials exhibiting both rotational and Gaussian symmetries are derived for both circular and elliptical geometries. They exhibit a close correspondence to the Zernike polynomials but are of Gaussian shape and orthogonal over the (x,y) plane. Consequently, they may be expressed in terms of Laguerre polynomials. Formulas for calculating the centroid of a real function are also presented and, along with the analytic expressions for the polynomials, may prove to be of especial use in reconstruction of the intensity distribution incident on a Shack-Hartmann wavefront sensor.

Topology Optimization Design of an Active Deformable Mirror Based on Discrete Orthogonal Zernike Polynomials

In order to design an active deformation mirror for projection objective aberration imaging quality control, a topology optimization design method of active deformation mirrors based on discrete orthogonal Zernike polynomials is proposed in this paper. Firstly, in order to solve the problem that continuous Zernike polynomials do not have orthogonality on the discrete coordinates inside the unit circle, which causes the instability of topology optimization results, discrete orthogonal Zernike polynomials are used to characterize the active deformation mirror wave aberrations. Then, the optical and structural deformations are combined to establish an optical-mechanical coupling topology optimization model with the help of the variable density method to derive the sensitivity of the mathematical model. Finally, a wave aberration corrected deformation mirror in an optical machine system is used as an arithmetic example for topology optimization, and the results show that the absolute value of the Zernike coefficient Z4 after optimization is improved by nearly one order of magnitude compared with the value before optimization, and the vibration characteristics of the optimized structure meet the design requirements. The optimization effect is significant, which improves the optical performance of the deformed mirror and provides a new scheme for the design of the deformed mirror structure which has a certain practical value for engineering.

Axisymmetric surface shape correction of optical elements by a wide-aperture ion beam.

Methods for calculating the process of axisymmetric surface shape correction with a wide-aperture ion beam through a forming diaphragm for optical elements with a round and arbitrary border shape are presented. In the case of circular optical elements, an approach based on the separation of the axisymmetric component of the surface shape error from its decomposition by Zernike polynomials is used. In the case of optical elements with an arbitrary border shape (not round), an algorithm for searching the axisymmetric component of the shape error is proposed. The search criterion is to minimize the residual volume of the deviation of the surface shape from the calculated one. The proposed algorithm makes it possible to separate the axisymmetric component of the shape error relative to an arbitrary axis, including one located outside the workpiece. The results of surface treatment in accordance with the above methods are presented.

Machine learning algorithms for classifying corneas by Zernike descriptors

Keratoconus is the most common primary ectasia, as the treatment is not easy, its early diagnosis is essential. The main goal of this study is to develop a method for classification of specific types of corneal shapes where 55 Zernike coefficients (angular index m = 9) are used as inputs. We describe and apply six Machine Learning (ML) classification methods and an ensemble of them to objectively discriminate between keratoconic and non-keratoconic corneal shapes. Earlier attempts by other authors have successfully implemented several Machine Learning models using different parameters (usually, indirect measurements) and have obtained positive results. Given the importance and ubiquity of Zernike polynomials in the eye care community, our proposal should be a suitable choice to incorporate to current methods which might serve as a prescreening test. In this project we work with 475 corneas, classified by experts in two groups, 50 keratoconics and 425 non-keratoconics. All six models yield high rated results with accuracies above 98%, precisions above 97%, or sensitivities above 93%. Also, by building an assembly with the models, we further improve the accuracy of our classification, for example we found an accuracy of 99.7%, a precision of 99.8% and sensitivity of 98.3%. The model can be easily implemented in any system, being very simple to use, thus providing ophthalmologists with a effortless and powerful tool to make a first diagnosis.