Orthonormal Basis Functions Research Articles

Statistical Model of Ocular Wavefronts With Accommodation.

The purpose of this study was to determine the minimum number of orthonormal basis functions, applying Principal Component Analysis (PCA), to represent the most wavefront aberrations at different accommodation stages. The study also aims to generate synthetic wavefront data using these functions. Monocular wavefront data from 191 subjects (26.15 ± 5.56years old) were measured with a Hartmann-Shack aberrometer, simulating accommodation from 0 diopters (D) to 5 D in 1 D steps. The wavefronts for each accommodative demand were rescaled for different pupil sizes: 4.66, 4.76, 4.40, 4.09, 4.07, and 3.68mm. PCA was applied to 150 wavefront parameters (25 Zernike coefficients × 6 accommodation levels) to obtain eigenvectors for dimensional reduction. A total of 49 eigenvectors were modeled as a sum of 2 multivariate Gaussians, from which 1000 synthetic data sets were generated. The first 49 eigenvectors preserved 99.97% of the original data variability. No significant differences were observed between the mean values and standard deviation of the generated and original 49 eigenvectors (two one-sided test [TOST], P > 0.05/49) and (F-test, P > 0.05/49), both with Bonferroni correction. The mean values of the generated parameters (1000) were statistically equal to those of the original data (TOST, P > 0.05/150). The variability of the generated data was similar to the original data for the most important Zernike coefficients (F-test, P > 0.05/150). PCA significantly reduces the dimensionality of wavefront aberration data across 6 accommodative demands, reducing the variable space by over 66%. The synthetic data generated by the proposed wavefront model for accommodation closely resemble the original clinical data.

An adaptive weighted magnetic anomaly detection based on full magnetic gradient orthonormal basis function

An adaptive weighted magnetic anomaly detection based on full magnetic gradient orthonormal basis function

Unbiased particle conformation extraction from scattering spectra using orthonormal basis expansions

A strategy is outlined for quantitatively evaluating the particle density profiles from small-angle scattering spectra of dilute solutions. The approach employs an orthonormal basis function expansion method, enabling the determination of characteristic mass distributions in self-assembled structures without the need for a specific structural model. Through computational benchmarking, the efficacy of this approach is validated by effectively reconstructing the density profile of soft-ball systems with varying fuzziness from their scattering signatures. The feasibility of the method is demonstrated by fitting small-angle neutron scattering data obtained from Pluronic L64 micelles at different temperatures. This proposed approach is both simple and analytical, eliminating the requirement for a presumptive structural model in scattering analysis. The new method could therefore facilitate quantitative descriptions of complex nanoscopic structures inherent to numerous soft-matter systems using small-angle scattering techniques.

Experimental and Simulation Investigation of an Adaptive Model Predictive Control Scheme: Model Parametrized by Orthonormal Basis Function.

The closed-loop system's performance in synthesizing model predictive control (MPC) heavily relies on the model used for prediction. In continuously operating plants, a linear model-based MPC is designed based on the operating point's linear model during the commissioning stage. However, if the plant requires significant transitions from its normal operating point, the linear model-based MPC may not be effective. Therefore, to maintain the MPC performance under changing nominal operating conditions, the model (deterministic and stochastic components) needs to be updated to predict every sampling instant. This study focuses on designing an adaptive MPC (AMPC) scheme based on the linear model estimated from the input-output perturbation data under nominal operating conditions. The OBF-ARX (generalized orthonormal basis filters with ARX structure) parametrizes the observer's dynamic components. The proposed fixed and variable pole AMPC schemes' efficacy is demonstrated using a simulation study on a binary distillation column and experimental evaluation studies on a benchmark two-tank heater setup. The efficacy of the proposed AMPC schemes in addressing both servo and regulator problems has been demonstrated through simulation and experimental results. Specifically, these schemes have been shown to effectively track set points while simultaneously rejecting disturbances. These findings suggest that the AMPC schemes hold promise for use in a variety of applications in which precise control is required.

Regularized Finite Impulse Response Models versus Laguerre Models: A Comparison

While in recent years, the estimation of finite impulse response (FIR) models has been improved by introducing new regularization schemes, also other orthonormal basis function (OBF) models are now becoming more prominent again. Although Laguerre models show very similar properties to the regularized FIR models, they are only rarely used for system identification. Therefore, in this paper, regularized FIR models and Laguerre models will be compared. This work focuses on the model structure and its similarities and differences, as well as the hyperparameter optimization utilizing the generalized cross-validation (GCV) error. Finally, the two model types are investigated using three different processes and the model performance is evaluated. Both model types show significant improvements compared to standard (unregularized) FIR models.

Frequency Domain Identification of Continuous-Time Hammerstein Systems With Adaptive Continuous-Time Rational Orthonormal Basis Functions

In this paper, we propose a new identification method for continuous-time Hammerstein systems. This method is based on the adaptive generalized rational orthonormal basis functions. In particular, we use the strategy in which poles of basis functions are selected one by one in terms of a criterion. By doing this, we not only take advantage of the variability of the generalized orthogonal basis functions, but also reduce the computational burden. The proposed algorithm is adaptive. Meanwhile, convergence analysis is performed with considering the sampling frequency, noise level, sampling length, and terms of basis functions. The obtained results guarantee the convergence of the whole Hammerstein system. Finally, a numerical example is presented to illustrate the usefulness of the proposed identification algorithm.

Continuous-Time Subspace Identification with Prior Information Using Generalized Orthonormal Basis Functions

This paper presents a continuous-time subspace identification method utilizing prior information and generalized orthonormal basis functions. A generalized orthonormal basis is constructed by a rational inner function, and the transformed noises have ergodic properties. The lifting approach and the Hambo system transform are used to establish the equivalent nature of continuous and transformed discrete-time stochastic systems. The constrained least squares method is adopted to investigate the incorporation of prior knowledge in order to further increase the subspace identification algorithm’s accuracy. The input–output algebraic equation derives an optimal multistep forward predictor, and prior knowledge is expressed as equality constraints. In order to solve an optimization problem with equality constraints characterizing the prior knowledge, the proposed method reduces the computational burden. The effectiveness of the proposed method is provided by numerical simulations.

Data-driven modal analysis of nonlinear quantities in turbulent plasmas using multi-field singular value decomposition

Nonlinear dynamics in the two-dimensional multi-component plasma turbulence described by the Hasegawa–Wakatani equation is investigated by using a data-driven modal analysis with the singular value decomposition (SVD). The conventional SVD is extended to ‘multi-field SVD’ which can decompose multiple turbulence fields simultaneously by a single set of orthonormal basis functions without imposing a priori scale separations. Then, in addition to the mode amplitude labeled by the singular value, the information on the phase relations in the nonlinear quantities such as a transport flux or a triad energy transfer is extracted in the mode space. Through applications to the two-dimensional plasma turbulence, it is revealed that the multi-field SVD can extract the dominant spatial structures for the turbulent transport and the nonlinear energy transfer, preserving the multi-scale nature of the original turbulent fields. It is also demonstrated that one can reduce the dimensionality or information using the multi-field SVD through comparisons with the conventional Fourier decomposition.

A Class of Algorithms for Recovery of Continuous Relaxation Spectrum from Stress Relaxation Test Data Using Orthonormal Functions.

The viscoelastic relaxation spectrum provides deep insights into the complex behavior of polymers. The spectrum is not directly measurable and must be recovered from oscillatory shear or relaxation stress data. The paper deals with the problem of recovery of the relaxation spectrum of linear viscoelastic materials from discrete-time noise-corrupted measurements of relaxation modulus obtained in the stress relaxation test. A class of robust algorithms of approximation of the continuous spectrum of relaxation frequencies by finite series of orthonormal functions is proposed. A quadratic identification index, which refers to the measured relaxation modulus, is adopted. Since the problem of relaxation spectrum identification is an ill-posed inverse problem, Tikhonov regularization combined with generalized cross-validation is used to guarantee the stability of the scheme. It is proved that the accuracy of the spectrum approximation depends both on measurement noises and the regularization parameter and on the proper selection of the basis functions. The series expansions using the Laguerre, Legendre, Hermite and Chebyshev functions were studied in this paper as examples. The numerical realization of the scheme by the singular value decomposition technique is discussed and the resulting computer algorithm is outlined. Numerical calculations on model data and relaxation spectrum of polydisperse polymer are presented. Analytical analysis and numerical studies proved that by choosing an appropriate model through selection of orthonormal basis functions from the proposed class of models and using a developed algorithm of least-square regularized identification, it is possible to determine the relaxation spectrum model for a wide class of viscoelastic materials. The model is smoothed and robust on measurement noises; small model approximation errors are obtained. The identification scheme can be easily implemented in available computing environments.

A New Data Processing Method for Magnetic Anomaly Detection and Localization Based on 2-D Orthonormal Basis Functions

This work proposes a new processing method based on orthonormal basis functions (OBFs) for two-dimensional (2D) magnetic anomaly data, which is an innovative extension of one-dimensional (1D) OBFs. In most surveys, ferromagnetic targets are usually regarded as magnetic dipoles. The magnetic anomaly field of a dipole in a horizontal plane is essentially a linear combination of linearly independent basis functions, which are then transformed into a set of 2D orthonormal basis functions (2D-OBFs) using the Gram–Schmidt algorithm. On the basis of the 2D-OBFs, a processing procedure for 2D magnetic anomaly data is established and named the 2D-OBF method. This method converts magnetic anomaly maps to energy maps, in which there is only one positive peak corresponding to the horizontal location of a magnetic dipole. As a consequence, the detection and localization are significantly simplified. To verify this new method, a total of 3000 Monte Carlo simulations were carried out, and the synthetic data were also processed using the traditional 1D-OBF method. The statistics of the 2D-OBF results showed that the localization accuracy and false alarm rate are 0.34 m and 1.9 %, respectively, at a data noise level of 10 nT, which was much better than the 1D-OBF method. Another synthetic data experiment verified the reliability of the 2D method for handling multiple targets simultaneously, even when the magnetic anomalies of adjacent targets are severely superimposed. For the measured aeromagnetic data, the 2D-OBF method also works well. Additionally, this method exhibits outstanding noise immunity in all tests. The signal-to-noise ratios (SNR) of the energy data are improved by approximately 14 dB compared to that of the magnetic anomaly data within the signal bandwidth. Developed from rigorous mathematical physics definitions, the 2D-OBF method is more conducive to the implementation of automatic data processing programs and has considerable practical promise.

HIGH-DIMENSIONAL STOCHASTIC DESIGN OPTIMIZATION UNDER DEPENDENT RANDOM VARIABLES BY A DIMENSIONALLY DECOMPOSED GENERALIZED POLYNOMIAL CHAOS EXPANSION

Newly restructured generalized polynomial chaos expansion (GPCE) methods for high-dimensional design optimization in the presence of input random variables with arbitrary, dependent probability distributions are reported. The methods feature a dimensionally decomposed GPCE (DD-GPCE) for statistical moment and reliability analyses associated with a high-dimensional stochastic response; a novel synthesis between the DD-GPCE approximation and score functions for estimating the first-order design sensitivities of the statistical moments and failure probability; and a standard gradient-based optimization algorithm, constructing the single-step DD-GPCE and multipoint single-step DD-GPCE (MPSS-DD-GPCE) methods. In these new design methods, the multivariate orthonormal basis functions are assembled consistent with the chosen degree of interaction between input variables and the polynomial order, thus facilitating to deflate the curse of dimensionality to the extent possible. In addition, when coupled with score functions, the DD-GPCE approximation leads to analytical formulae for calculating the design sensitivities. More importantly, the statistical moments, failure probability, and their design sensitivities are determined concurrently from a single stochastic analysis or simulation. Numerical results affirm that the proposed methods yield accurate and computationally efficient optimal solutions of mathematical problems and design solutions for simple mechanical systems. Finally, the success in conducting stochastic shape optimization of a bogie side frame with 41 random variables demonstrates the power of the MPSS-DD-GPCE method in solving industrial-scale engineering design problems.

Optimal pupil basis set for telescope-coronagraph design and perturbation analysis based on the method of moments.

This paper concerns optimization and analysis of telescope-coronagraph systems for direct imaging of exoplanets. In this paper, the coronagraph system, with arbitrary pupil geometry, is theoretically considered, and the governing equation for the pupil design is derived. The method of moments is applied to solve the generalized energy-concentration eigenvalue problem to obtain the optimal pupil apodization and complete sets of orthonormal basis functions for arbitrary pupil geometries. The method yields eigenvalues indicating the fraction of starlight energy encircled in the area of the focal-plane mask (FPM), where starlight can be occulted and/or nulled. In other words, a higher eigenvalue implies less leakage/spillover of light outside of the FPM region and into the planet-discovery zone. Thus, a higher eigenvalue supports better starlight suppression for a given type of coronagraph. This methodology is useful for semi-quantitatively ranking different modes of perturbation with respect to energy spillage in the focal plane independent of coronagraph design details. A model-order-reduction-based sensitivity analysis is conducted to investigate the coupling between different pupil modes induced by aberrations. A pupil mode recovery scheme is presented to offer a theoretically rigorous and computationally efficient approach to reconstruct the optimal pupil mode under an arbitrary phase perturbation. The reconstruction coefficients and recovery-effectiveness factors are derived theoretically and demonstrated numerically. Several numerical examples, including the LUVOIR A and B pupils, are provided to validate and demonstrate the applicability of the proposed methods. The reported methodology enables model-order reduction based on degree of focal-plane energy concentration and reconstruction of optimal pupil apodization vis-á-vis phase aberrations using a precomputed basis set. These features should enhance computational efficiency for coronagraph design and sensitivity analysis.

Light-front wavefunctions of mesons by design

We develop a mechanism to build the light-front wavefunctions (LFWFs) of meson bound states on a small-sized basis function representation. Unlike in a standard Hamiltonian formalism, the Hamiltonian in this method is implicit, and the information of the system is carried directly by the functional form and adjustable parameters of the LFWFs. In this work, we model the LFWFs for four charmonium states, eta _c, J/psi , psi ', and psi (3770) as superpositions of orthonormal basis functions. We choose the basis functions as eigenfunctions of an effective Hamiltonian, which has a longitudinal confining potential in addition to the transverse confining potential from light-front holographic QCD. We determine the basis function parameters and superposition coefficients by employing both guidance from the nonrelativistic description of the meson states and the experimental measurements of the meson decay widths. With the obtained wavefunctions, we study the features of those meson states, including charge radii and parton distribution functions. We use the J/psi LFWF to calculate the meson production in diffractive deep inelastic scattering and ultra-peripheral heavy-ion collisions, and the eta _c LFWF to calculate its diphoton transition form factor. Both results show good agreement with experiments. The obtained LFWFs have simple-functional forms and can be readily used to predict additional experimental observables.

A New Fault Diagnosis Approach for Heavy-Duty Gas Turbines

Effective fault detection, estimation, and isolation are essential for the safety and reliability of gas turbines. In this article, a hybrid fault detection and isolation (FDI) approach is presented for condition monitoring of heavy-duty gas turbines. First, nonlinear dynamical models are constructed using an orthonormal basis function and an adaptive neuro-fuzzy inference system through experimental data. Following that, a fuzzy inference system is employed to estimate the fault severity. For this aim, fuzzy rules are extracted from the fault patterns by defining appropriate features. Later, various faults are isolated using an ensemble decision tree classifier. The proposed nonlinear modeling compensates for disturbances and uncertainty in the system and leads to adaptive thresholds for fault detection, which reduces the false alarm rate. Moreover, the proposed FDI method brings high accuracy in fault estimation by properly modeling a bounded uncertainty using the adaptive threshold. Experimental data are applied to validate the gas turbine model. Test results indicate that the proposed hybrid FDI method via adaptive threshold overwhelms the other FDI methods, where the misclassified data are 5.6%.

Magnetic anomaly detection via a combination approach of minimum entropy and gradient orthogonal functions

Aiming at the problem of magnetic anomaly detection under low signal-to-noise ratio, a full magnetic gradient detection via minimum entropy and gradient orthonormal basis function is proposed in this paper. Firstly, the mathematical model of magnetic dipole gradient tensor and the experimental method of collecting magnetic gradient tensor signals are introduced. Secondly, a new minimum entropy detector is designed, the basic functions which are processed by new minimum entropy detector can be orthogonalized easily. Thirdly, a group of new standard orthogonal basis functions are constructed according to the characteristics of magnetic gradient anomaly signals. Finally, the detector and orthogonal basis functions are used to detect the magnetic anomaly. Both numerical simulation signals and experiment signals are used to prove the effectiveness of the methods.

Implementation of fractional-order difference via Takenaka-Malmquist functions

The paper presents a new definition of nabla fractional-order difference, equivalent to the Grünwald-Letnikov difference. The difference is based on the general approach of orthonormal basis functions in terms of discrete-time Takenaka-Malmquist filters. The main advantage of the proposed definition is that for finite model length, the model quickly converges to the actual difference. The paper proposes the method of selecting the poles of the Takenaka-Malmquist functions. It also proposes the implementation of the Takenaka-Malmquist-based difference in non-commensurate state-space system and fractional-order integrator. Simulation experiments show the proposed methodology’s high effectiveness in modeling fractional-order difference, integrator, and non-commensurate state-space systems.

A spectral method based on Bernstein orthonormal basis functions for solving an inverse Roseneau equation

A spectral method based on Bernstein orthonormal basis functions for solving an inverse Roseneau equation

Low-rank tensor reconstruction of concentrated densities with application to Bayesian inversion

This paper presents a novel method for the accurate functional approximation of possibly highly concentrated probability densities. It is based on the combination of several modern techniques such as transport maps and low-rank approximations via a nonintrusive tensor train reconstruction. The central idea is to carry out computations for statistical quantities of interest such as moments based on a convenient representation of a reference density for which accurate numerical methods can be employed. Since the transport from target to reference can usually not be determined exactly, one has to cope with a perturbed reference density due to a numerically approximated transport map. By the introduction of a layered approximation and appropriate coordinate transformations, the problem is split into a set of independent approximations in seperately chosen orthonormal basis functions, combining the notions h- and p-refinement (i.e. “mesh size” and polynomial degree). An efficient low-rank representation of the perturbed reference density is achieved via the Variational Monte Carlo method. This nonintrusive regression technique reconstructs the map in the tensor train format. An a priori convergence analysis with respect to the error terms introduced by the different (deterministic and statistical) approximations in the Hellinger distance and the Kullback–Leibler divergence is derived. Important applications are presented and in particular the context of Bayesian inverse problems is illuminated which is a main motivation for the developed approach. Several numerical examples illustrate the efficacy with densities of different complexity and degrees of perturbation of the transport to the reference density. The (superior) convergence is demonstrated in comparison to Monte Carlo and Markov Chain Monte Carlo methods.

Active Electric Anomaly Detection Method for Underwater Targets Based on the Orthonormal Basis Function

Electric anomaly detection (EAD) has been widely used for target detection in underwater areas. However, due to the high path loss in the water, an electric anomaly is usually submerged in environmental noise and interference, which affects the detection performance of traditional anomaly detection methods. To address this problem and improve the detection accuracy in a low signal-to-noise ratio (SNR) environment, this paper proposes an active electric anomaly detection (AEAD) method based on the orthonormal basis function (OBF). First, a four-electrode active detection system was designed. Then, a set of OBFs based on the electric field disturbance model were derived to describe the detection system characteristic, linearly and effectively. Finally, an AEAD system was designed, and the proposed method was verified experimentally using a electromagnetic simulation tool to detect a spherical anomaly target. The experimental results show that, compared with the traditional AEAD methods, the proposed method has a better performance.

Effect assessment of spatial sampling length on the orthonormal basis functions detector

Magnetic anomaly detection (MAD) is a passive way to detect hidden magnetic objects. As the major way of MAD, the orthonormal basis function (OBF) detector has been widely researched and applied. During the practical signal processing process, we found that the performance of the OBF detector is mainly affected by the space sampling length. However, there is almost no literature studying the relationship between the OBF detector and the space sampling length. In our work, the effect of the spatial sampling length on the performance of the OBF detector is assessed from the perspective of the statistical distribution of independent and correlated background noise. The theoretical and experimental results both turn out to be that the smaller the noise correlation, the more evident the gain effect of the spatial sampling length on the performance of the OBF detector under the same signal-to-noise ratio. Therefore, the spatial sampling length should be fitted with the noise environment to attain a better trade-off between detector performance and computing resources.