Error Norm Research Articles

Improved 3D human face reconstruction from 2D images using blended hard edges

This study reports an effective and robust edge-based scheme for the reconstruction of 3D human faces from input of single images, addressing drawbacks of existing methods in case of large face pose angles or noisy input images. Accurate 3D face reconstruction from 2D images is important, as it can enable a wide range of applications, such as face recognition, animations, games and AR/VR systems. Edge features extracted from 2D images contain wealthy and robust 3D geometric information, which were used together with landmarks for face reconstruction purpose. However, the accurate reconstruction of 3D faces from contour features is a challenging task, since traditional edge or contour detection algorithms introduce a great deal of noise, which would adversely affect the reconstruction. This paper reports on the use of a hard-blended face contour feature from a neural network and a Canny edge extractor for face reconstruction. The quantitative results indicate that our method achieves a notable improvement in face reconstruction with a Euclidean distance error of 1.64 mm and a normal vector distance error of 1.27 mm when compared to the ground truth, outperforming both traditional and other deep learning-based methods. These metrics show particularly significant advancements, especially in face shape reconstruction under large pose angles. The method also achieved higher accuracy and robustness on in-the-wild images under conditions of blurring, makeup, occlusion and poor illumination.

A novel perspective for simulations of the Modified Equal-Width Wave equation by cubic Hermite B-spline collocation method

In the current study, the Modified Equal-Width (MEW) equation will be handled numerically by a novel technique using collocation finite element method where cubic Hermite B-splines are used as trial functions. To test the accuracy and efficiency of the method, four different experimental problems; namely, the motion of a single solitary wave, interaction of two solitary waves, interaction of three solitary waves and the birth of solitons with the Maxwellian initial condition will be investigated. In order to verify, the validity and reliability of the proposed method, the newly obtained error norms L2 and L∞ as well as three conservation constants have been compared with some of the other numerical results given in the literature at the same parameters. Furthermore, some wave profiles of the newly obtained numerical results have been given to demonstrate that each test problem exhibits accurate physical simulations. The advantage of the proposed method over other methods is the usage of inner points at Legendre and Chebyshev polynomial roots. This advantage results in better accuracy with less number of elements in spatial direction. The results of the numerical experiments clearly reveal that the presented scheme produces more accurate results even with comparatively coarser grids.

Regular variation in Hilbert spaces and principal component analysis for functional extremes

Motivated by the increasing availability of data of functional nature, we develop a general probabilistic and statistical framework for extremes of regularly varying random elements X in L2[0,1]. We place ourselves in a Peaks-Over-Threshold framework where a functional extreme is defined as an observation X whose L2-norm ‖X‖ is comparatively large. Our goal is to propose a dimension reduction framework resulting into finite dimensional projections for such extreme observations. Our contribution is double. First, we investigate the notion of Regular Variation for random quantities valued in a general separable Hilbert space, for which we propose a novel concrete characterization involving solely stochastic convergence of real-valued random variables. Second, we propose a notion of functional Principal Component Analysis (PCA) accounting for the principal ‘directions’ of functional extremes. We investigate the statistical properties of the empirical covariance operator of the angular component of extreme functions, by upper-bounding the Hilbert–Schmidt norm of the estimation error for finite sample sizes. Numerical experiments with simulated and real data illustrate this work.

Non-central m-point formula in method of lines for solving the Korteweg-de Vries (KdV) equation

AbstractThe present study is committed to devising efficient spatial discretization with two non-central difference formulae incorporated in the method of lines (MOL). The method is then implemented numerically on the renowned dispersive evolution equation, the Korteweg-de Vries (KdV) model while infusing Euler and fourth-order Rung-Kutta (RK4) methods, respectively. The resulting schemes are proven to be numerically stable using Fourier’s stability approach, with the MOL matrix admitting negative real eigenvalues. Moreover, this proposal has been assessed on certain initial-boundary value problems of the KdV model by examining so many factors, like the percentage errors, absolute error differences, the error norms, and the invariants $$I_{k},$$ I k , for $$k=1,2,3,$$ k = 1 , 2 , 3 , among others. In this work, we solve KdV by applying a new approach of MOL to improve the results presented in some published articles. Lastly, the assessment revealed that our proposal was found to be better than those of the exponential finite-difference method (Exponential) and the Heat Balance Integral (HBI) methods, respectively, which serve as competing approaches for validation, in terms of both the accuracy and efficiency. The numerical examples are obtained by using MATLAB.

Decentralized Implicit Inverse Control for Large-Scale Hysteretic Nonlinear Time-Delay Systems and Its Application on Triple-Axis Giant Magnetostrictive Actuators.

This article proposes fuzzy-logic systems (FLSs)-based decentralized adaptive implicit inverse control scheme for a class of large-scale nonlinear systems with time delays and multihysteretic loops. Our novel algorithms feature hysteretic implicit inverse compensators designed to effectively mitigate multihysteretic loops in large-scale systems. In this article, hysteretic implicit inverse compensators can replace the traditional hysteretic inverse models, which are exceedingly difficult to construct, and no longer necessary. The authors provide three contributions: 1) a searching mechanism to obtain the approximate value of the practical input signal from the so-called hysteretic temporary control law; 2) the arbitrarily small L∞ norm of the tracking error attained by utilizing the proposed initializing technique, which applies the combination of FLSs and a finite covering lemma to deal with time delays; and 3) the construction of a triple-axis giant magnetostrictive motion control platform, which validates the effectiveness of the proposed control scheme and algorithms.

Development of elbow rehabilitation device with iterative learning control and internet of things

In this study, we present a novel approach for rehabilitation devices through the design of an active elbow joint orthosis, inspired by the fundamental principles of robotic exoskeletons. The device not only enables home-based usage but also facilitates the transmission of exercise data from patients to physiotherapists via the Internet of Things (IoT) device. This approach offers the possibility of increased therapy sessions for each patient while allowing physiotherapists access to data for real-time or subsequent analyses, thereby establishing a database. This permits a single physiotherapist to manage multiple patients more effectively. The developed mobile application within this research incorporates a distinct entry interface for both patients and physiotherapists. Maximum force and position values generated during each exercise period are displayed within the application. The device enables active exercise with a single degree of freedom at the elbow joint and is equipped with force sensors to ensure safety against potential high-shear forces. Furthermore, it can be worn on the upper extremity using adjustable Velcro straps to accommodate users with varying arm circumferences. Specifically, this system amalgamates two primary components: a microcontroller operating control algorithms and IoT technology, and a smartphone application containing interfaces for physiotherapists and users undergoing therapy. The control design of the device employs a P-Type Iterative Learning Control (ILC) due to periodic exercise movements, reducing the error norm by approximately 20% during each exercise period (excluding the initial period). The controller consistently diminishes error values with each iteration, ultimately converging to zero. Throughout an exercise lasting around 3 minutes, the average error norm is recorded as 0.229⁰. In essence, this study presents a pioneering approach that sets itself apart from other research by minimizing shear forces and errors through a specialized controller, all while enabling remote, home-based rehabilitation under expert supervision.

Exploring the role of spending on malaria incidence in Uganda using the auto-regressive distributed lag approach.

Malaria has remained a persistent global health problem. Despite multiple government and donor initiatives to eradicate malaria and its detrimental effects on Uganda's health outcomes, the incidence of malaria is worrying as it appears higher than the average of 219 cases per 1000 for sub-Saharan Africa for the period 2017-2018. This study investigated the effect of public and private healthcare spending on the incidence of malaria in Uganda. Employing time series data spanning over 20years from the first quarter of 2000 to the last quarter of 2019, the study builds a model based on the Grossman framework for analysing demand for health. The estimation technique used was the ARDL approach that takes into account reverse causality and incidental relationships. Prior to the adoption of the technique, a bounds test was performed to determine whether the variables contained in the model have a long-term relationship. Several diagnostic tests for serial correlation, functional normality, and heteroskedastic specification error were carried out to verify the ARDL model's goodness of fit. Additionally, the cumulative sum of recursive (CUSUM) and cumulative sum of squares of recursive residuals (CUSUMSQ) were used to test model stability. The results indicate that in the long run, an increase in public spending of one percent significantly reduces malaria incidence by 0.196 at the 10 percent level of significance. On the other hand, there is no significant evidence of private health expenditure's effect on malaria incidence. However, in the short run, public spending reduces malaria incidence by a smaller magnitude of 0.158 percent relative to the long-run. Still, private expenditure is found to exhibit no significant effect. Additional findings point to the importance of GDP per capita and urban population growth in reducing malaria incidence, whereas female unemployment, income inequality, as well as female-headed household. In the short run, however, the female-headed households and urban population growth are found to significantly reduce malaria incidence while an improvement in regulatory quality decreases malaria incidence by 0.129 percent. There is need for further government interventions to reduce malaria incidence in the country via budget allocation, as well as the strengthening of programmes to raise household income to support private health spending, in addition to the development of strategies to promote well-planned and organized urban centres.

A numerical investigation of a well-known nonlinear Newell-Whitehead-Segel equation using the rank polynomial of the star graph

Mathematical models of pattern formation are indispensable tools in various fields, from developmental biology to ecology, providing insights into complex phenomena and contributing to our understanding of the natural world. These patterns have been extensively studied using reaction-diffusion and NewellWhiteheadSegel models. This article intended to find an approximate numerical solution to the NewellWhiteheadSegel equation. The appearance of stripe patterns in two-dimensional systems is explained in nonlinear systems using the NewellWhiteheadSegel equation. Based on the function basis of rank polynomials of star graphs and the well-posed operational matrices, the rank polynomial collocation method is constructed. The alleged rank polynomial collocation method created a system of nonlinear algebraic equations from the nonlinear NewellWhiteheadSegel equation. The nonlinear NewellWhiteheadSegel equation solution is approximated by solving the resulting system via Newton’s Raphson method. Numerical instances are provided to illustrate the validity and effectiveness of the technique. Verification of accuracy is accomplished by calculating error norms. The obtained numerical results show a reasonable degree of consistency with the findings reported in the current literature. The scheme’s primary benefit is the algorithm’s ease of implementation.

Back-and-forth nudging moving horizon estimation for discrete-time linear systems

In this paper, we propose a novel moving horizon estimation algorithm for discrete-time linear systems with a limited number of measurements. Motivated by the idea of the back-and-forth nudging algorithm, we design the back-and-forth nudging in time moving-horizon estimation method by deploying two moving horizon estimators that move backward and forward iteratively. Based on a finite number of measurements, the proposed algorithm can be used for moving-horizon state estimation with guaranteed convergence. By using the proposed method, we show that the norm of the state estimation error is upper bounded by a sequence that converges to its steady-state in finite-time (i.e., using a finite number of measurements), provided that suitable parameters are selected. The unbiasedness properties and comparison results with the conventional (forward-in-time) moving horizon estimation are discussed. The effectiveness of the proposed approach is validated through a numerical example.

Distributed moving horizon fusion estimation for linear constrained uncertain systems

AbstractIn this paper, the distributed moving horizon fusion estimation of uncertain systems with constraints of system noise and state variables is studied. Firstly, relying on the basic idea of consensus algorithm, the cost function in the performance index is reconstructed by weighted fusion of the state prediction values. Secondly, considering the performance index with uncertain parameters, the min‐max optimization problem of the algorithm is transformed into the least squares optimization problem based on 2‐norm regularization method. Thirdly, the scalar‐weighted linear minimum variance fusion estimation strategy is used to realize the weighted fusion of local state estimation values. Then, on the premise of minimum network connectivity and collective observability, the stability of the proposed algorithm is studied, and the sufficient conditions for the expected convergence of the fused estimation error norm square are given. Finally, the effectiveness of the algorithm is verified by numerical simulation.

Adaptive backstepping controller based on a novel framework for dynamic solution of an ankle rehabilitation spherical parallel robot

AbstractThis research offers an adaptive model-based methodology for autonomous control of 3-RRR spherical parallel manipulator (RSPM) based on a novel modeling framework. RSPM is an overconstrained parallel mechanism that has a variety of applications in medical procedures such as ankle rehabilitation because of its precision and accuracy. However, obtaining a complete explicit dynamic model of these mechanisms for tracking purposes has been a problematic challenge due to their inherent singularities, coupling effects of the limbs, and redundant constraints imposed by the intermediate joints. This paper presents a novel algorithm to obtain the analytical kinematic solutions of RSPMs based on the closed-loop vector method, which includes constraint analysis. By incorporating constrained kinematics into the dynamic model, a comprehensive explicit dynamic solution of the non-overconstrained version 3-RCC of RSPM is developed in task space, based on screw theory and the linear homogeneous property of algebraic equations on the manipulator twist. Based on the proposed computational framework, a robust self-tuning backstepping control (STBC) strategy is applied to the robot to overcome the effect of external disturbances and time-varying uncertainties. Furthermore, an observer-based compensation (OBC) method is presented for dealing with the nonlinear hysteresis loops of the ankle during trajectory tracking purposes. The closed-loop stability of the whole system including STBC and OBC is theoretically performed by Lyapunov methods. The proposed methodologies are validated by realistic co-simulations in different scenarios. For instant, in the presence of external disturbances, the maximum tracking error norm of STBC is 37.5% less than the sliding mode approach.

Spinet-QSM: model-based deep learning with schatten p-norm regularization for improved quantitative susceptibility mapping.

Quantitative susceptibility mapping (QSM) provides an estimate of the magnetic susceptibility of tissue using magnetic resonance (MR) phase measurements. The tissue magnetic susceptibility (source) from the measured magnetic field distribution/local tissue field (effect) inherent in the MR phase images is estimated by numerically solving the inverse source-effect problem. This study aims to develop an effective model-based deep-learning framework to solve the inverse problem of QSM. This work proposes a Schatten -norm-driven model-based deep learning framework for QSM with a learnable norm parameter to adapt to the data. In contrast to other model-based architectures that enforce the l -norm or l -norm for the denoiser, the proposed approach can enforce any -norm ( ) on a trainable regulariser. The proposed method was compared with deep learning-based approaches, such as QSMnet, and model-based deep learning approaches, such as learned proximal convolutional neural network (LPCNN). Reconstructions performed using 77 imaging volumes with different acquisition protocols and clinical conditions, such as hemorrhage and multiple sclerosis, showed that the proposed approach outperformed existing state-of-the-art methods by a significant margin in terms of quantitative merits. The proposed SpiNet-QSM showed a consistent improvement of at least 5% in terms of the high-frequency error norm (HFEN) and normalized root mean squared error (NRMSE) over other QSM reconstruction methods with limited training data.

Comparative analysis of tracer studies in the determination of longitudinal dispersion coefficient.

The research presented in this paper is to determine the best tracer studies that will give acceptable estimates of longitudinal dispersion coefficient for Orashi river using rhodamine WT dye and sodium chloride as water tracer. Estimated results obtained for longitudinal dispersion coefficient for the case of rhodamine WT experiment ranges between 71 and 104.4 m2s-1 while that of sodium chloride experiment ranges between 20.1 and 34.71 m2s-1. These results revealed lower dispersion coefficient using sodium chloride as water tracer (WT) indicating that for larger rivers, sodium chloride should not be used as water tracer. The usage of sodium chloride as water tracer in the estimation of longitudinal dispersion coefficient is recommended in smaller streams as NaCl is relatively conservative. The established equations for both cases of investigation are proving satisfactory upon validation as degree of accuracy of 100.0% was obtained using discrepancy ratio (Dr). Standard error (SE), normal mean error (NME) and mean multiplication error (MME) of the developed equations is better when compared with other existing equations. However, Equation (17) is satisfactorily recommended.

Numerical Analysis of the Discrete MRLW Equation for a Nonlinear System Using the Cubic B-Spline Collocation Method

By employing the cubic B-spline functions, a collocation approach was devised in this study to address the Modified Regularized Long Wave (MRLW) equation. Then, we derived the corresponding nonlinear system and easily solved it using Newton’s iterative approach. It was established that the cubic B-spline collocation technique exhibits unconditional stability. The dynamics of solitary waves, including their pairwise and triadic interactions, were meticulously investigated utilizing the proposed numerical method. Additionally, the transformation of the Maxwellian initial condition into solitary wave formations is presented. To validate the current work, three distinct scenarios were compared against the analytical solution and outcomes from alternative methods under both L2- and L∞-error norms. Primarily, the key strength of the suggested scheme lies in its capacity to yield enhanced numerical resolutions when employed to solve the MRLW equation, and these conservation laws show that the solitary waves have time and space translational symmetry in the propagation process. Finally, this paper concludes with a summary of our findings.

A new output feedback adaptive control method for fractional order systems with inaccessible state

In this paper, the output tracking control problem for fractional-order (FO) systems with inaccessible state is investigated, and a new fractional-order model reference adaptive control (FOMRAC) method is proposed. The designed FO adaptive update law is able to adjust the controller parameters online in real time according to the system output information. Compared with the classical output feedback FOMRAC method, the proposed FOMRAC strategy only needs the online updating of a scalar function, which significantly reduces the complexity of the controller design and the online computational burden, and can achieve better control effect. Furthermore, the scheme ensures the boundedness of the closed-loop signals and that the mean value of the squared norm of the output error converges to zero. Finally, simulation results are carried out to verify the effectiveness of the established approach.

New Exact and Numerical Experiments for the Caudrey-Dodd-Gibbon Equation

In this study, an exact and a numerical method namely direct algebraic method and collocation finite element method are proposed for solving soliton solutions of a special form of fifth-order KdV (fKdV) equation that is of particular importance: Caudrey-Dodd-Gibbon (CDG) equation. For these aims, homogeneous balance method and septic B-spline functions are used for exact and numerical solutions, respectively. Next, it is proved by applying von-Neumann stability analysis that the numerical method is unconditionally stable. The error norms $L_{2}$ and $L_{\infty }$ have been computed to control proficiency and conservation properties of the suggested algorithm. The obtained numerical results have been listed in the tables. The graphs are modelled so that easy visualization of properties of the problem. Also, the obtained results indicate that our method is favourable for solving such problems.

A Combined Model of Diffusion Model and Enhanced Residual Network for Super-Resolution Reconstruction of Turbulent Flows

In this study, we introduce a novel model, the Combined Model, composed of a conditional denoising diffusion model (SR3) and an enhanced residual network (EResNet), for reconstructing high-resolution turbulent flow fields from low-resolution flow data. The SR3 model is adept at learning the distribution of flow fields. The EResNet architecture incorporates a long skip connection extending from the input directly to the output. This modification ensures the preservation of essential features learned by the SR3, while simultaneously enhancing the accuracy of the flow field. Additionally, we incorporated physical gradient constraints into the loss function of EResNet to ensure that the flow fields reconstructed by the Combined Model are consistent with the direct numerical simulation (DNS) data. Consequently, the high-resolution flow fields reconstructed by the Combined Model exhibit high conformity with the DNS results in terms of flow distribution, details, and accuracy. To validate the effectiveness of the model, experiments were conducted on two-dimensional flow around a square cylinder at a Reynolds number (Re) of 100 and turbulent channel flow at Re = 4000. The results demonstrate that the Combined Model can reconstruct both high-resolution laminar and turbulent flow fields from low-resolution data. Comparisons with a super-resolution convolutional neural network (SRCNN) and an enhanced super-resolution generative adversarial network (ESRGAN) demonstrate that while all three models perform admirably in reconstructing laminar flows, the Combined Model excels in capturing more details in turbulent flows, aligning the statistical outcomes more closely with the DNS results. Furthermore, in terms of L2 norm error, the Combined Model achieves an order of magnitude lower error compared to SRCNN and ESRGAN. Experimentation also revealed that SR3 possesses the capability to learn the distribution of flow fields. This work opens new avenues for high-fidelity flow field reconstruction using deep learning methods.

Convergence analysis of high-order IMEX-BDF schemes for the incompressible Navier–Stokes equations

In this paper, we consider developing high-order temporal integration schemes for the unsteady incompressible Navier–Stokes equations in bounded two-dimensional domain subjected to the periodic boundary conditions. Utilizing the k-step (k=3,4,5) backward differentiation formula (BDF) coupled with the implicit–explicit (IMEX) treatment of the nonlinear convective term in an anti-symmetry form, a class of IMEX-BDFk schemes up to fifth-order in time are constructed and analyzed. By imposing a zero-mean constrain on the finite-dimensional space for the pressure, the proposed numerical schemes are proven to be uniquely solvable. Based on the recent theoretical framework consisting of a class of discrete orthogonal convolution kernels, rigorous L2 norm error estimates for both the velocity and the pressure are established by using a novel divergence free projection system. The proposed schemes are then implemented in two benchmark experiments, including a Taylor–Green vortex problem and a double shear layer flow at various high Reynolds numbers. Numerical results demonstrate the expected solution accuracy and the computational effectiveness in simulating the realistic flow dynamics.

Symmetric Q-learning: Reducing Skewness of Bellman Error in Online Reinforcement Learning

In deep reinforcement learning, estimating the value function to evaluate the quality of states and actions is essential. The value function is often trained using the least squares method, which implicitly assumes a Gaussian error distribution. However, a recent study suggested that the error distribution for training the value function is often skewed because of the properties of the Bellman operator, and violates the implicit assumption of normal error distribution in the least squares method. To address this, we proposed a method called Symmetric Q-learning, in which the synthetic noise generated from a zero-mean distribution is added to the target values to generate a Gaussian error distribution. We evaluated the proposed method on continuous control benchmark tasks in MuJoCo. It improved the sample efficiency of a state-of-the-art reinforcement learning method by reducing the skewness of the error distribution.

Error Analysis of Normal Surface Measurements Based on Multiple Laser Displacement Sensors.

The robotic drilling of assembly holes is a crucial process in aerospace manufacturing, in which measuring the normal of the workpiece surface is a key step to guide the robot to the correct pose and guarantee the perpendicularity of the hole axis. Multiple laser displacement sensors can be used to satisfy the portable and in-site measurement requirements, but there is still a lack of accurate analysis and layout design. In this paper, a simplified parametric method is proposed for multi-sensor normal measurement devices with a symmetrical layout, using three parameters: the sensor number, the laser beam slant angle, and the laser spot distribution radius. A normal measurement error distribution simulation method considering the random sensor errors is proposed. The measurement error distribution laws at different sensor numbers, the laser beam slant angle, and the laser spot distribution radius are revealed as a pyramid-like region. The influential factors on normal measurement accuracy, such as sensor accuracy, quantity and installation position, are analyzed by a simulation and verified experimentally on a five-axis precision machine tool. The results show that increasing the laser beam slant angle and laser spot distribution radius significantly reduces the normal measurement errors. With the laser beam slant angle ≥15° and the laser spot distribution radius ≥19 mm, the normal measurement error falls below 0.05°, ensuring normal accuracy in robotic drilling.