B-spline Approximation Research Articles

SplineGen: Approximating unorganized points through generative AI

This paper presents a learning-based method to solve the traditional parameterization and knot placement problems in B-spline approximation. Different from conventional heuristic methods or recent AI-based methods, the proposed method does not assume ordered or fixed-size data points as input. There is also no need for manually setting the number of knots. Parameters and knots are generated in an associative way to attain better parameter-knot alignment, and therefore a higher approximation accuracy. These features are attained by using a new generative model SplineGen, which casts the parameterization and knot placement problems as a sequence-to-sequence translation problem. It first adopts a shared autoencoder model to learn a 512-D embedding for each input point, which has the local neighborhood information implicitly captured. Then these embeddings are autoregressively decoded into parameters and knots by two associative decoders, a generative process automatically determining the number of knots, their placement, parameter values, and their ordering. The two decoders are made to work in a coordinated manner by a new network module called internal cross-attention. Once trained, SplineGen demonstrates a notable improvement over existing methods, with one to two orders of magnitude increase in approximation accuracy on test data.

A Semiparametric Bayesian Approach to Heterogeneous Spatial Autoregressive Models.

Many semiparametric spatial autoregressive (SSAR) models have been used to analyze spatial data in a variety of applications; however, it is a common phenomenon that heteroscedasticity often occurs in spatial data analysis. Therefore, when considering SSAR models in this paper, it is allowed that the variance parameters of the models can depend on the explanatory variable, and these are called heterogeneous semiparametric spatial autoregressive models. In order to estimate the model parameters, a Bayesian estimation method is proposed for heterogeneous SSAR models based on B-spline approximations of the nonparametric function. Then, we develop an efficient Markov chain Monte Carlo sampling algorithm on the basis of the Gibbs sampler and Metropolis-Hastings algorithm that can be used to generate posterior samples from posterior distributions and perform posterior inference. Finally, some simulation studies and real data analysis of Boston housing data have demonstrated the excellent performance of the proposed Bayesian method.

Generalized multilevel B-spline approximation for scattered data interpolation in image processing

This paper proposes a Generalized Multilevel B-spline Approximation (GMBA) method, which addresses scattered data interpolation problems in image processing. Mathematically, the GMBA provides a better solution for the B-spline control lattice by superimposing identical level B-splines compared with traditional Multilevel B-spline Approximation (MBA). Specifically, the GMBA allows the spacing of next control lattice to be subdivided arbitrarily or remained unchanged, which is determined by a predefined spacing set or the current error level. These improvements bring higher approximation accuracy and more flexibility for algorithm design to avoid over-fitting. In this paper, basic GMBA algorithm and its refined algorithm are compiled for image processing. Finally, six relevant cases are involved to test the GMBA, including surface approximation, image enlargement, image completion, and Salt-and-Pepper (SAP) noise removal. The experimental results show that the GMBA has better performance than the MBA in surface approximation and image processing, performs comparatively fast with the best performance on more than half of the standard test images compared with traditional algorithms, and has partially better performance even than deep learning algorithms. The GMBA can effectively recover meaningful details in images contaminated with even extremely high SAP noise level (up to 99%).

EC-QCL sensing system–incorporated adaptive baseline correction algorithm for simultaneous detection of multiple gas components

In this study, we presented a multi-gas sensor developed by employing an external cavity quantum cascade laser (EC-QCL)-integrated adaptive baseline correction algorithm for the simultaneous detection of ammonia (NH3), ozone (O3), and carbon dioxide (CO2). The EC-QCL sensing system was selected due to its high wavelength resolution and wide tuning range (1033–1068 cm−1), enabling the simultaneous measurement of multi-gases. The system complexity was effectively decreased and the superimposed noise was avoided by employing an adaptive baseline correction algorithm based on differential transformation and B-Spline approximation (ADTBA), which enabled the resolution of the uneven EC-QCL baseline signals. The performance validation of the ADTBA algorithm by comparing it with the conventional polynomial fitting for detecting the spectra of NH3, O3, and CO2 showed that the former outperformed the latter. The results showed a sensitivity enhancement factor of 3.7, indicating that the newly developed algorithm can generate a high-quality gas absorption spectrum. Allan deviation analysis of the system's responsiveness revealed an averaging time of 120 s resulting in sensitivities of 1.3, 26.7, and 37.0 ppm for NH3, O3, and CO2, respectively.

B-spline curve approximation with transformer neural networks

Approximating a curve with a B-spline is a well-known problem with many challenges. Computing parametric values and knot vector that leads to the best approximation of a point sequence is an open problem. Existing methods are usually based on heuristics, genetic algorithms, or meta-heuristics. Nowadays, Deep Neural Networks have demonstrated their usefulness as shown in the use of a Multi-Layer Perceptron in the existing literature. Since its inception, the Transformer architecture has achieved state-of-the-art in multiple domains, like Natural Language Processing and Computer Vision. In this paper, we propose a method for knot placement that focuses on using a Transformer neural network architecture for B-spline approximation. We present and compare the results of our ongoing experimentations with Transformers for B-spline curve approximation. We conclude with possible improvements and modifications to our method for future experiments.

Individualized dynamic latent factor model for multi-resolutional data with application to mobile health

Summary Mobile health has emerged as a major success for tracking individual health status, due to the popularity and power of smartphones and wearable devices. This has also brought great challenges in handling heterogeneous, multi-resolution data that arise ubiquitously in mobile health due to irregular multivariate measurements collected from individuals. In this paper, we propose an individualized dynamic latent factor model for irregular multi-resolution time series data to interpolate unsampled measurements of time series with low resolution. One major advantage of the proposed method is the capability to integrate multiple irregular time series and multiple subjects by mapping the multi-resolution data to the latent space. In addition, the proposed individualized dynamic latent factor model is applicable to capturing heterogeneous longitudinal information through individualized dynamic latent factors. Our theory provides a bound on the integrated interpolation error and the convergence rate for B-spline approximation methods. Both the simulation studies and the application to smartwatch data demonstrate the superior performance of the proposed method compared to existing methods.

Unfolding neutron spectra using splines

Organic scintillators detect neutrons by scattering reactions with hydrogen nuclei (protons) in organic compounds. The recoil protons then produce the light output from the scintillator. The energy spectrum of neutrons impinging on the scintillator has to be determined from this light output using a so-called unfolding algorithm. However, this problem is ill-posed and therefore difficult to solve. In this study, a new method for unfolding neutron spectra is presented. This method uses B-splines approximation of neutron energy spectrum. Results of this method are shown for test data and experimental data and compared to the results of the previously developed unfolding method.

Map-aiding of the additional secondary factors isosurface by the spline approximation method as a condition of improving the e-Loran observations accuracy

The benefit of the Loran ground-based radio navigation system as an alternative to global navigation satellite systems is argued. The motivation of searching for a duplicate variant is caused by the necessity to ensure the reliability of positioning as a subject of countering spoofing and jamming risks when implementing the concept of cybernetic awareness on water transport. Hypothetically, the possibility of using e-Loran on the Northern Sea Route as a backup navigation system is considered. The experience of operating pilot projects of the enhanced Loran in the issue of the predicted potential of positioning accuracy is analyzed. The optimal method of compensation for the system error of geolocation based on the use of additional secondary factors is investigated. Taking into account the principle of the d-Loran functional, based on comparing the measured values of the radio signal passage delay with published analogues for transmitting clarifications to marine consumers, the necessity of using integrity as a criterion for a confidence assessment of the navigation information processing reliability has been determined. A critical analysis of the attempts effectiveness to use the linear interpolation method to calculate intermediate representative values of the delay in the radio signal transmission in order to simulate the map of additional secondary factors is performed. The hypothesis of the prospects for the isosurface corrections synthesis based on B-spline approximation is put forward. The South Korean experiment of synthesizing a chart of additional secondary factors with measured indicators of the radio signal propagation time delay in nanoseconds from the Pohang transmitting station has been repeated at the correct algorithmic level. In order to demonstrate the practical feasibility of the spline algorithm, a computer visualization of a cartographic fragment of additional secondary factors of Yongil Bay is performed. It is suggested that three-dimensional representation of the additional secondary factor for situational perception by the Officer on watch of map-aided correction as a process in alternative positioning in order to increase the reliability of location control through visual evaluation of proper use of corrections of the e-Loran differential variant. The prospect of using the developed package of applied Pascal-programs with implementations on display-type monitors as intellectual support for decision-making by the navigator in a posteriori assessment of the observation accuracy due to the visual representation of the correction field is noted.

Modified hybrid B-spline estimation based on spatial regulator tensor network for burger equation with nonlinear fractional calculus

In this paper, a modified hybrid B-spline approximation based on spatial regulator tensor network (MHBA-SRTN) is proposed for solving the application and the feasibility problems of fractional calculus-based Burger equation. The main innovation points include: (1) a dual singular kernel based fractional derivative operator is employed on the Burger’s equation with external force term to analyze the solutions and properties under perturbation condition for further realism simulation; (2) a modified hybrid B-spline basis function and its corresponding topological tensor network structure are further proposed to solve the provided Burger’s equation with more favorable approximation accuracy and solution uniqueness; (3) the tensor network transformed the hybrid B-spline construction process into the tensor solving with the introduced spatially regulated tensor network decomposition for a more robust solution procedure; (4) an acceleration approach for decomposition process is introduced to significantly reduce its computational and storage complexity. Extensive experiments are also carried out to verify the performance of MHBA-SRTN.

The deep neural network solver for B-spline approximation

This paper introduces a novel unsupervised deep learning approach to address the knot placement problem in the field of B-spline approximation, called deep neural network solvers (DNN-Solvers). Given discrete points, the DNN acts as a solver for calculating knot positions in the case of a fixed knot number. The input can be any initial knots and the output provides the desirable knots. The loss function is based on the approximation error. The DNN-Solver converts the lower-dimensional knot placement problem, characterized as a nonconvex nonlinear optimization problem, into a search for suitable network parameters within a high-dimensional space. Owing to the over-parameterization nature, DNN-Solvers are less likely to be trapped in local minima and robust against initial knots. Moreover, the unsupervised learning paradigm of DNN-Solvers liberates us from constructing high-quality synthetic datasets with labels. Numerical experiments demonstrate that DNN-Solvers are excellent in both approximation results and efficiency under the premise of an appropriate number of knots.

Chaotic dynamics of diatomic systems in an electromagnetic field: Dynamical and topological invariants

An advanced combined quantum-dynamic and chaos-geometric method for analysis, modelling, and forecasting of the chaotic dynamics of diatomic molecules in an intense electromagnetic field is presented. The method is based on the use of the non-stationary theory of the Schrödinger equation in the approximation of the density functional and the methods of the theory of chaos and dynamic systems for the analysis of time series of polarization and other characteristics of diatomic molecules in an intense electromagnetic field. In particular, the latter includes the Gottwald-Melbourne test, the correlation integral method , fractal and multifractal formalism, average mutual information, false nearest neighbours, surrogate data algorithms, analysis on the basis of the Lyapunov's exponents, Kolmogorov entropy, nonlinear forecast models based on algorithms of optimized predicted trajectories, B-spline approximations. As an illustration, the advanced data for the dynamical and topological invariants (correlation dimension, embedding dimension, Kaplan-York dimension, Lyapunov's exponents, Kolmogorov entropy, etc.) for the diatomic ZrO molecule in a linearly polarized electromagnetic field are listed.

An adaptive B-spline representation of topology optimization design for Additive Manufacturing

Topology optimization is a renowned structural optimization approach used to compute the optimal topology for the enhancement of structural performance. It has been in common practice in different engineering fields such as automobile and aerospace. However, there still exist gaps between topology optimization and its engineering operations, which considerably impedes topology optimization’s applications. One of these gaps is how to make (especially finite element-based) topology optimization results machine-readable, which means, how to transform these into computer-aided design (CAD) models, which are ready-to-use models for manufacturing. In the proposed work the authors adopted a unique methodology that plugs the gap between topology optimization and manufacturing. This methodology seamlessly integrates the structural analysis tool, boundary description tool and standard CAD geometry representation; equipped with the Hausdorff distance approach for 3D printing/additive manufacturing of optimal structures. Initially, we extract the skeleton of the optimal design (in the form of a points cloud) using the level set method, followed by an interpolation technique with shape preservation property, the extracted skeleton is made dense. Incorporating the shape and curvature information of the data set, an adaptive B-spline approximation is devised for fitting a smooth curve through the points cloud. This approach interprets topology optimization results as a parametric CAD model with minimum possible control points and minimum approximation error. The CAD-based optimal designs are then used for additive manufacturing. The numerical results exhibit in detail the validity and accuracy of the proposed method.

Optimizing the Mechanical Performance of Green Composite Materials Using Muti-Integrated Optimization Solvers

Natural fiber composites are potential alternatives for synthetic materials due to environmental issues. The overall performance of the fiber composites depends on the reinforcement conditions. Thus, this work aimed to optimize the reinforcement conditions of the natural fiber composites to improve their mechanical performance via applying an integrated scheme of Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and differential evolution (DE) methods considering various reinforcement conditions including fiber length, fiber loading, and treatment time for optimal characteristics of the composite mechanical performance. The B-Spline approximation function was adopted to predict the experimental performance of green composites. The B-Spline approximation function demonstrated incomparable accuracy compared to linear or quadratic regressions. The function is then optimized using an integrated optimization method. Results have demonstrated that optimal reinforcement conditions for the maximized desired mechanical performance of the composite were achieved with high accuracy. The robustness of the proposed approach was approved using various surface plots of the considered input-output parameter relations. Pareto front or the non-dominated solutions of the desired output mechanical properties were also obtained to demonstrate the interaction between the desired properties to facilitate finding the optimal reinforcement conditions of the composite materials.

A Novel Quintic B-Spline Technique for Numerical Solutions of the Fourth-Order Singular Singularly-Perturbed Problems

Singular singularly-perturbed problems (SSPPs) are a powerful mathematical tool for modelling a variety of real phenomena, such as nuclear reactions, heat explosions, mechanics, and hydrodynamics. In this paper, the numerical solutions to fourth-order singular singularly-perturbed boundary and initial value problems are presented using a novel quintic B-spline (QBS) approximation approach. This method uses a quasi-linearization approach to solve SSPNL initial/boundary value problems. And the non-linear problems are transformed into a sequence of linear problems by applying the quasi-linearization approach. The QBS functions produce more accurate results when compared to other existing approaches because of their local support, symmetry, and partition of unity features. This method can be applied to immediately solve the SSPPs without reducing the order in which they are presented. It has been demonstrated that the suggested numerical approach converges uniformly over the whole domain. The proposed approach is implemented on a few problems to validate the scheme. The computational results are compared, and they illustrate that the proposed approach performs better.

SIMEX Estimation of Partially Linear Multiplicative Regression Model with Mismeasured Covariates

In many practical applications, such as the studies of financial and biomedical data, the response variable usually is positive, and the commonly used criteria are based on absolute errors, which is not desirable. Rather, the relative errors are more of concern. We consider statistical inference for a partially linear multiplicative regression model when covariates in the linear part are measured with error. The simulation–extrapolation (SIMEX) estimators of parameters of interest are proposed based on the least product relative error criterion and B-spline approximation, where two kinds of relative errors are both introduced and the symmetry emerges in the loss function. Extensive simulation studies are conducted and the results show that the proposed method can effectively eliminate the bias caused by the measurement errors. Under some mild conditions, the asymptotic normality of the proposed estimator is established. Finally, a real example is analyzed to illustrate the practical use of our proposed method.

A composite Bayesian approach for quantile curve fitting with non-crossing constraints

To fit a set of quantile curves, Bayesian simultaneous quantile curve fitting methods face some challenges in properly specifying a feasible formulation and efficiently accommodating the non-crossing constraints. In this article, we propose a new minimization problem and develop its corresponding Bayesian analysis. The new minimization problem imposes two penalties to control not only the smoothness of fitted quantile curves but also the differences between quantile curves. This enables a direct inference on differences of quantile curves and facilitates improved information sharing among quantiles. After adopting B-spline approximation for the positive smoothing functions in the minimization problem, we specify the pseudo composite asymmetric Laplace likelihood and derive its priors. The computation algorithm, including partially collapsed Gibbs sampling for model parameters and Monte Carlo Expectation-Maximization algorithm for penalty parameters, are provided to carry out the proposed approach. The extensive simulation studies show that, compared with other candidate methods, the proposed approach yields more robust estimation. More advantages of the proposed approach are observed for the extreme quantiles, heavy-tailed random errors, and inference on the differences of quantiles. We also demonstrate the relative performances of the proposed approach and other competing methods through two real data analyses.

B-spline surface approximation method for achieving optimum dwell time in deterministic polishing

In the computer-controlled optical surfacing (CCOS) process, the accurate calculation of the dwell time map is essential for achieving the desired amount of material removal distribution. However, it remains challenging for current methods to generate a smooth dwell time map that reduces the dynamic requirements of machine tools while maintaining the high finishing capability on freeform optics. This paper proposes a B-spline surface approximation method for obtaining the optimum dwell time in deterministic polishing. The B-spline surface representation allows the dwell time map to be expressed by a small number of control points instead of a large number of discrete dwell points. The dwell time deconvolution then becomes a linear least-squares problem with only a few control points as variables, which significantly reduces the scale of the problem. To achieve a more accurate and smoother dwell time map, a knot position optimization algorithm is proposed to adapt the knots to the geometric features of the target removal surface. Simulation results demonstrate that the proposed B-spline-based dwell time algorithm significantly outperforms common dwell time algorithms in terms of the computational accuracy, efficiency, as well as smoothness of the dwell time map. To validate the effectiveness and practicability of the proposed method, experimental polishing of a bi-sinusoidal optical surface has been carried out. The results show that the developed algorithm has greatly reduced the tracking error of the machine tool caused by dwell time fluctuations, and consequently, the corresponding residual error of material removal in deterministic polishing.

Numerical approximation with the splitting algorithm to a solution of the modified regularized long wave equation

In this article, a Lie-Totter splitting algorithm, which is highly reliable, flexible and convenient, is proposed along with the collocation finite element method to approximate solutions of the modified regular long wave equation. For this article, quintic B-spline approximation functions are used in the implementation of collocation methods. Four numerical examples including a single solitary wave, the interaction of two- three solitary waves, and a Maxwellian initial condition are presented to test the closeness of the solutions obtained by the proposed algorithm to the exact solutions. The solutions produced are compared with those in some studies with the same parameters that exist in the literature. The fact that the present algorithm produces results as intended is a proof of how useful, accurate and reliable it is. It can be stated that this fact will be very useful the application of the presented technique for other partial differential equations, with the thought that it may lead the reader to obtain superior results from this study.

Fourier-informed knot placement schemes for B-spline approximation

Fitting B-splines to scientific data is especially challenging when the given data contain noise, jumps, or corners. Here, we describe how periodic data sets with these features can be efficiently approximated with B-splines by analyzing the Fourier spectrum of the data. Our method uses a collection of spectral filters to compute high-order derivatives, smoothed versions of noisy data, and the locations of jump discontinuities. These quantities are then combined to choose knots that capture the qualitative features of the data, leading to accurate B-spline approximations with few knots. The method we introduce is direct and does not require any intermediate B-spline fitting before choosing the final knot distribution. Aside from fast Fourier transforms to transfer to and from Fourier space, the method runs in linear time with very little communication. We assess performance on several test cases in one and two dimensions, including data sets with jump discontinuities and noise. These tests show the method fits discontinuous data without spurious oscillations and remains effective in the presence of noise.

Optimal Maneuvers Around Binary Asteroids Using Particle Swarm Optimization and Machine Learning

Designing optimal transfer trajectories and reference orbit tracking in binary asteroid systems is both challenging and computationally expensive. This paper proposes a method of bypassing the high computational overhead by leveraging a collection of known techniques. Indeed, the proposed framework is based on the combination of artificial intelligence techniques, such as the particle swarm optimization and neural networks, along with the inverse dynamics and the B-splines approximation of the trajectory. The real irregular shapes of the asteroids are considered in the free dynamics of the system, which are obtained via the mutual polyhedral model. The gravitational accelerations of the single asteroids acting on the spacecraft are approximated by using two single-layer neural networks trained via an extreme learning machine. By using a combination of these techniques, the computational time of the whole optimization is decreased from hours to minutes. The proposed approach is applied to the optimal trajectory design around the binary asteroid system, 1999 KW4, showing the feasibility of the proposed optimization approach, reducing the computational effort and time, and increasing the reliability of the obtained results. It is shown through a Monte Carlo analysis that our optimization strategy yields more accurate solutions than other optimization algorithms, such as the interior point and sequential quadratic programming methods, when a random initial guess is provided. Finally, the proposed optimization approach can be used in combination with other techniques to provide a feasible and reliable initial guess for a better solution refinement.t