B-spline Approximation Research Articles

Intelligent monitoring and residual analysis of tunnel point cloud data based on free-form approximation

With the rapid development of urban rail transit and the aging of transportation infrastructure, the demand for shield tunnel disaster detection is about to break out. Most of the traditional monitoring methods require considerable manpower and time costs, which cannot satisfy the increasing requirements of tunnel operation and maintenance. Free-form model construction and deviation mechanism analysis are investigated in this paper to monitor the geometric deformation information of shield tunnels quickly and accurately. A method for identifying geometric features of tunnel sections based on the free-form B-spline approximation is proposed. The innovation of this paper lies in the intelligent recognition of common interference targets by the residual classification method. Furthermore, various Root Mean Squared Error (RMSE) distributions are investigated, which successfully realizes and verifies the clustering analysis of certain point cloud features.

A Cubic B-Splines Approximation Method Combined with DWT and IBP for Single Image Super-resolution

The process of converting low-resolution images into high-resolution images by removing noise and estimating high-frequency information is known as image super-resolution. Aliased and decimated versions of the actual scenes are considered low-resolution images. The edges of high-resolution images produced by super-resolution from a single image are typically blurred. This paper proposes an approach to generate high-resolution image with sharp edges by combining a cubic B-Splines approximation, a discrete wavelet transform (DWT), and an iterative back-projection (IBP) edge-preserving weighted guided filter. A two-stage cubic B-Splines approximation, which includes pre-filtering and interpolation, is employed to up-sample the low-resolution image. The pre-filtering approach is used to transform pixel values to B-Splines coefficients. This approach minimizes blurring in the up-sampled image. The lost high-frequency information is then estimated using a one-level discrete wavelet transform based on the db1 wavelet. Finally, using a weighted guided filter, the resulting image is subjected to back-projection to obtain a high-resolution image. The proposed single-image super-resolution approach is applied on RGB colour images. The proposed method outperforms other selected approaches for comparison objectively in terms of PSNR and SSIM and also in visual quality.

Varying coefficient linear discriminant analysis for dynamic data

Linear discriminant analysis (LDA) is an important classification tool in statistics and machine learning. This paper investigates the varying coefficient LDA model for dynamic data, with Bayes’ discriminant direction being a function of some exposure variable to address the heterogeneity. We propose a new least-square estimation method based on the B-spline approximation. The data-driven discriminant procedure is more computationally efficient than the dynamic linear programming rule [21]. We also establish the convergence rates for the corresponding estimation error bound and the excess misclassification risk. The estimation error in L2 distance is optimal for the low-dimensional regime and is near optimal for the high-dimensional regime. Numerical experiments on synthetic data and real data both corroborate the superiority of our proposed classification method.

DIC deformation analysis using B-spline smoothing with consideration of characteristic noise properties

This paper deals with 3D Digital Image Correlation (DIC) analysis including smoothing of the raw data and subsequent strain determination. The evaluation method is based on B-spline approximation of the 3D coordinates with optimization of the essential smoothing parameters. Thereby, a noise analysis of the reference and deformed state by 2D Fast Fourier Transform is performed with adjustment of smoothing parameters during the optimization. In the procedure, the similarity of noise characteristic is evaluated using the Pearson correlation coefficient. The presented evaluation method is demonstrated by an example of a DIC deformation analysis on bending loaded hybrid laminates. As a result, a smooth and plausible strain distribution is achieved.

A novel energy demodulation method using B-spline approximation for bearing fault detection

The vibration fault signal recorded from a faulty bearing is generally non-stationary in nature and the envelope of such a signal is usually time-varying. This may bring challenges to bearing fault detection. To overcome such difficulties, a novel energy demodulation technique is proposed in this work, which first interpolates the raw signals with nonuniform B-spline approximation and then utilizes an energy demodulation tool based on an enhanced energy measure called envelope-derivative operator (EDO). The B-spline approximation and the EDO are integrated into a new energy operator for extracting the fault signatures. The B-spline EDO enjoys high time resolution and performs well in the case of strong noise. The superiority of the spline-based EDO in dealing with strong noise is demonstrated by simulation and real experiments.

Unified statistical inference for a nonlinear dynamic functional/longitudinal data model

Studying massive functional/longitudinal data, we adopt a flexible nonlinear dynamic regression method named the Semi-Varying Coefficient Additive Model, in which the response can be a functional/longitudinal variable, and the explanatory variables can be a mixture of functional/longitudinal and scalar variables. With the aid of an initial B-spline approximation, a local linear smoothing is proposed to estimate the unknown functional effects in the model. Existing methods of statistical inference for sparse data and dense data are significantly different. We therefore develop the asymptotic theories of the resultant pilot estimation based local linear estimators (PEBLLE) on a unified framework of sparse, dense and ultra-dense cases of data. Remarkably, we obtain the oracle properties as if other functions were known in advance. Extensive Monte Carlo simulation studies investigating the finite sample performance of the proposed methodologies confirm our asymptotic results. We further illustrate our methodologies by analyzing COVID-19 data from China.

Quantile estimation of semiparametric model with time-varying coefficients for panel count data.

Panel count data frequently occurs in follow-up studies, such as medical research, social sciences, reliability studies, and tumorigenicity experiences. This type data has been extensively studied by various statistical models with time-invariant regression coefficients. However, the assumption of invariant coefficients may be violated in some reality, and the temporal covariate effects would be of great interest in research studies. This motivates us to consider a more flexible time-varying coefficient model. For statistical inference of the unknown functions, the quantile regression approach based on the B-spline approximation is developed. Asymptotic results on the convergence of the estimators are provided. Some simulation studies are presented to assess the finite-sample performance of the estimators. Finally, two applications of bladder cancer data and US flight delay data are analyzed by the proposed method.

New trigonometric B-spline approximation for numerical investigation of the regularized long-wave equation

Abstract The objective of this work is to propose a collocation technique based on new cubic trigonometric B-spline (NCTB-spline) functions to approximate the regularized long-wave (RLW) equation. This equation is used for modelling numerous problems occurring in applied sciences. The NCTB-spline collocation method is used to integrate the spatial derivatives. We use the Rubin–Graves linearization technique to linearize the non-linear term. The accuracy and efficiency of the technique are examined by employing it on three important numerical examples which have three invariants of motion viz. mass, momentum, and energy. It is observed that the error norms of the present method are less than the error norms of the methods available in the literature. The numerical values of these invariants have also been approximated, which remain conserved during the program run which shows that the propagation of the solitary wave is represented perfectly. The propagation of one and two solitary waves and undulations of waves are depicted graphically. The stability analysis shows that the method is unconditionally stable.

On semi-analytical Stochastic Boundary Element Method and its application to eigenproblem of thin elastic plate immersed into a fluid

The main objective in this work is to study an application of the B-spline approximations in the Least Squares Method recovery of the structural response for boundary value problems in some linear elastic systems. These responses were approximated before using polynomial bases in the Stochastic perturbation-based Boundary Element Method and now are replaced with second or third order B-spline functions. A majority of such an approach is that the resulting expected values and standard deviations of structural response are obtained using analytical calculus of probability integrals, so that neither statistical nor expansion methods are necessary. Numerical illustration delivered in this paper includes fundamental eigenfrequencies of elastic thin homogeneous and isotropic plate immersed into Newtonian fluid subjected to various boundary conditions at its external edges.

An isogeometric scaled boundary plate formulation for the analysis of ionic electroactive paper

In recent years, electroactive paper emerged as a new alternative in the field of smart actuators. It is based on a cellulose material which is able to bend under the influence of an external electric field similarly as ionic polymer metal composites. The bending mechanism is mainly attributed to the migration of ionic charges over the thickness of a thin sheet of paper. The present contribution proposes a numerical framework for the simulation of electroactive paper. It is based on a scaled boundary plate formulation for isogeometric analysis. In contrast to the standard scaled boundary plate approach, the scaling direction is solved numerically by a B-Spline approximation. This allows to render nonlinear effects over the plate thickness as well as displacement fields of higher continuity. The model is applicable to very thin structures such as electroactive paper, and it also captures the nonlinear ionic charge distribution which is coupled to the bending mechanism of the actuator.

The Unimodality of Initial B-Spline Approximations in Spline Fitting

The Unimodality of Initial B-Spline Approximations in Spline Fitting

An efficient technique based on cubic B-spline functions for solving time-fractional advection diffusion equation involving Atangana-Baleanu derivative.

The present paper deals with cubic B-spline approximation together with theta -weighted scheme to obtain numerical solution of the time fractional advection diffusion equation using Atangana–Baleanu derivative. To discretize the Atangana–Baleanu time derivative containing a non-singular kernel, finite difference scheme is utilized. The cubic basis functions are associated with spatial discretization. The current discretization scheme used in the present study is unconditionally stable and the convergence is of order O(h^2+Delta t^{2}). The proposed scheme is validated through some numerical examples which reveal the current scheme is feasible and quite accurate.

Approximation of the first eigenpair of the p(x)-Laplacian using WEB-spline based mesh-free method

Our objective in this article is to develop weighted extended B-spline (WEB-spline) approximation of the first eigenpair of p(x)-Laplacian problem. One of the reason for choosing WEB-spline based mesh-free method for solving p(x)-Laplacian problem is that, it has additional advantages compared to usual finite element method. For example, it does not require any mesh generation procedure and hence eliminates the difficult, time consuming preprocessing step in finite element method. Also, it combines the computational advantages of B-splines and geometric flexibility of standard mesh-based elements. To compute the first eigenpair of p(x)-Laplacian problem, we examine the problem numerically by applying an inverse power method with three different minimization techniques and we compare the results obtained. Finally, we study the asymptotic behaviour of the eigenvalue problem of p(x)-Laplacian and validate the results comparing with those given in the literature.

A Local Approach for Computing Smooth B-Spline Surfaces for Arbitrary Quadrilateral Base Meshes

Abstract We present a method for approximating surface data of arbitrary topology by a model of smoothly connected B-spline surfaces. Most of the existing solutions for this problem use constructions with limited degrees-of-freedom or they address smoothness between surfaces in a post-processing step, often leading to undesirable surface behavior in proximity of the boundaries. Our contribution is the design of a local method for the approximation process. We compute a smooth B-spline surface approximation without imposing restrictions on the topology of a quadrilateral base mesh defining the individual B-spline surfaces, the used B-spline knot vectors, or the number of B-spline control points. Exact tangent plane continuity can generally not be achieved for a set of B-spline surfaces for an arbitrary underlying quadrilateral base mesh. Our method generates a set of B-spline surfaces that lead to a nearly tangent plane continuous surface approximation and is watertight, i.e., continuous. The presented examples demonstrate that we can generate B-spline approximations with differences of normal vectors along shared boundary curves of less than one degree. Our approach can also be adapted to locally utilize other approximation methods leading to higher orders of continuity.

High-Order Data-Driven Spatial Simulation of Categorical Variables

Modern approaches for the spatial simulation of categorical variables are largely based on multi-point statistical methods, where a training image is used to derive complex spatial relationships using relevant patterns. In these approaches, simulated realizations are driven by the training image utilized, while the spatial statistics of the actual sample data are ignored. This paper presents a data-driven, high-order simulation approach based on the approximation of high-order spatial indicator moments. The high-order spatial statistics are expressed as functions of spatial distances that are similar to variogram models for two-point methods, while higher-order statistics are connected with lower-orders via boundary conditions. Using an advanced recursive B-spline approximation algorithm, the high-order statistics are reconstructed from the available data and are subsequently used for the construction of conditional distributions using Bayes’ rule. Random values are subsequently simulated for all unsampled grid nodes. The main advantages of the proposed technique are its ability to (a) simulate without a training image to reproduce the high-order statistics of the data, and (b) adapt the model’s complexity to the information available in the data. The practical intricacies and effectiveness of the proposed approach are demonstrated through applications at two copper deposits.

The cubic B-spline least-squares finite-element method for the numerical solutions of regularized long-wave equation

A new least-squares method based on cubic B-spline functions is presented to solve regularized long-wave (RLW) equation numerically. An algebraic system of equations is obtained by employing the least-squares method based on cubic B-spline approximation for spatial discretization and the Crank–Nicolson technique for time discretization for integrating the RLW equation. The two test problems are implemented to show suitability of the proposed scheme. New results have been compared with those of some previous studies available in the literature.

Model detection and variable selection for mode varying coefficient model

Varying coefficient model is often used in statistical modeling since it is more flexible than the parametric model. However, model detection and variable selection of varying coefficient model are poorly understood in mode regression. Existing methods in the literature for these problems are often based on mean regression and quantile regression. In this paper, we propose a novel method to solve these problems for mode varying coefficient model based on the B-spline approximation and SCAD penalty. Moreover, we present a new algorithm to estimate the parameters of interest, and discuss the parameters selection for the tuning parameters and bandwidth. We also establish the asymptotic properties of estimated coefficients under some regular conditions. Finally, we illustrate the proposed method by some simulation studies and an empirical example.

An approximation method combined with conformal transformation for the fast measurement of blade section curves

To realize high-precision machining of free-form blade surfaces, it is necessary to measure and reconstruct the blade morphology several times. Sampling measurements and data compression through free-curve approximation technology are of great benefit in shortening the processing time. A novel B-spline approximation method is proposed in this paper, which can determine a few dominant points based on curvature characteristics to approximate dense and noisy measured points with high precision. We adopt a unique adaptive knot placement strategy combined with a curvature dependent evaluation parameter and a conformal transformation method. This strategy is error-bounded in the initial knot placement, which greatly reduces the time consumed by subsequent iterative insertions. The proposed method shows advantages in approximation accuracy, compression rate, and especially in computational efficiency, in comparison with four traditional knot placement methods. Experiments were carried out and the results show that the dominant point selected by this method can replace the fine measurement results, and the compression rate of the measurement points can reach more than 90%. This method is suitable for the sampling measurement and compression of the data of blades, aircraft wings, and other similar free-form surfaces, and is especially suitable for solving the inefficiency problem caused by repeated measurements made in the process of iterative machining of aero-engine blades.

New Cubic B-Spline Approximation for Solving Linear Two-Point Boundary-Value Problems

In this study, we introduce a new cubic B-spline (CBS) approximation method to solve linear two-point boundary value problems (BVPs). This method is based on cubic B-spline basis functions with a new approximation for the second-order derivative. The theoretical new approximation for a second-order derivative and the error analysis have been successfully derived. We found that the second-order new approximation was O(h3) accurate. By using this new second-order approximation, the proposed method was O(h5) accurate. Four numerical problems consisting of linear ordinary differential equations and trigonometric equations with different step sizes were performed to validate the accuracy of the proposed methods. The numerical results were compared with the least squares method, finite difference method, finite element method, finite volume method, B-spline interpolation method, extended cubic B-spline interpolation method and the exact solutions. By finding the maximum errors, the results consistently showed that the proposed method gave the best approximations among the existing methods. We also found that our proposed method involved simple implementation and straightforward computations. Hence, based on the results and the efficiency of our method, we can say that our method is reliable and a promising method for solving linear two-point BVPs.

Semiparametric inference for the scale-mixture of normal partial linear regression model with censored data

In the censored data exploration, the classical linear regression model which assumes normally distributed random errors is perhaps one of the commonly used frameworks. However, practical studies have often criticized the classical linear regression model because of its sensitivity to departure from the normality and partial nonlinearity. This paper proposes to solve these potential issues simultaneously in the context of the partial linear regression model by assuming that the random errors follow a scale-mixture of normal (SMN) family of distributions. The postulated method allows us to model data with great flexibility, accommodating heavy tails and outliers. By implementing the B-spline approximation and using the convenient hierarchical representation of the SMN distributions, a computationally analytical EM-type algorithm is developed for obtaining maximum likelihood (ML) parameter estimates. Various simulation studies are conducted to investigate the finite sample properties, as well as the robustness of the model in dealing with the heavy tails distributed datasets. Real-world data examples are finally analyzed for illustrating the usefulness of the proposed methodology.