B-spline Approximation Research Articles

Functional Data Analysis and Design of Experiments as Efficient Tools to Determine the Dynamical Design Space of Food and Biotechnological Batch Processes

A dynamical design space for the batch milling process of a hazelnut-and-cocoa-based paste in a stirred ball mill was obtained through functional data analysis (FDA) combined with design of experiments (DOE). A face-centred central composited design with two functional responses, fineness and energy, and three factors (rotational speed, mass of balls and ball diameter) was used. The functional responses were pre-processed, smoothed through B-spline approximation and subjected to functional principal component analysis (FPCA). The scores of the FPCA were modelled as a function of the experimental factors through response surface models and used to build a final model for the functional responses including only one principal component for energy and two for fineness. The developed models were able to predict with good accuracy the functional responses as a function of the experimental factors and time and allowed to build a dynamical design space. FDA combined with DOE appears to be an efficient and easy-to-use tool to model batch processes and obtain their dynamical design space.

Rank-based estimation in varying coefficient partially functional linear regression models

This paper focuses on robust estimation for varying coefficient partially functional linear regression model (VCPFLM) without specification of the error distributions. The proposed estimators of the nonparametric coefficients and slope function are based on B-spline approximation and rank regression, which demonstrates that the proposed estimation is robust for non-normal error distributions compared with that of least square estimate. In addition, the optimal convergence rates of the estimators are established under some mild regularity conditions. Finally, simulation study and a real data analysis are conducted to illustrate the finite sample performance of the proposed method.

Automatic identification of curve shapes with applications to ultrasonic vocalization

Like human beings, many animals produce sounds for communication and social interactions. The vocalizations of mice have the characteristics of songs, consisting of syllables of different types determined by the frequency modulations and structure variations. To characterize the impact of social environments and genotypes on vocalizations, it is important to identify the patterns of syllables based on the shapes of frequency contours. Using existing hypothesis testing methods to determine the shape classes would require testing various null and alternative hypotheses for each curve, and is impractical for vocalization studies where the interest is on a large number of frequency contours. A new penalization-based method is proposed, which provides function estimation and automatic shape identification simultaneously. The method estimates the functional curve through quadratic B-spline approximation, and captures the shape feature by penalizing the positive and negative parts of the first two derivatives of the spline function in a group manner. It is shown that under some regularity conditions, the proposed method can identify the correct shape with probability approaching one, and the resulting nonparametric estimator can achieve the optimal convergence rate. Simulation shows that the proposed method gives more stable curve estimation and more accurate curve classification than the unconstrained B-spline estimator, and it is competitive to the shape-constrained estimator assuming prior knowledge of the curve shape. The proposed method is applied to the motivating vocalization study to examine the effect of Methyl-CpG binding protein 2 gene on the vocalizations of mice during courtship.

A new optic disc segmentation method using a modified Dolph-Chebyshev matched filter

In this paper, a novel algorithm for automatic optic disc segmentation on fundus images is presented. In the proposed algorithm, a modified Dolph-Chebyshev type I function matched filter is utilized to detect optic disc boundary candidates. Subsequently, the Circular Hough Transform and B-spline approximation are employed to obtain the final optic disc boundary. The matched filter, Circular Hough Transform and B-spline approximation work on a localized region of interest that is obtained based on three independent methods: template-based, vessels density map-based, and maximum entropy-based methods. The performance of the proposed algorithm is evaluated using fundus images from the MESSIDOR and DRIVE datasets. From the experiments, it can be observed that the proposed algorithm successfully locates optic discs with an average success rate of above 99%. Also, the proposed algorithm can achieve best scores in terms of Area Overlap, Dices coefficient, Accuracy, and False Positive Fraction, which makes the algorithm suitable for practical applications such as an automatic detection tool for retinal diseases.

Robust model reconstruction for intelligent health monitoring of tunnel structures

Advanced robotic systems will encounter a rapid breakthrough opportunity and become increasingly important, especially with the aid of the accelerated development of artificial intelligence technology. Nowadays, advanced robotic systems are widely used in various fields. However, the development of artificial intelligence-based robot systems for structural health monitoring of tunnels needs to be further investigated, especially for data modeling and intelligent processing for noises. This research focuses on integrated B-spline approximation with a nonparametric rank method and reveals its advantages of high efficiency and noise resistance for the automatic health monitoring of tunnel structures. Furthermore, the root-mean-square error and time consumption of the rank-based and Huber’s M-estimator methods are compared based on various profiles. The results imply that the rank-based method to model point cloud data has a comparative advantage in the monitoring of tunnel, as well as the large-area structures, which requires high degrees of efficiency and robustness.

An error-bounded B-spline curve approximation scheme using dominant points for CNC interpolation of micro-line toolpath

• The bi-chord error test fully considers the geometric properties of the micro-line toolpath. • The bi-chord errors test greatly reduces the impact of noisy points on the number and distribution of the dominant points. • The chord error estimation for the B-spline curve approximation analytically determines whether or not the geometric deviation between the polygon formed by the micro-line toolpath and the initial B-spline curve is under the chord error constraint. This paper presents a unified framework for computing a B-spline curve to approximate the micro-line toolpath within the desired fitting accuracy. First, a bi-chord error test extended from our previous work is proposed to select the dominant points that govern the overall shape of the micro-line toolpath. It fully considers the geometric characteristics of the micro-line toolpath, i.e., the curvature, the curvature variation and the torsion, appropriately determining the distribution of the dominant points. Second, an initial B-spline curve is constructed by the dominant points in the least square sense. The fitting error is unpredictable and uncontrollable. It is classified into two types: (a) the geometric deviations between the vertices of the polygon formed by the data points and the constructed B-spline curve; (b) those between the edges of the polygon and the constructed B-spline curve. Herein, an applicable dominant point insertion is employed to keep the first geometric deviation within the specified tolerance of fitting error. A geometric deviation model extended from our previous work is developed to estimate the second geometric deviation. It can be effectively integrated into global toolpath optimization. Computational results demonstrate that the bi-chord error test applies to both the planar micro-line toolpath and the spatial micro-line toolpath, and it can greatly reduce the number of the control points. Simulation and experimental results demonstrate that the proposed B-spline approximation approach can significantly improve machining efficiency while ensuring the surface quality.

Discontinuous Galerkin isogeometric analysis for elliptic problems with discontinuous diffusion coefficients on surfaces

This paper is concerned with using discontinuous Galerkin isogeometric analysis (dG-IGA) as a numerical treatment of diffusion problems on orientable surfaces ${\Omega } \subset \mathbb {R}^{3}$ . The computational domain or surface considered consists of several non-overlapping subdomains or patches which are coupled via an interior penalty scheme. In Langer and Moore [13], we presented a priori error estimate for conforming computational domains with matching meshes across patch interface and a constant diffusion coefficient. However, in this article, we generalize the a priori error estimate to non-matching meshes and discontinuous diffusion coefficients across patch interfaces commonly occurring in industry. We construct B-spline or NURBS approximation spaces which are discontinuous across patch interfaces. We present a priori error estimate for the symmetric discontinuous Galerkin scheme and numerical experiments to confirm the theory.

A repair strategy based on tool path modification for damaged turbine blade

Turbine blade plays an extremely important role in the aerodynamic performance of aero-engine. And the repair of damaged turbine blades is of great interest for aerospace industries due to continual increase in raw material and manufacturing costs. This manuscript presents a new repair strategy for damaged turbine blades based on tool path modification. The strategy basically involves three crucial procedures: rigid registration of nominal curve and measured points adjacent to damaged region, B-spline approximation of residual errors based on feasible region, and modification of the tool paths used for repairing machining. Rigid registration of nominal curve to measured points is accomplished using improved ICP algorithm adjacent to damaged region. B-spline approximation of residual errors is used to calculate smooth modification amount corresponding to cutter location points. And finally, modified tool paths are used for repair machining, including grinding and polishing. Tip repairing of a rotor turbine blade has been successfully performed to demonstrate the effectiveness of the proposed methodology. The error of repair machined trace between the repaired area and undamaged area is within 0.01 mm which commendably meets the repair machined requirement. And the repaired results have shown that the repairing strategy proposed in this paper is a reliable solution for repairing tip of turbine blades and can guarantee a surface smooth restoration with higher accuracy.

On the BIC for determining the number of control points in B-spline surface approximation in case of correlated observations

Abstract B-spline curves are a linear combination of control points (CP) and B-spline basis functions. They satisfy the strong convex hull property and have a fine and local shape control as changing one CP affects the curve locally, whereas the total number of CP has a more general effect on the control polygon of the spline. Information criteria (IC), such as Akaike IC (AIC) and Bayesian IC (BIC), provide a way to determine an optimal number of CP so that the B-spline approximation fits optimally in a least-squares (LS) sense with scattered and noisy observations. These criteria are based on the log-likelihood of the models and assume often that the error term is independent and identically distributed. This assumption is strong and accounts neither for heteroscedasticity nor for correlations. Thus, such effects have to be considered to avoid under-or overfitting of the observations in the LS adjustment, i.e. bad approximation or noise approximation, respectively. In this contribution, we introduce generalized versions of the BIC derived using the concept of quasi- likelihood estimator (QLE). Our own extensions of the generalized BIC criteria account (i) explicitly for model misspecifications and complexity (ii) and additionally for the correlations of the residuals. To that aim, the correlation model of the residuals is assumed to correspond to a first order autoregressive process AR(1). We apply our general derivations to the specific case of B-spline approximations of curves and surfaces, and couple the information given by the different IC together. Consecutively, a didactical yet simple procedure to interpret the results given by the IC is provided in order to identify an optimal number of parameters to estimate in case of correlated observations. A concrete case study using observations from a bridge scanned with a Terrestrial Laser Scanner (TLS) highlights the proposed procedure.

Dynamic Centripetal Parameterization Method for B-Spline Curve Interpolation

B-spline data interpolation and approximation require parameterization at the first step. For this purpose, many algorithms have been developed, such as the uniform, centripetal, chord length, Foley and universal methods. The uniform method works well if the input data points are distributed regularly. The chord length method produces large deflections if long chords exist in the data polygon. To remove this effect, the centripetal method was developed. The traditional centripetal method uses a fixed power for chord lengths for parameter distribution. In this paper, we propose an improved version of the centripetal parameterization method for B-spline data interpolation. Our experiments show that individual dynamic power calculation can be possible for each chord length. This new parameterization method produces better behavior when compared to the traditional centripetal method and is more robust against fast changes in chord lengths since it uses the natural logarithm of chord lengths to calculate the parameters.

Strain analysis of cord-rubber composites using DIC

This contribution deals with the experimental investigation of cord-rubber composites with polyamide cord reinforcement. A loading procedure including internal pressure, compression and torsion was applied to the specimens. For deformation analyses at the surface of the specimens, 3D Digital Image Correlation was applied. Tangential strain values were determined by means of a special evaluation procedure using B-spline approximation of the DIC point data. The results show large strain differences between cord and elastomer sections. Based on the obtained experimental data, validation of numerical simulations of the cord-rubber composites can be performed in further investigations.

High-Precision Trajectory Data Reconstruction for TT&C Systems Using LS B-Spline Approximation

Existing trajectory data reconstruction algorithms for tracking, telemetry, and command (TT&C) channel simulators cannot achieve continuity and noise reduction simultaneously. By combining the B-spline interpolation algorithm with the least-square (LS) approximation algorithm, we propose a novel trajectory data reconstruction algorithm using LS B-spline approximation. In addition, a polyphase structure is proposed to implement the algorithm. Compared with existing algorithms, the proposed algorithm achieves higher reconstruction precision when the known trajectory data include measurement errors; moreover, it preserves continuity. Numerical simulation results verified the validity and effectiveness of our proposed algorithm.

Prediction of working memory ability based on EEG by functional data analysis

BackgroundThere is always a demand for fast and accurate algorithms for EEG signal processing. Owing to the high sample rate, EEG signals usually come with a large number of sample points, making it difficult to predict the working memory ability in cognitive research with EEG. New MethodFollowing well-designed experiments, the functional linear model provides a simple framework for regressions involving EEG signal predictors. The use of a data-driven basis in a linear structure naturally extends the standard linear regression model. The proposed approach utilizes B-spline approximation of functional principal components that greatly facilitates implementation. ResultsUsing LASSO feature selection, critical features have been extracted from eight frontal electrodes, and the R-square of 0.72 indicates rather strong linear association of actual observations and out-of-sample predictions. Comparison with Existing MethodsThere does not seem to be any existing methods of predicting working memory ability from N-back task tests via EEG signals; the data-driven functional linear regression method proposed in this work is, to the best of our knowledge, the first of its kind. ConclusionsThe data analytics suggest that a multiple functional linear regression model for the predictive relationship between working memory ability and frontal activity of the brain is both feasible and accurate via EEG signal processing.

Optimized free-form surface modeling of point clouds from laser-based measurement

Freeform parameterizations to reproduce structure deformation are increasingly important topics in laser-scanner-based deformation analyses. High-accuracy assurance of free-form surface approximation is extremely critical for reliable deformation analysis. One main challenge in this field is the model selection. Improper model complexity could result in under-fitting the real object shape or overfitting data noises, and thus a failure of deformation analysis. A multi-sensor system could integrate advantages of different sensors and improve the quality of mission completed. This paper combines terrestrial laser scanning (TLS) and laser tracker (LT) technologies, to enhance high-accuracy surface modeling in deformation analysis. A surface-based B-spline approximation and a multi-sensor system are investigated, the latter of which focuses mainly on the combination of TLS and LT technologies. The innovation of this paper is that the surface-based B-spline approximation is validated and optimized with LT corner cube reflectors. Hypothesis testing is adopted to select the best parameter setting by judging most consistency of TLS and LT in various epochs. In the B-spline surface modeling, both instrumental and numerical uncertainties are considered. We use the instrumental uncertainty model based on intensity value, as well as numerical uncertainty based on adjustment theories. A sampling strategy is proposed to avoid data gaps and obtain even distributed data points.

Robust and automatic modeling of tunnel structures based on terrestrial laser scanning measurement

The terrestrial laser scanning technology is increasingly applied in the deformation monitoring of tunnel structures. However, outliers and data gaps in the terrestrial laser scanning point cloud data have a deteriorating effect on the model reconstruction. A traditional remedy is to delete the outliers in advance of the approximation, which could be time- and labor-consuming for large-scale structures. This research focuses on an outlier-resistant and intelligent method for B-spline approximation with a rank (R)-based estimator, and applies to tunnel measurements. The control points of the B-spline model are estimated specifically by means of the R-estimator based on Wilcoxon scores. A comparative study is carried out on rank-based and ordinary least squares methods, where the Hausdorff distance is adopted to analyze quantitatively for the different settings of control point number of B-spline approximation. It is concluded that the proposed method for tunnel profile modeling is robust against outliers and data gaps, computationally convenient, and it does not need to determine extra tuning constants.

Support and approximation properties of Hermite splines

In this paper, we formally investigate two mathematical aspects of Hermite splines that are relevant to practical applications. We first demonstrate that Hermite splines are maximally localized, in the sense that the size of their support is minimal among pairs of functions with identical reproduction properties. Then, we precisely quantify the approximation power of Hermite splines for the reconstruction of functions and their derivatives. It is known that the Hermite and B-spline approximation schemes have the same approximation order. More precisely, their approximation error vanishes as O(T4) when the step size T goes to zero. In this work, we show that they actually have the same asymptotic approximation error constants, too. Therefore, they have identical asymptotic approximation properties. Hermite splines combine optimal localization and excellent approximation power, while retaining interpolation properties and closed-form expression, in contrast to existing similar functions. These findings shed a new light on the convenience of Hermite splines in the context of computer graphics and geometrical design.

Feature screening for ultrahigh-dimensional additive logistic models

This paper introduces a sure screening method for ultrahigh-dimensional additive logistic models. With binary response variable, additive logistic model is very useful in social and biological research such as disease detection. The proposed feature screening procedure, ALNIS (nonparametric independence screening for additive logistic models), employs B-spline approximation to model the marginal effect, transforming nonparametric problems into parametric ones. This screening process ranks the nonparametric components according to their norms of the marginal likelihood estimate. Under appropriate conditions, the proposed method is shown to possess sure screening property with a vanishing false selection rate. In numerical studies, we use simulated data to compare the performance of the proposed approach with other seven methods that allow the existence of binary response. We further illustrate the proposed procedure by a real data analysis. Numerical comparison indicates that the proposed approach enjoys robustness and effectiveness under ultrahigh-dimensional additive logistic models.

Automated Longitudinal Control Based on Nonlinear Recursive B-Spline Approximation for Battery Electric Vehicles

This works presents a driver assistance system for energy-efficient ALC of a BEV. The ALC calculates a temporal velocity trajectory from map data. The trajectory is represented by a cubic B-spline function and results from an optimization problem with respect to travel time, driving comfort and energy consumption. For the energetic optimization we propose an adaptive model of the required electrical traction power. The simple power train of a BEV allows the formulation of constraints as soft constraints. This leads to an unconstrained optimization problem that can be solved with iterative filter-based data approximation algorithms. The result is a direct trajectory optimization method of which the effort grows linearly with the trajectory length, as opposed to exponentially as with most other direct methods. We evaluate ALC in real test drives with a BEV. We also investigate the energy-saving potential in driving simulations with ALC compared to MLC. On the chosen reference route the ALC saves up to 3.4% energy compared to MLC at same average velocity, and achieves a 2.6% higher average velocity than MLC at the same energy consumption.

Cubic B-spline approximation for linear stochastic integro-differential equation of fractional order

In this paper, the cubic B-spline collocation method is used for solving the stochastic integro-differential equation of fractional order. we show that stochastic integro-differential equation of fractional order is equivalent to a modified stochastic integral equation. Then we apply the proposed method to obtain a numerical scheme of the modified stochastic integral equation. Using this method, the problem solving turns into a linear system solution of equations. Also, the convergence analysis of this numerical approach has been discussed. In the end, examples are given to test the accuracy and the implementation of the method. The results are compared with the results obtained by other methods to verify that this method is accurate and efficient.

A new quintic B-spline approximation for numerical treatment of Boussinesq equation

A new quintic B-spline approximation for numerical treatment of Boussinesq equation