B-spline Approximation Method Research Articles

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%).

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.

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.

A Genetic Algorithm Application in Planning Path Using B-Spline Model for Autonomous Underwater Vehicle (AUV)

This paper proposes a method of using the B-spline mathematical model to plan high smoothness curve trajectories with heading condition through given waypoints for autonomous underwater vehicles (AUVs) in particular and ships with rudder systems in general. In addition, this paper examines some of the physical limitations of this vehicle, which lead to some binding conditions of the trajectory. Besides, the paper applies B-spline approximation method to reduce the curvature of the trajectory, when waypoints are too close and we do not need to go through exactly these waypoints in the 2D plane. Finally, this paper also proposes the genetic algorithm application with some modifications to solve the optimization of B-spline path length with constraint in turning radius of the vehicle.

Iterative modeling of wind turbine power curve based on least‐square B‐spline approximation

AbstractPower curve plays a vital role in design, operation, control, and condition monitoring of wind turbines. Especially, it provides necessary information for wind farm maintenance decisions, allowing for not only comparison among the same type of wind turbines installed in different places, but also the same one at different time stamps. A standardized power curve iterative modeling procedure of wind turbines is proposed based on actual supervisory control and data acquisition (SCADA) data for performance evaluation in an online manner with guaranteed smoothness. The data are firstly preprocessed for anomaly detection by following normalization correction and then sliced into different wind speed bins. The least‐square B‐spline approximation method is iteratively implemented for power curve construction based on dominant points selected from the bin centroids, and turbulence intensity correction and outlier detection are conducted iteratively for refining identification results. Finally, the test results based on actual operating SCADA data demonstrate better performance in comparison with two counterparts.

New Cubic B-spline Approximation for Solving Non-linear Singular Boundary Value Problems Arising in Physiology

In this article, a new cubic B-spline approximation method is presented for solving second order singular boundary value problems. The proposed numerical technique is based on cubic B-spline collocation method equipped with a new approximation for second order derivative. Several test problems arising in the field of Physiological sciences are considered. The approximate results are compared with the numerical techniques existing in literature. It is found that our new approximation performs superior to current methods due to its simplicity, straight forward interpolation and less computational cost.

A point projection approach for improving the accuracy of the multilevel B-spline approximation

Abstract In this study, we present a method for improving the accuracy of the multilevel B-spline approximation (MBA) method. We combine a point projection method with the MBA method for reducing the approximation error by directly adjusting the control points in the local area. An initial surface is generated by the MBA method, and grid points are produced on the surface. These grid points are projected onto the scattered point set, and the distances between the grid points and the projected points are computed. The control points are then modified based on the distances. The proposed method shows better approximations even with the same number of control points and ensures C2-continuity. The experimental results with examples verify the validity of the proposed method. Highlights We propose a method for improving the multilevel B-spline approximation method. We use a point projection method for computing the amount of errors. The computed errors are directly applied to the control points for reducing the approximation error.

Path smoothing for five-axis machine tools using dual quaternion approximation with dominant points

Nowadays, G01 blocks are still the most widely utilized representation form for the tool paths in the five-axis machine tools. Tangency and curvature discontinuities of the linear tool paths may lead to frequent feedrate fluctuation and acceleration variation, which inevitably deteriorate the machining efficiency and quality. To solve this problem, this paper proposed a dual quaternion B-spline approximation method of dominant points to generate smooth tool paths for five-axis CNC machine tools. First, the dominant points, which can characterize the shape of the linear path, are selected for the tool tip location and the tool orientation simultaneously. Such scheme can significantly compress the number of the control points and release the computational load. Then, the linear tool paths are converted to dual quaternions and approximated to dual quaternion B-spline, which can realize the parameter synchronization for the tool tip and tool orientation, and avoid the fitting oscillatory as well. To verify the proposed method, a trajectory generation method for the dual quaternion B-spline is developed. Simulations and experiments on a five-axis machine tool are conducted, and the results demonstrate the feasibility and efficiency of the proposed method.

Spacecraft Orbit Determination with The B-spline Approximation Method

It is known that the dynamical orbit determination is the most common way to get the precise orbits of spacecraft. However, it is hard to build up the precise dynamical model of spacecraft sometimes. In order to solve this problem, the technique of the orbit determination with the B-spline approximation method based on the theory of function approximation is presented in this article. In order to verify the effectiveness of this method, simulative orbit determinations in the cases of LEO (Low Earth Orbit), MEO (Medium Earth Orbit), and HEO (Highly Eccentric Orbit) satellites are performed, and it is shown that this method has a reliable accuracy and stable solution. The approach can be performed in both the conventional celestial coordinate system and the conventional terrestrial coordinate system. The spacecraft's position and velocity can be calculated directly with the B-spline approximation method, it needs not to integrate the dynamical equations, nor to calculate the state transfer matrix, thus the burden of calculations in the orbit determination is reduced substantially relative to the dynamical orbit determination method. The technique not only has a certain theoretical significance, but also can serve as a conventional algorithm in the spacecraft orbit determination.

B-spline approximation methods for digital image reconstruction in strain measurement

A non-invasive digital imaging procedure has been successfully developed to quantify the strain history during electric resistance welded (ERW) pipe forming. A curve fitting method was used to reconstruct a continuous and smooth pipe profile from digital image data. Three B-spline approximation methods (one-dimensional, two-dimensional, and trigonometric B-spline) were evaluated using the same set of digital image data, and validated against the strain gauge measurements. Trigonometric B-spline provides superior curve fitting and strain prediction. Using the trigonometric B-spline approximation method, a 3D strain history contour plot was generated to optimize the mill setup and identify the strain reversal.

Optimization of the Composition of High-Temperature Coatings Using a B-Spline Approximation Method

The operational characteristics (fracture energy, apparent density, and open porosity) of multicomponent phosphate-hardened coatings are studied as a function of the component concentration using a B-spline approximation method and optimum compositions for the coating material are obtained.

Automatic cardiac LV boundary detection and tracking using hybrid fuzzy temporal and fuzzy multiscale edge detection.

This paper describes a new fully automatic fuzzy multiresolution-based algorithm for cardiac left ventricular (LV) epicardial and endocardial boundary detection and tracking on a sequence of short axis (SA) echocardiographic images of a complete cardiac cycle. This is a necessary step for automatic quantification of cardiac function using echo images. The proposed method is a "center-based" approach in which epicardial and endocardial boundary edge points are searched for on radial lines emanating from the LV center point. The central point of the LV cavity is estimated using a fuzzy-based technique in which the "uncertain" spatial, morphological, and intensity information of the image are represented as fuzzy sets and then combined by fuzzy operators. Edge-detection stage uses multiscale spatial and temporal information in a fuzzy multiresolution framework to identify a single moving edge point for each one of the epicardial and endocardial boundaries over the M radii in the N frames of a complete cardiac cycle. The raw extracted edge points are then processed in the wavelet domain to reduce the effects of noise from the boundaries and papillary muscles from the endocardial boundary extraction process. Finally, a uniform cubic B-spline approximation method is used to define the closed LV boundaries. Experiments with simulated and real echocardiographic images are presented.

Structural analysis of shells by the b-spline field approximation method

The B-spline field approximation method is used to construct discrete models of thick shells. The shell formulation is developed from first principles, using assumptions that are useful in this numerical context. The transverse shear strains, rather than the rotation variables, are used as independent variables for the purpose of discretization. Examples of the application of this formulation are presented.

Predicting and avoiding shear locking in beam vibration problems using the B-spline field approximation method

The B-spline field approximation method is used to construct discrete models of straight Timoshenko beams. The traditional Timoshenko beam discretization treats deflection and rotation as the independent field variables and permits the use of C 0 discretizations, but can sometimes exhibit unwanted shear locking behaviour. In this paper a new explanation of shear locking is presented and is used in selecting a variety of B-spline interpolation fields that can be predicted not to suffer from shear locking. It is also exploited in an adaptation of a Timoshenko model where deflection and shear strain are independently discretized field variables. This requires a C 1 discretization of the deflection, but avoids shear locking in every case, and allows for a more economical solution than previous implementations of this formulation. As a means of illustrating the performance characteristics of the formulation presented in the paper the vibration characteristics of these models are compared with traditional finite element models; for this problem the B-spline method outperforms the finite element method and has the advantage of an easy predictability of the locking behaviour that can be expected.

Collision Free Quasi-Minimum-Time Trajectory Planning for Skid-Steer Platforms.

This paper addresses a problem of time-optimal trajectory planning for skid-steer platforms considering obstacle avoidance. Considering both time-optimality and obstacle avoidance for mobile robot is not easy, because the two problems are coupled in the trajectory planning problem. Using an idea of path parameter, the trajectory planning problem is divided into two sub-problems, i.e., (1) time-optimization of trajectory along specified path, (2) optimal path planning considering obstacle avoidance. Then an efficient search method is proposed for each problem, using the idea of path parameter and B-Spline approximation method. The idea of path parameter simplifies the optimization problem for specified path planning problem of mobile robot. Simulation results show the effectiveness of the proposed trajectory planning method.