Journal articles: 'B-spliny'

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Dube, Mridula, and Reenu Sharma. "Cubic TP B-Spline Curves with a Shape Parameter." International Journal of Engineering Research in Africa 11 (October 2013): 59–72. http://dx.doi.org/10.4028/www.scientific.net/jera.11.59.

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In this paper a new kind of splines, called cubic trigonometric polynomial B-spline (cubic TP B-spline) curves with a shape parameter, are constructed over the space spanned by As each piece of the curve is generated by three consecutive control points, they posses many properties of the quadratic B-spline curves. These trigonometric curves with a non-uniform knot vector are C1 and G2 continuous. They are C2 continuous when choosing special shape parameter for non-uniform knot vector. These curves are closer to the control polygon than the quadratic B-spline curves when choosing special shape parameters. With the increase of the shape parameter, the trigonometric spline curves approximate to the control polygon. The given curves posses many properties of the quadratic B-spline curves. The generation of tensor product surfaces by these new splines is straightforward.

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Istiqomatul Fajriyah Yuliati and Pardomuan Sihombing. "Pemodelan Fertilitas Di Indonesia Tahun 2017 Menggunakan Pendekatan Regresi Nonparametrik Kernel dan Spline." Jurnal Statistika dan Aplikasinya 4, no. 1 (June 30, 2020): 48–60. http://dx.doi.org/10.21009/jsa.04105.

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Tujuan dari penelitian ini adalah untuk menganalisis pola hubungan Total Fertility Rate (TFR) dengan Contraceptive Prevalence Rate (CPR). Analisis yang sering digunakan untuk pemodelan adalah analisis regresi. Analisis regresi menurut pendekatannya dapat dibedakan menjadi dua, parametrik dan nonparametrik. Metode regresi nonparametrik yang sering digunakan adalah regresi kernel dan spline. Pada penelitian ini untuk regresi kernel yang digunakan adalah regresi kernel dengan metode penaksir Nadaraya-Watson (NWE) dan penaksir polinomial lokal (LPE), sedangkan untuk regresi spline yang digunakan adalah smoothing spline dan b-splines. Hasil pengepasan kurva (fitting curve) menunjukkan bahwa model regresi nonparametrik terbaik adalah model regresi b-splines dengan degree 2 dan jumlah knot 5. Hal ini dikarenakan model regresi b-splines memiliki kurva yang halus dan terlihat lebih mengikuti sebaran data dibandingkan kurva model regresi lainnya. Model regresi b-splines terpilih memiliki nilai koefisien determinasi R2 sebesar 76.86%, artinya besarnya variasi variabel TFR yang dijelaskan oleh model regresi b-splines sebesar 76.86%, sedangkan sisanya 23.14% dijelaskan oleh variabel lainnya yang tidak dimasukkan ke dalam model.

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Budakçı, Gülter, and Halil Oruç. "Further Properties of Quantum Spline Spaces." Mathematics 8, no. 10 (October 14, 2020): 1770. http://dx.doi.org/10.3390/math8101770.

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We construct q-B-splines using a new form of truncated power functions. We give basic properties to show that q-B-splines form a basis for quantum spline spaces. On the other hand, we derive algorithmic formulas for 1/q-integration and 1/q-differentiation for q-spline functions. Moreover, we show a way to find the polynomial pieces on each interval of a q-spline function.

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Journal, Baghdad Science. "B-splines Algorithms for Solving Fredholm Linear Integro-Differential Equations." Baghdad Science Journal 1, no. 2 (June 6, 2004): 340–46. http://dx.doi.org/10.21123/bsj.1.2.340-346.

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Algorithms using the second order of B -splines [B (x)] and the third order of B -splines [B,3(x)] are derived to solve 1' , 2nd and 3rd linear Fredholm integro-differential equations (F1DEs). These new procedures have all the useful properties of B -spline function and can be used comparatively greater computational ease and efficiency.The results of these algorithms are compared with the cubic spline function.Two numerical examples are given for conciliated the results of this method.

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Ezhov, Nikolaj, Frank Neitzel, and Svetozar Petrovic. "Spline Approximation, Part 2: From Polynomials in the Monomial Basis to B-splines—A Derivation." Mathematics 9, no. 18 (September 8, 2021): 2198. http://dx.doi.org/10.3390/math9182198.

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In a series of three articles, spline approximation is presented from a geodetic point of view. In part 1, an introduction to spline approximation of 2D curves was given and the basic methodology of spline approximation was demonstrated using splines constructed from ordinary polynomials. In this article (part 2), the notion of B-spline is explained by means of the transition from a representation of a polynomial in the monomial basis (ordinary polynomial) to the Lagrangian form, and from it to the Bernstein form, which finally yields the B-spline representation. Moreover, the direct relation between the B-spline parameters and the parameters of a polynomial in the monomial basis is derived. The numerical stability of the spline approximation approaches discussed in part 1 and in this paper, as well as the potential of splines in deformation detection, will be investigated on numerical examples in the forthcoming part 3.

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Tsay, D. M., and C. O. Huey. "Application of Rational B-Splines to the Synthesis of Cam-Follower Motion Programs." Journal of Mechanical Design 115, no. 3 (September 1, 1993): 621–26. http://dx.doi.org/10.1115/1.2919235.

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A procedure employing rational B-spline functions for the synthesis of cam-follower motion programs is presented. It differs from earlier techniques that employ spline functions by using rational B-spline basis functions to interpolate motion constraints. These rational B-splines permit greater flexibility in refining motion programs. Examples are provided to illustrate application of the approach.

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Rahayu, Putri Indi, and Pardomuan Robinson Sihombing. "PENERAPAN REGRESI NONPARAMETRIK KERNEL DAN SPLINE DALAM MEMODELKAN RETURN ON ASSET (ROA) BANK SYARIAH DI INDONESIA." JURNAL MATEMATIKA MURNI DAN TERAPAN EPSILON 14, no. 2 (March 2, 2021): 115. http://dx.doi.org/10.20527/epsilon.v14i2.2968.

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Sharia Bank Return On Assets (ROA) modeling in Indonesia in 2018 aims to analyze the relationship pattern of Retturn On Assets (ROA) with interest rates. The analysis that is often used for modeling is regression analysis. Regression analysis is divided into two, namely parametric and nonparametric. The most commonly used nonparametric regression methods are kernel and spline regression. In this study, the nonparametric regression used was kernel regression with the Nadaraya-Watson (NWE) estimator and Local Polynomial (LPE) estimator, while the spline regression was smoothing spline and B-splines. The fitting curve results show that the best model is the B-splines regression model with a degree of 3 and the number of knots 5. This is because the B-splines regression model has a smooth curve and more closely follows the distribution of data compared to other regression curves. The B-splines regression model has a determination coefficient of R ^ 2 of 74.92%,%, meaning that the amount of variation in the ROA variable described by the B-splines regression model is 74.92%, while the remaining 25.8% is explained by other variables not included in the model.

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Dokken, T., V. Skytt, and O. Barrowclough. "LOCALLY REFINED SPLINES REPRESENTATION FOR GEOSPATIAL BIG DATA." ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XL-3/W3 (August 20, 2015): 565–70. http://dx.doi.org/10.5194/isprsarchives-xl-3-w3-565-2015.

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When viewed from distance, large parts of the topography of landmasses and the bathymetry of the sea and ocean floor can be regarded as a smooth background with local features. Consequently a digital elevation model combining a compact smooth representation of the background with locally added features has the potential of providing a compact and accurate representation for topography and bathymetry. The recent introduction of Locally Refined B-Splines (LR B-splines) allows the granularity of spline representations to be locally adapted to the complexity of the smooth shape approximated. This allows few degrees of freedom to be used in areas with little variation, while adding extra degrees of freedom in areas in need of more modelling flexibility. In the EU fp7 Integrating Project IQmulus we exploit LR B-splines for approximating large point clouds representing bathymetry of the smooth sea and ocean floor. A drastic reduction is demonstrated in the bulk of the data representation compared to the size of input point clouds. The representation is very well suited for exploiting the power of GPUs for visualization as the spline format is transferred to the GPU and the triangulation needed for the visualization is generated on the GPU according to the viewing parameters. The LR B-splines are interoperable with other elevation model representations such as LIDAR data, raster representations and triangulated irregular networks as these can be used as input to the LR B-spline approximation algorithms. Output to these formats can be generated from the LR B-spline applications according to the resolution criteria required. The spline models are well suited for change detection as new sensor data can efficiently be compared to the compact LR B-spline representation.

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WULANDARY, SEPTIE, and DRAJAT INDRA PURNAMA. "PERBANDINGAN REGRESI NONPARAMETRIK KERNEL DAN B-SPLINES PADA PEMODELAN RATA-RATA LAMA SEKOLAH DAN PENGELUARAN PERKAPITA DI INDONESIA." Jambura Journal of Probability and Statistics 1, no. 2 (November 18, 2020): 89–97. http://dx.doi.org/10.34312/jjps.v1i2.7501.

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Analisis regresi merupakan salah satu alat statistik yang banyak digunakan untuk mengetahui hubungan antara dua variabel acak atau lebih. Metode penaksiran model regresi terbagi atas regresi parametrik dan nonparametrik. Penelitian ini bertujuan menganalisis pola hubungan pengeluaran perkapita terhadap rata-rata lama sekolah di Indonesia tahun 2018 melalui perbandingan regresi nonparametrik, yaitu regresi kernel dan spline. Regresi kernel yang digunakan adalah regresi kernel dengan metode penaksir Nadaraya-Watson (NWE), sedangkan regresi spline yang digunakan adalah B-Splines. Berdasarkan nilai Generalized Cross Validation (GCV) yang minimum dari model regresi B-Splines, digunakan model dengan degree 2. Perbandingan model terbaik antara model NWE dan B-Splines dilakukan berdasarkan nilai RMSE terkecil dan kurva yang dihasilkan. Pada penelitian ini, model yang terbaik adalah model B-Splines karena memiliki RMSE 0,705, lebih kecil dibandingkan NWE dengan RMSE 1,854. Selain itu, regresi B-Splines memiliki kurva yang halus dan mengikuti sebaran data dibandingkan kurva NWE.

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XING, JUN, JIAHAN LI, RUNQING YANG, XIAOJING ZHOU, and SHIZHONG XU. "Bayesian B-spline mapping for dynamic quantitative traits." Genetics Research 94, no. 2 (April 2012): 85–95. http://dx.doi.org/10.1017/s0016672312000249.

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SummaryOwing to their ability and flexibility to describe individual gene expression at different time points, random regression (RR) analyses have become a popular procedure for the genetic analysis of dynamic traits whose phenotypes are collected over time. Specifically, when modelling the dynamic patterns of gene expressions in the RR framework, B-splines have been proved successful as an alternative to orthogonal polynomials. In the so-called Bayesian B-spline quantitative trait locus (QTL) mapping, B-splines are used to characterize the patterns of QTL effects and individual-specific time-dependent environmental errors over time, and the Bayesian shrinkage estimation method is employed to estimate model parameters. Extensive simulations demonstrate that (1) in terms of statistical power, Bayesian B-spline mapping outperforms the interval mapping based on the maximum likelihood; (2) for the simulated dataset with complicated growth curve simulated by B-splines, Legendre polynomial-based Bayesian mapping is not capable of identifying the designed QTLs accurately, even when higher-order Legendre polynomials are considered and (3) for the simulated dataset using Legendre polynomials, the Bayesian B-spline mapping can find the same QTLs as those identified by Legendre polynomial analysis. All simulation results support the necessity and flexibility of B-spline in Bayesian mapping of dynamic traits. The proposed method is also applied to a real dataset, where QTLs controlling the growth trajectory of stem diameters in Populus are located.

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Rajashekar, Naraveni, Sudhakar Chaudhary, and V. V. K. Srinivas Kumar. "Approximation of p-Biharmonic Problem using WEB-Spline based Mesh-Free Method." International Journal of Nonlinear Sciences and Numerical Simulation 20, no. 6 (October 25, 2019): 703–12. http://dx.doi.org/10.1515/ijnsns-2018-0298.

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Abstract We describe and analyze the weighted extended b-spline (WEB-Spline) mesh-free finite element method for solving the p-biharmonic problem. The WEB-Spline method uses weighted extended b-splines as basis functions on regular grids and does not require any mesh generation which eliminates a difficult, time consuming preprocessing step. Accurate approximations are possible with relatively low-dimensional subspaces. We perform some numerical experiments to demonstrate the efficiency of the WEB-Spline method.

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12

MacCarthy, B. L., and N. D. Burns. "An Evaluation of Spline Functions for use in Cam Design." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 199, no. 3 (July 1985): 239–48. http://dx.doi.org/10.1243/pime_proc_1985_199_118_02.

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This paper shows how spline functions can be employed for kinematic motion specification in cam design. The polynomial spline is introduced as a special case of a continuous piecewise function. Cubic and quintic splines are derived and their properties are discussed in the cam design context. It is shown how standard cam laws can be approximated extremely accurately with a small number of points and appropriate boundary conditions. The modified sinusoidal acceleration cam law is used as an example. The application of quintic splines to non-standard and special motions is discussed. The algebraic and B-spline representations of spline functions are compared. The former is considered preferable in this context and a list of useful algorithms is given. The real power of the spline function, in particular the algebraic quintic spline, is its simplicity, ease of computation and adaptability to non-standard design problems. The use of parametrized, deficient and exponential splines is proposed for specific applications.

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Sissouno, N. "Anisotropic spline approximation with non-uniform B-splines." Applicable Analysis 97, no. 2 (November 14, 2016): 135–44. http://dx.doi.org/10.1080/00036811.2016.1254774.

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14

Ezhov, Nikolaj, Frank Neitzel, and Svetozar Petrovic. "Spline approximation, Part 1: Basic methodology." Journal of Applied Geodesy 12, no. 2 (April 25, 2018): 139–55. http://dx.doi.org/10.1515/jag-2017-0029.

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Abstract In engineering geodesy point clouds derived from terrestrial laser scanning or from photogrammetric approaches are almost never used as final results. For further processing and analysis a curve or surface approximation with a continuous mathematical function is required. In this paper the approximation of 2D curves by means of splines is treated. Splines offer quite flexible and elegant solutions for interpolation or approximation of “irregularly” distributed data. Depending on the problem they can be expressed as a function or as a set of equations that depend on some parameter. Many different types of splines can be used for spline approximation and all of them have certain advantages and disadvantages depending on the approximation problem. In a series of three articles spline approximation is presented from a geodetic point of view. In this paper (Part 1) the basic methodology of spline approximation is demonstrated using splines constructed from ordinary polynomials and splines constructed from truncated polynomials. In the forthcoming Part 2 the notion of B-spline will be explained in a unique way, namely by using the concept of convex combinations. The numerical stability of all spline approximation approaches as well as the utilization of splines for deformation detection will be investigated on numerical examples in Part 3.

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Strelkovskaya, Irina, Irina Solovskaya, and Anastasiya Makoganiuk. "Spline-Extrapolation Method in Traﬃc Forecasting in 5G Networks." Journal of Telecommunications and Information Technology 3 (September 30, 2019): 8–16. http://dx.doi.org/10.26636/jtit.2019.134719.

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This paper considers the problem of predicting self-similar traﬃc with a signiﬁcant number of pulsations and the property of long-term dependence, using various spline functions. The research work focused on the process of modeling self-similar traﬃc handled in a mobile network. A splineextrapolation method based on various spline functions (linear, cubic and cubic B-splines) is proposed to predict selfsimilar traﬃc outside the period of time in which packet data transmission occurs. Extrapolation of traﬃc for short- and long-term forecasts is considered. Comparison of the results of the prediction of self-similar traﬃc using various spline functions has shown that the accuracy of the forecast can be improved through the use of cubic B-splines. The results allow to conclude that it is advisable to use spline extrapolation in predicting self-similar traﬃc, thereby recommending this method for use in practice in solving traﬃc prediction-related problems.

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Wang, Zhihua, Falai Chen, and Jiansong Deng. "Evaluation Algorithm of PHT-Spline Surfaces." Numerical Mathematics: Theory, Methods and Applications 10, no. 4 (September 12, 2017): 760–74. http://dx.doi.org/10.4208/nmtma.2017.0003.

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AbstractPHT-splines are a type of polynomial splines over hierarchical T-meshes which posses perfect local refinement property. This property makes PHT-splines useful in geometric modeling and iso-geometric analysis. Current implementation of PHT-splines stores the basis functions in Bézier forms, which saves some computational costs but consumes a lot of memories. In this paper, we propose a de Boor like algorithm to evaluate PHT-splines provided that only the information about the control coefficients and the hierarchical mesh structure is given. The basic idea is to represent a PHT-spline locally in a tensor product B-spline, and then apply the de-Boor algorithm to evaluate the PHT-spline at a certain parameter pair. We perform analysis about computational complexity and memory costs. The results show that our algorithm takes about the same order of computational costs while requires much less amount of memory compared with the Bézier representations. We give an example to illustrate the effectiveness of our algorithm.

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Gou, Zhi Jian, and Cheng Wang. "The Trajectory Planning and Simulation for Industrial Robot Based on Fifth-Order B-Splines." Applied Mechanics and Materials 538 (April 2014): 367–70. http://dx.doi.org/10.4028/www.scientific.net/amm.538.367.

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The trajectory is planned with fifth-order uniform B-splines for the industrial robot aimed to assure the motion is smooth and the trajectory is fourth-order continuous. Under the premise to satisfy the initial kinematic parameters of the robot as zero, its speed, acceleration and jerk are continuous. Based on B-spline theory, process five B-spline curve function is calculated inversely in joint space. Under the robot kinematics parameter constraints, using fifth-order B-spline interpolates to plan robot trajectory when known interpolation points and the kinematic parameters are simulated and validated by the software of ADAMS.So it provides an effective new method for the trajectory planning.

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JENA, M. K. "CONSTRUCTION OF COMPACTLY SUPPORTED WAVELETS FROM TRIGONOMETRIC B-SPLINES." International Journal of Wavelets, Multiresolution and Information Processing 09, no. 05 (September 2011): 843–65. http://dx.doi.org/10.1142/s021969131100433x.

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We construct a class of semiorthogonal wavelets by taking a normalized trigonometric B-spline of any order as the scaling function. The construction is based on generalized Euler–Frobenius polynomial and generalized autocorrelation function. We also show that the odd order normalized trigonometric B-spline satisfies convex hull property as well as partition of unity property. Moreover, we also present a subdivision algorithm for the convolution of normalized trigonometric B-splines. Several examples of wavelet are also provided.

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Sana, Madiha, and Muhammad Mustahsan. "Finite Element Approximation of Optimal Control Problem with Weighted Extended B-Splines." Mathematics 7, no. 5 (May 20, 2019): 452. http://dx.doi.org/10.3390/math7050452.

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In this research article, an optimal control problem (OCP) with boundary observations is approximated using finite element method (FEM) with weighted extended B-splines (WEB-splines) as basis functions. This type of OCP has a distinct aspect that the boundary observations are outward normal derivatives of state variables, which decrease the regularity of solution. A meshless FEM is proposed using WEB-splines, defined on the usual grid over the domain, R 2 . The weighted extended B-spline method (WEB method) absorbs the regularity problem as the degree of the B-splines is increased. Convergence analysis is also performed by some numerical examples.

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TSIANOS, KONSTANTINOS I., and RON GOLDMAN. "BEZIER AND B-SPLINE CURVES WITH KNOTS IN THE COMPLEX PLANE." Fractals 19, no. 01 (March 2011): 67–86. http://dx.doi.org/10.1142/s0218348x11005221.

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We extend some well known algorithms for planar Bezier and B-spline curves, including the de Casteljau subdivision algorithm for Bezier curves and several standard knot insertion procedures (Boehm's algorithm, the Oslo algorithm, and Schaefer's algorithm) for B-splines, from the real numbers to the complex domain. We then show how to apply these polynomial and piecewise polynomial algorithms in a complex variable to generate many well known fractal shapes such as the Sierpinski gasket, the Koch curve, and the C-curve. Thus these fractals also have Bezier and B-spline representations, albeit in the complex domain. These representations allow us to change the shape of a fractal in a natural manner by adjusting their complex Bezier and B-spline control points. We also construct natural parameterizations for these fractal shapes from their Bezier and B-spline representations.

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Lamnii, A., M. Lamnii, and H. Mraoui. "Constructing B-spline representation of quadratic Sibson–Thomson splines." Computer Aided Geometric Design 33 (February 2015): 66–81. http://dx.doi.org/10.1016/j.cagd.2015.02.001.

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Le-Thi-Thu, Nga, Khoi Nguyen-Tan, and Thuy Nguyen-Thanh. "Reconstruction of Low Degree B-spline Surfaces with Arbitrary Topology Using Inverse Subdivision Scheme." Journal of Science and Technology: Issue on Information and Communications Technology 3, no. 1 (March 31, 2017): 82. http://dx.doi.org/10.31130/jst.2017.41.

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Multivariate B-spline surfaces over triangular parametric domain have many interesting properties in the construction of smooth free-form surfaces. This paper introduces a novel approach to reconstruct triangular B-splines from a set of data points using inverse subdivision scheme. Our proposed method consists of two major steps. First, a control polyhedron of the triangular B-spline surface is created by applying the inverse subdivision scheme on an initial triangular mesh. Second, all control points of this B-spline surface, as well as knotclouds of its parametric domain are iteratively adjusted locally by a simple geometric fitting algorithm to increase the accuracy of the obtained B-spline. The reconstructed B-spline having the low degree along with arbitrary topology is interpolative to most of the given data points after some fitting steps without solving any linear system. Some concrete experimental examples are also provided to demonstrate the effectiveness of the proposed method. Results show that this approach is simple, fast, flexible and can be successfully applied to a variety of surface shapes.

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Singh, Kumar Amerender, Hitesh Shah, and Benjamin Joseph. "Comparison of plaster-of-Paris casts and Woodcast splints for immobilization of the limb during serial manipulation and casting for idiopathic clubfoot in infants." Bone & Joint Journal 102-B, no. 10 (October 1, 2020): 1399–404. http://dx.doi.org/10.1302/0301-620x.102b10.bjj-2020-0181.r4.

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Aims The study was undertaken to compare the efficacy of Woodcast splints and plaster-of-Paris casts in maintaining correction following sequential manipulation of idiopathic clubfeet. Methods In this randomized prospective trial, 23 idiopathic clubfeet were immobilized with plaster-of-Paris casts and 23 clubfeet were immobilized with a splint made of Woodcast that encircled only two-thirds the circumference of the limb. The number of casts or splints needed to obtain full correction, the frequency of cast or splint-related complications, and the time taken for application and removal of the casts and splints were compared. Results The mean number of casts required to obtain full correction of the deformity (Pirani Score 0) was 4.35 (95% confidence interval (CI) 3.74 to 4.95) when plaster-of-Paris was used and 4.87 (95% CI 4.33 to 5.41) when the Woodcast splint was used (p = 0.190). The time required for application and removal of the Woodcast splint were significantly less than that required for application and removal of plaster-of-Paris casts (p < 0.001). Woodcast splint-related complications were not more frequent than plaster-of-Paris cast related complications. Conclusion Though Woodcast splints covering two-thirds of the circumference of the lower limbs of infants were effective in maintaining the correction of clubfoot deformity during serial manipulation and casting treatment, the superiority of Woodcasts over plaster-of-Paris could not be established. Cite this article: Bone Joint J 2020;102-B(10):1399–1404.

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Fujioka, Hiroyuki, and Hiroyuki Kano. "Monotone Smoothing Spline Curves Using Normalized Uniform Cubic B-splines." Transactions of the Institute of Systems, Control and Information Engineers 26, no. 11 (2013): 389–97. http://dx.doi.org/10.5687/iscie.26.389.

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Fujioka, Hiroyuki, and Hiroyuki Kano. "Monotone Smoothing Spline Curves Using Normalized Uniform Cubic B-splines." Proceedings of the ISCIE International Symposium on Stochastic Systems Theory and its Applications 2013 (May 5, 2013): 152–57. http://dx.doi.org/10.5687/sss.2013.152.

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DUBE, MRIDULA, and REENU SHARMA. "PIECEWISE QUARTIC TRIGONOMETRIC POLYNOMIAL B-SPLINE CURVES WITH TWO SHAPE PARAMETERS." International Journal of Image and Graphics 12, no. 04 (October 2012): 1250028. http://dx.doi.org/10.1142/s0219467812500283.

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Analogous to the quartic B-splines curve, a piecewise quartic trigonometric polynomial B-spline curve with two shape parameters is presented in this paper. Each curve segment is generated by three consecutive control points. The given curve posses many properties of the B-spline curve. These curves are closer to the control polygon than the different other curves considered in this paper, for different values of shape parameters for each curve. With the increase of the value of shape parameters, the curve approach to the control polygon. For nonuniform and uniform knot vector the given curves have C0, G3; C1, G3; C1, G7; and C3 continuity for different choice of shape parameters. A quartic trigonometric Bézier curves are also introduced as a special case of the given trigonometric spline curves. A comparison of quartic trigonometric polynomial curve is made with different other curves. In the last, quartic trigonometric spline surfaces with two shape parameters are constructed. They have most properties of the corresponding curves.

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Liu, Xinyue, Xingce Wang, Zhongke Wu, Dan Zhang, and Xiangyuan Liu. "Extending Ball B-spline by B-spline." Computer Aided Geometric Design 82 (October 2020): 101926. http://dx.doi.org/10.1016/j.cagd.2020.101926.

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Gondegaon, Sangamesh, and Hari K. Voruganti. "Spline Parameterization of Complex Planar Domains for Isogeometric Analysis." Journal of Theoretical and Applied Mechanics 47, no. 1 (March 1, 2017): 18–35. http://dx.doi.org/10.1515/jtam-2017-0002.

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Abstract Isogeometric Analysis (IGA) involves unification of modelling and analysis by adopting the same basis functions (splines), for both. Hence, spline based parametric model is the starting step for IGA. Representing a complex domain, using parametric geometric model is a challenging task. Parameterization problem can be defined as, finding an optimal set of control points of a B-spline model for exact domain modelling. Also, the quality of parameterization, too has significant effect on IGA. Finding the B-spline control points for any given domain, which gives accurate results is still an open issue. In this paper, a new planar B-spline parameterization technique, based on domain mapping method is proposed. First step of the methodology is to map an input (non-convex) domain onto a unit circle (convex) with the use of harmonic functions. The unique properties of harmonic functions: global minima and mean value property, ensures the mapping is bi-jective and with no self-intersections. Next step is to map the unit circle to unit square to make it apt for B-spline modelling. Square domain is re-parameterized by using conventional centripetal method. Once the domain is properly parameterized, the required control points are computed by solving the B-spline tensor product equation. The proposed methodology is validated by applying the developed B-spline model for a static structural analysis of a plate, using isogeometric analysis. Different domains are modelled to show effectiveness of the given technique. It is observed that the proposed method is versatile and computationally efficient.

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Tsay, Der Min, and Guan Shyong Hwang. "The Synthesis of Follower Motions of Camoids Using Nonparametric B-Splines." Journal of Mechanical Design 118, no. 1 (March 1, 1996): 138–43. http://dx.doi.org/10.1115/1.2826845.

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This paper proposes a tool to synthesize the motion functions of the camoid-follower mechanisms. The characteristics of these kinds of motion functions are that they possess two independent parameters. To implement the work, this study applies the nonparametric B-spline surface interpolation, whose spline functions are constructed by the closed periodic B-splines and the de Boor’s knot sequences in the two parametric directions of the motion function, respectively. The rules and the restrictions needed to be noticed in the process of synthesis are established. Numerical examples are also given to verify the feasibility of the proposed method.

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Gomes, Lorrayne, Milena Vieira Lima, Jeferson Corrêa Ribeiro, Andreia Santos Cezário, Eliandra Maria Bianchini Oliveira, Wallacy Barbacena Rosa dos Santos, Tiago Neves Pereira Valente, Crislaine Messias de Souza, and Aline Sousa Camargos. "FUNÇÕES SPLINES APLICADAS EM DADOS DE CRESCIMENTO." COLLOQUIUM AGRARIAE 13, Especial 2 (June 1, 2017): 222–34. http://dx.doi.org/10.5747/ca.2017.v13.nesp2.000229.

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In animal breeding, new methodologies can be applied in statistical analysis to improve the genetic evaluation and, for this reason, they have been the subject in several studies. In the last years, several research works have intended the model development with more adjustable functions to the distinct variables. A set of functions known as Spline functions has called the attention of researches. Then, the purpose of this review is to discuss the use of Spline functions that are applied to growth data in animal breeding. Splines are segmented regression functions that are united by points known as joint points and have the ability to improve the curvature of models and, therefore, the function adjustment. These functions have interesting properties such as the interpolatory nature, less multicolinearity problems, parameter linearity and the ability of increasing the approximation domain, all of which provide estimates in a wide range of possible values. There are three types of Spline functions: natural spline functions, smoothing spline 223 Colloquium Agrariae, vol. 13, n. Especial 2, Jan–Jun, 2017, p. 222-234. ISSN: 1809-8215. DOI: 10.5747/ca.2017.v13.nesp2.000229 functions or nonparametric regression and B-splines functions. These latter functions are more applied to animal breeding, mainly as alternatives to random regression models (RRM) that use the Legendre polynomials. The matrices formed by RRMs with the use of B-spline functions or Legendre polynomials are more scarce and easier to be inverted. Then, the use of Spline functions has been more intensified in the last years because studies have had the purpose of improving the adjustment with less model parameters in functions. New studies will allow improving the methodology and finding out new applications to the Spline functions.

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Du, Xiaogang, Jianwu Dang, Yangping Wang, Song Wang, and Tao Lei. "A Parallel Nonrigid Registration Algorithm Based on B-Spline for Medical Images." Computational and Mathematical Methods in Medicine 2016 (2016): 1–14. http://dx.doi.org/10.1155/2016/7419307.

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The nonrigid registration algorithm based on B-spline Free-Form Deformation (FFD) plays a key role and is widely applied in medical image processing due to the good flexibility and robustness. However, it requires a tremendous amount of computing time to obtain more accurate registration results especially for a large amount of medical image data. To address the issue, a parallel nonrigid registration algorithm based on B-spline is proposed in this paper. First, the Logarithm Squared Difference (LSD) is considered as the similarity metric in the B-spline registration algorithm to improve registration precision. After that, we create a parallel computing strategy and lookup tables (LUTs) to reduce the complexity of the B-spline registration algorithm. As a result, the computing time of three time-consuming steps including B-splines interpolation, LSD computation, and the analytic gradient computation of LSD, is efficiently reduced, for the B-spline registration algorithm employs the Nonlinear Conjugate Gradient (NCG) optimization method. Experimental results of registration quality and execution efficiency on the large amount of medical images show that our algorithm achieves a better registration accuracy in terms of the differences between the best deformation fields and ground truth and a speedup of 17 times over the single-threaded CPU implementation due to the powerful parallel computing ability of Graphics Processing Unit (GPU).

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Wang, Jieying, Jiří Kosinka, and Alexandru Telea. "Spline-Based Dense Medial Descriptors for Lossy Image Compression." Journal of Imaging 7, no. 8 (August 19, 2021): 153. http://dx.doi.org/10.3390/jimaging7080153.

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Medial descriptors are of significant interest for image simplification, representation, manipulation, and compression. On the other hand, B-splines are well-known tools for specifying smooth curves in computer graphics and geometric design. In this paper, we integrate the two by modeling medial descriptors with stable and accurate B-splines for image compression. Representing medial descriptors with B-splines can not only greatly improve compression but is also an effective vector representation of raster images. A comprehensive evaluation shows that our Spline-based Dense Medial Descriptors (SDMD) method achieves much higher compression ratios at similar or even better quality to the well-known JPEG technique. We illustrate our approach with applications in generating super-resolution images and salient feature preserving image compression.

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Wang, Aizeng, Gang Zhao, and Chuan He. "Unified Representation of Curves and Surfaces." Mathematics 9, no. 9 (April 30, 2021): 1019. http://dx.doi.org/10.3390/math9091019.

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In conventional modeling, shared control points can be employed to realize a unified representation for an object consisting of only curves or only surfaces touching one another. However, this method fails in treating the following two cases: (a) a system consisting of detached curves or surfaces; (b) a system having both curves and surfaces. The purpose of the present paper is to develop a new theoretical tool to solve such problems. By introducing the definitions of naked knot and I-mesh, the concept of I-spline is put forth, which is, in essence, an expanded B-spline or T-spline. It is verified by examples that the naked knots make I-splines flexible and effective in transforming different surfaces and/or curves into a unified one, especially in the above two cases.

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Garcia-Capulin, C. H., F. J. Cuevas, G. Trejo-Caballero, and H. Rostro-Gonzalez. "Hierarchical Genetic Algorithm for B-Spline Surface Approximation of Smooth Explicit Data." Mathematical Problems in Engineering 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/706247.

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B-spline surface approximation has been widely used in many applications such as CAD, medical imaging, reverse engineering, and geometric modeling. Given a data set of measures, the surface approximation aims to find a surface that optimally fits the data set. One of the main problems associated with surface approximation by B-splines is the adequate selection of the number and location of the knots, as well as the solution of the system of equations generated by tensor product spline surfaces. In this work, we use a hierarchical genetic algorithm (HGA) to tackle the B-spline surface approximation of smooth explicit data. The proposed approach is based on a novel hierarchical gene structure for the chromosomal representation, which allows us to determine the number and location of the knots for each surface dimension and the B-spline coefficients simultaneously. The method is fully based on genetic algorithms and does not require subjective parameters like smooth factor or knot locations to perform the solution. In order to validate the efficacy of the proposed approach, simulation results from several tests on smooth surfaces and comparison with a successful method have been included.

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Wan, Neng, Ke Du, Tao Chen, Sentang Zhang, and Gongnan Xie. "Stabilized Discretization in Spline Element Method for Solution of Two-Dimensional Navier-Stokes Problems." Abstract and Applied Analysis 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/350682.

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In terms of the poor geometric adaptability of spline element method, a geometric precision spline method, which uses the rational Bezier patches to indicate the solution domain, is proposed for two-dimensional viscous uncompressed Navier-Stokes equation. Besides fewer pending unknowns, higher accuracy, and computation efficiency, it possesses such advantages as accurate representation of isogeometric analysis for object boundary and the unity of geometry and analysis modeling. Meanwhile, the selection of B-spline basis functions and the grid definition is studied and a stable discretization format satisfying inf-sup conditions is proposed. The degree of spline functions approaching the velocity field is one order higher than that approaching pressure field, and these functions are defined on one-time refined grid. The Dirichlet boundary conditions are imposed through the Nitsche variational principle in weak form due to the lack of interpolation properties of the B-splines functions. Finally, the validity of the proposed method is verified with some examples.

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Goss, Andreas, Manuel Hernández-Pajares, Michael Schmidt, David Roma-Dollase, Eren Erdogan, and Florian Seitz. "High-Resolution Ionosphere Corrections for Single-Frequency Positioning." Remote Sensing 13, no. 1 (December 22, 2020): 12. http://dx.doi.org/10.3390/rs13010012.

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The ionosphere is one of the main error sources in positioning and navigation; thus, information about the ionosphere is mandatory for precise modern Global Navigation Satellite System (GNSS) applications. The International GNSS Service (IGS) and its Ionosphere Associated Analysis Centers (IAAC) routinely provide ionospheric information in terms of global ionosphere maps (final GIM). Typically, these products are modeled using series expansion in terms of spherical harmonics (SHs) with a maximum degree of n=15 and are based on post processed observations from Global Navigation Satellite Systems (GNSS), as well as final satellite orbits. However, precise applications such as autonomous driving or precision agriculture require real-time (RT) information about the ionospheric electron content with high spectral and spatial resolution. Ionospheric RT-GIMs are disseminated via Ntrip protocol using the SSR VTEC message of the RTCM. This message can be streamed in RT, but it is limited for the dissemination of coefficients of SHs of lower degrees only. It allows the dissemination of SH coefficients up to a degree of n=16. This suits to most the SH models of the IAACs, but higher spectral degrees or models in terms of B-spline basis functions, voxels, splines and many more cannot be considered. In addition to the SHs, several alternative approaches, e.g., B-splines or Voxels, have proven to be appropriate basis functions for modeling the ionosphere with an enhanced resolution. Providing them using the SSR VTEC message requires a transfer to SHs. In this context, the following questions are discussed based on data of a B-spline model with high spectral resolution; (1) How can the B-spline model be transformed to SHs in order to fit to the RTCM requirements and (2) what is the loss of detail when the B-spline model is converted to SHs of degree of n=16? Furthermore, we discuss (3) what is the maximum necessary SH degree n to convert the given B-spline model and (4) how can the transformation be performed to make it applicable for real-time applications? For a final assessment, we perform both, the dSTEC analysis and a single-frequency positioning in kinematic mode, using the transformed GIMs for correcting the ionospheric delay. The assessment shows that the converted GIMs with degrees n≥30 coincide with the original B-spline model and improve the positioning accuracy significantly.

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Fujioka, Hiroyuki, Hiroyuki Kano, and Clyde F. Martin. "Constrained smoothing and interpolating spline surfaces using normalized uniform B-splines." Communications in Information and Systems 14, no. 1 (2014): 23–56. http://dx.doi.org/10.4310/cis.2014.v14.n1.a2.

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Seefried, Andreas, and Andreas Pfeiffer. "Trajektoriengenerator zur Erzeugung beschränkter B-Splines." at - Automatisierungstechnik 68, no. 1 (January 28, 2020): 72–85. http://dx.doi.org/10.1515/auto-2019-0067.

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ZusammenfassungDer Beitrag stellt einen Algorithmus zur Generierung von mehrfach stetig differenzierbaren Trajektorien auf Basis von B-Splines vor, deren Funktionswerte und Ableitungen innerhalb von konstanten Grenzen liegen. Die Trajektorien können für Systeme genutzt werden, bei denen Grenzen in den Ableitungen des Systemeingangs vorliegen. Dies trifft beispielsweise auf Modelle zu, bei denen ein Antrieb mit Begrenzungen in der Position, Geschwindigkeit und Beschleunigung als Systemeingang genutzt wird.Zur Generierung werden Parameter, Startwerte der Trajektorie sowie die konstanten Grenzen benötigt. Eine quadratische Optimierung liefert die Stützpunkte der B-Spline-Kurve, die dadurch definiert ist. Die implizit begrenzten B-Splines können ideal in Optimalsteuerungsproblemen und modellprädiktiven Regelungen als Steuertrajektorie eingesetzt werden. Durch die implizite Berücksichtigung von Nebenbedingungen können ebensolche aus dem eigentlichen Optimierungsproblem entfallen. Daraus resultiert eine Verringerung der Komplexität des Optimalsteuerungsproblems bzw. der modellprädiktiven Regelung was wiederum zu einer schnelleren Berechnungszeit führt.Die implizit begrenzten B-Splines werden im Beitrag vorgestellt und exemplarisch in einer modellprädiktiven Regelung mit einem robotischen System angewandt. Sie führen zu einer deutlich schnelleren Berechnungszeit im Vergleich zur Nutzung von B-Splines mit expliziten Nebenbedingungen im Optimierungsproblem.

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Loe, K. F. "?B-spline: a linear singular blending B-spline." Visual Computer 12, no. 1 (January 1, 1996): 18–25. http://dx.doi.org/10.1007/s003710050044.

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Mittal, R. C., and Rachna Bhatia. "Numerical Solution of Nonlinear Sine-Gordon Equation by Modified Cubic B-Spline Collocation Method." International Journal of Partial Differential Equations 2014 (August 10, 2014): 1–8. http://dx.doi.org/10.1155/2014/343497.

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Modified cubic B-spline collocation method is discussed for the numerical solution of one-dimensional nonlinear sine-Gordon equation. The method is based on collocation of modified cubic B-splines over finite elements, so we have continuity of the dependent variable and its first two derivatives throughout the solution range. The given equation is decomposed into a system of equations and modified cubic B-spline basis functions have been used for spatial variable and its derivatives, which gives results in amenable system of ordinary differential equations. The resulting system of equation has subsequently been solved by SSP-RK54 scheme. The efficacy of the proposed approach has been confirmed with numerical experiments, which shows that the results obtained are acceptable and are in good agreement with earlier studies.

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Powell, Mark, William R. Post, Jay Keener, and Stanley Wearden. "Effective Treatment of Chronic Plantar Fasciitis with Dorsiflexion Night Splints: A Crossover Prospective Randomized Outcome Study." Foot & Ankle International 19, no. 1 (January 1998): 10–18. http://dx.doi.org/10.1177/107110079801900103.

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Chronic plantar fasciitis frustrates patients and treating physicians. Our hypothesis was that use of a dorsiflexion night splint for 1 month would effectively treat patients with recalcitrant plantar fasciitis. A 6-month randomized crossover study included 37 patients with chronic plantar fasciitis. Patients were treated with dorsiflexion night splints for 1 month. Group A wore splints for the 1st month and group B for the 2nd month. No splints were used in either group for the final 4 months of the study. No other medications, stretching, or strengthening exercises were prescribed. Eighty-eight percent of patients who completed the study improved. Eighty percent of the involved feet improved subjectively. Results of the AOFAS Ankle-Hindfoot Rating System 17 and the Mayo Clinical Scoring System 7 demonstrated significant improvement for both groups during the period of splint wear. Improvements were maintained at study completion. Response to splinting did not correlate with foot type, degree of obesity, or the presence of heel spur on radiographs. We believe dorsiflexion splints provide relief from the symptoms of recalcitrant plantar fasciitis in the majority of patients.

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Rosłoniec, Stanisław. "An Example of Two-Dimensional Interpolation Using a Linear Combination of Bicubic B-Splines." International Journal of Electronics and Telecommunications 57, no. 3 (September 1, 2011): 293–99. http://dx.doi.org/10.2478/v10177-011-0039-2.

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An Example of Two-Dimensional Interpolation Using a Linear Combination of Bicubic B-Splines The paper describes how a linear combination of bicubic B-splines can be effectively used in a two-dimensional interpolation. It is assumed that values of a function to be interpolated are evaluated at the uniformly located nodes of a corresponding rectangular grid. All formulae of importance have been derived step by step and are presented in a form convenient for computer implementations. To ensure clarity of considerations a short description of one-dimensional B-spline is also given in Appendix 1. The usefulness of the presented interpolation algorithm has been confirmed by the real engineering example of applications.

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Janecki, Dariusz, Leszek Cedro, and Jarosław Zwierzchowski. "Separation Of Non-Periodic And Periodic 2D Profile Features Using B-Spline Functions." Metrology and Measurement Systems 22, no. 2 (June 1, 2015): 289–302. http://dx.doi.org/10.1515/mms-2015-0016.

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Abstract The form, waviness and roughness components of a measured profile are separated by means of digital filters. The aim of analysis was to develop an algorithm for one-dimensional filtering of profiles using approximation by means of B-splines. The theory of B-spline functions introduced by Schoenberg and extended by Unser et al. was used. Unlike the spline filter proposed by Krystek, which is described in ISO standards, the algorithm does not take into account the bending energy of a filtered profile in the functional whose minimization is the principle of the filter. Appropriate smoothness of a filtered profile is achieved by selecting an appropriate distance between nodes of the spline function. In this paper, we determine the Fourier transforms of the filter impulse response at different impulse positions, with respect to the nodes. We show that the filter cutoff length is equal to half of the node-to-node distance. The inclination of the filter frequency characteristic in the transition band can be adjusted by selecting an appropriate degree of the B-spline function. The paper includes examples of separation of 2D roughness, as well as separation of form and waviness of roundness profiles.

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Jetter, K., S. D. Riemenschneider, and N. Sivakumar. "Schoenberg's exponential Euler spline curves." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 118, no. 1-2 (1991): 21–33. http://dx.doi.org/10.1017/s0308210500028869.

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SynopsisThe exponential Euler spline curves of Schoenberg are used to derive the correctness of cardinal interpolation by shifted univariate B-splines and the “metric condition” on the bi-infinite Toeplitz matrix of interpolation. Additional monotonicity properties of the associated symbol for interpolation in each of its parameters are also given.

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ZHENG, TIANXIANG, and LIHUA YANG. "B-SPLINE ANALYTICAL REPRESENTATION OF THE MEAN ENVELOPE FOR EMPIRICAL MODE DECOMPOSITION." International Journal of Wavelets, Multiresolution and Information Processing 08, no. 02 (March 2010): 175–95. http://dx.doi.org/10.1142/s0219691310003420.

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This paper investigates how the mean envelope, the subtrahend in the sifting procedure for the Empirical Mode Decomposition (EMD) algorithm, represents as an expansion in terms of basis. To this end, a novel approach that gives an alternative analytical expression using B-spline functions is presented. The basic concept lies mainly on the idea that B-spline functions form a basis for the space of splines and have refined-node representations by knot insertion. This newly-developed expression is essentially equivalent to the conventional one, but gives a more explicit formulation on this issue. For the purpose of establishing the mathematical foundation of the EMD methodology, this study may afford a favorable opportunity in this direction.

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Marsden, M. J. "A Voronovskaya Theorem for Variation-Diminishing Spline Approximation." Canadian Journal of Mathematics 38, no. 5 (October 1, 1986): 1081–93. http://dx.doi.org/10.4153/cjm-1986-053-0.

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In [7] Schoenberg introduced the following variation-diminishing spline approximation methods.Let m > 1 be an integer and let Δ = {xi} be a biinfinite sequence of real numbers with xi ≧ xi + l < xi+m. To a function f associate the spline function Vf of order m with knots Δ defined by(1.1)whereand the Nj(x) are B-splines with support xj < x < xj+m normalized so that ΣjNj(x) = 1. See, e.g., [2] for a precise definition of the Nj(x) and a discussion of the properties of Vf.

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Le, Tat-Hien, Dong-Joon Kim, Kyong-Cheol Min, and Sang-Woo Pyo. "B-spline Surface Fitting using Genetic Algorithm." Journal of the Society of Naval Architects of Korea 46, no. 1 (February 20, 2009): 87–95. http://dx.doi.org/10.3744/snak.2009.46.1.087.

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Ryu, Chun-Ha, Tae-Hun Kim, Byung-Gook Lee, Byung-Jae Choi, and Kil-Houm Park. "ECG signal compression based on B-spline approximation." Journal of Korean Institute of Intelligent Systems 21, no. 5 (October 25, 2011): 653–59. http://dx.doi.org/10.5391/jkiis.2011.21.5.653.

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Che, Xiang Jiu, Gerald Farin, Zhan Heng Gao, and Dianne Hansford. "The Product of Two B-Spline Functions." Advanced Materials Research 186 (January 2011): 445–48. http://dx.doi.org/10.4028/www.scientific.net/amr.186.445.

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A method for calculating the product of two B-spline functions is presented. The product is computed by solving a linear system. The coefficient matrix of the system is a Gramian, which guarantees that the system has a unique solution. Every element of the coefficient matrix and the righthand vector of the system is an inner product of B-splines. The inner product can be computed accurately by making use of numerical methods.

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XU, Gang. "Extended Cubic Uniform B-spline and α-B-spline." Acta Automatica Sinica 34, no. 8 (March 2, 2009): 980–83. http://dx.doi.org/10.3724/sp.j.1004.2008.00980.

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Journal articles on the topic 'B-spliny'

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