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Journal articles on the topic 'Cubic Spline'

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Published: 4 June 2021

Last updated: 20 February 2023

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1

Xie, Jin, and Xiaoyan Liu. "The EH Interpolation Spline and Its Approximation." Abstract and Applied Analysis 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/745765.

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A new interpolation spline with two parameters, called EH interpolation spline, is presented in this paper, which is the extension of the standard cubic Hermite interpolation spline, and inherits the same properties of the standard cubic Hermite interpolation spline. Given the fixed interpolation conditions, the shape of the proposed splines can be adjusted by changing the values of the parameters. Also, the introduced spline could approximate to the interpolated function better than the standard cubic Hermite interpolation spline and the quartic Hermite interpolation splines with single parameter by a new algorithm.

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2

Syafwan, Elvathna, Mahdhivan Syafwan, and Shandy Tresnawati. "Pengembangan Metode Interpolasi Splin Kubik Terapit dan Aplikasinya pada Masalah Pelacakan Trajektori Objek." Jurnal Teknologi Informasi dan Ilmu Komputer 9, no. 5 (October 31, 2022): 943. http://dx.doi.org/10.25126/jtiik.2022954612.

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<p class="Abstrak">Interpolasi splin kubik merupakan sebuah metode pencocokan kurva yang sangat populer karena mudah diterapkan dan menghasilkan kurva yang mulus. Pada artikel ini dibahas pengembangan metode interpolasi splin kubik untuk syarat batas terapit yang diambil dari rumus eksplisit beda hingga dengan ketelitian orde lebih tinggi. Pengembangan metode ini diterapkan pada masalah pelacakan trajektori objek (<em>object tracking</em>). Secara khusus, masalah ini diujikan untuk splin kubik terapit orde dua, dan hasil interpolasinya dibandingkan dengan hasil pada splin kubik alami dan splin kubik terapit orde satu. Dari simulasi data trajektori yang dibangkitkan dari kurva spiral Archimedean, diperoleh nilai galat total untuk splin kubik alami, terapit orde satu dan terapit orde dua masing-masing sebagai berikut: , dan . Berdasarkan hasil tersebut, disimpulkan bahwa interpolasi splin kubik terapit orde dua yang dikembangkan pada artikel ini dapat menghasilkan trajektori objek yang lebih akurat dibandingkan splin kubik alami dan splin kubik terapit orde satu.</p><p class="Abstrak"> </p><p class="Abstrak"><em><strong>Abstrract</strong></em></p><p class="Abstract"><em>Cubic spline interpolation is a very popular curve fitting method since it is easy to implement and produces a smooth curve. This article discusses the development of the cubic spline interpolation method for a clamped boundary condition taken from finite-difference explicit formulas with higher-order accuracy. The development of this method is applied to an object tracking problem. In particular, this problem is examined for second-order clamped cubic spline, and the interpolated results are compared with those for natural and first-order clamped cubic splines. From the simulation of trajectory data generated from the Archimedean spiral curve, the total error values for natural, first-order, and second-order clamped cubic splines are respectively , and . Based on these results, it is concluded that the second-order clamped cubic spline interpolation developed in this article can produce a more accurate object trajectory than the natural and first-order clamped cubic splines.</em></p><p class="Abstrak"><em><strong><br /></strong></em></p>

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3

Kumar, Arun, and L. K. Govil. "Interpolation of natural cubic spline." International Journal of Mathematics and Mathematical Sciences 15, no. 2 (1992): 229–34. http://dx.doi.org/10.1155/s0161171292000292.

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From the result in [1] it follows that there is a unique quadratic spline which bounds the same area as that of the function. The matching of the area for the cubic spline does not follow from the corresponding result proved in [2]. We obtain cubic splines which preserve the area of the function.

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4

Kirsiaed, Evely, Peeter Oja, and Gul Wali Shah. "CUBIC SPLINE HISTOPOLATION*." Mathematical Modelling and Analysis 22, no. 4 (July 3, 2017): 514–27. http://dx.doi.org/10.3846/13926292.2017.1329756.

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Cubic spline histopolation with arbitrary placement of histogram knots and spline knots between them is studied. Classical boundary conditions are used. Histopolating spline is represented with the help of second moments and particular integrals. The systems determining these parameters are investigated in different cases where diagonal dominance in matrices takes place or may be absent.

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5

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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6

Strelkovskaya, Irina, Irina Solovskaya, and Juliya Strelkovska. "Application of real and complex splines in infocommunication problems." Problemi telekomunìkacìj, no. 1(28) (December 22, 2021): 3–19. http://dx.doi.org/10.30837/pt.2021.1.01.

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The work offers the solution to problems of analysis and synthesis of infocommunication systems with the help of real and complex spline functions. The use of the spline approximation method for solving problems of recovery of random signals and self-similar traffic, management of network objects and network as a whole, and procedures of infocommunication objects and networks functioning is offered. To solve the problems of forecasting, in particular, forecasting the characteristics of network traffic and maintaining the QoS characteristics in its service and formation of requirements for network buffer devices, developed spline extrapolation based on different types of real spline functions, namely: linear, quadratic, quadratic B-splines, cubic, cubic B-splines, cubic Hermite splines. As a criterion for choosing the type of spline function, the prediction error is selected, the accuracy of which can be increased by using a particular kind of spline, depending on the object being predicted. The use of complex flat spline functions is considered to solve the class of user positioning problems in the radio access network. In general, the use of real and complex spline functions allows obtaining the results of improving the Quality of Service in the infocommunication network and ensuring the scalability of the obtained solutions.

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7

Papamichael, N., and M. J. Soares. "Cubic and quintic spline-on-spline interpolation." Journal of Computational and Applied Mathematics 20 (November 1987): 359–66. http://dx.doi.org/10.1016/0377-0427(87)90153-1.

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8

Kim, Jung-Min, Eun-Kook Jung, and Sun-Shin Kim. "Simplification of Face Image using Cubic Spline Interpolation." Journal of Korean Institute of Intelligent Systems 20, no. 5 (October 25, 2010): 722–27. http://dx.doi.org/10.5391/jkiis.2010.20.5.722.

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9

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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10

Rana, S. S., and M. Purohit. "Deficient cubic spline interpolation." Proceedings of the Japan Academy, Series A, Mathematical Sciences 64, no. 4 (1988): 111–14. http://dx.doi.org/10.3792/pjaa.64.111.

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11

Dyer, S. A., and J. S. Dyer. "Cubic-spline interpolation. 1." IEEE Instrumentation & Measurement Magazine 4, no. 1 (March 2001): 44–46. http://dx.doi.org/10.1109/5289.911175.

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12

Nazren, A. R. A., Shahrul Nizam Yaakob, R. Ngadiran, N. M. Wafi, and M. B. Hisham. "Cubic Polynomial as Alternatives Cubic Spline Interpolation." Advanced Science Letters 23, no. 6 (June 1, 2017): 5069–72. http://dx.doi.org/10.1166/asl.2017.7311.

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13

Truong, Nghia, Cem Yuksel, and Larry Seiler. "Quadratic Approximation of Cubic Curves." Proceedings of the ACM on Computer Graphics and Interactive Techniques 3, no. 2 (August 26, 2020): 1–17. http://dx.doi.org/10.1145/3406178.

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We present a simple degree reduction technique for piecewise cubic polynomial splines, converting them into piecewise quadratic splines that maintain the parameterization and C1 continuity. Our method forms identical tangent directions at the interpolated data points of the piecewise cubic spline by replacing each cubic piece with a pair of quadratic pieces. The resulting representation can lead to substantial performance improvements for rendering geometrically complex spline models like hair and fiber-level cloth. Such models are typically represented using cubic splines that are C1-continuous, a property that is preserved with our degree reduction. Therefore, our method can also be considered a new quadratic curve construction approach for high-performance rendering. We prove that it is possible to construct a pair of quadratic curves with C1 continuity that passes through any desired point on the input cubic curve. Moreover, we prove that when the pair of quadratic pieces corresponding to a cubic piece have equal parametric lengths, they join exactly at the parametric center of the cubic piece, and the deviation in positions due to degree reduction is minimized.

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14

Pool, Maryam Khazaei, and Lori Lewis. "A SURVEY ON RECENT HIGHER ORDER SPLINE TECHNIQUES FOR SOLVING BURGERS EQUATION USING B-SPLINE METHODS AND VARIATION OF B-SPLINE TECHNIQUES." Journal of Mathematical Sciences: Advances and Applications 70, no. 1 (April 10, 2022): 1–26. http://dx.doi.org/10.18642/jmsaa_7100122245.

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This is a summary of articles based on higher order B-splines methods and the variation of B-spline methods such as Quadratic B-spline finite elements method, Exponential cubic B-spline method, Septic B-spline technique, Quintic B-spline Galerkin method, and B-spline Galerkin method based on the Quadratic B-spline Galerkin method (QBGM) and Cubic B-spline Galerkin method (CBGM). In this paper, we study the B-spline methods and variations of B-spline techniques to find a numerical solution to the Burgers’ equation. A set of fundamental definitions including Burgers equation, spline functions, and B-spline functions are provided. For each method, the main technique is discussed as well as the discretization and stability analysis. A summary of the numerical results is provided and the efficiency of each method presented is discussed. A general conclusion is provided where we look at a comparison between the computational results of all the presented schemes. We describe the effectiveness and advantages of these methods.

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15

László, Lajos. "Cubic spline interpolation with quasiminimal B-spline coefficients." Acta Mathematica Hungarica 107, no. 1-2 (February 2005): 77–87. http://dx.doi.org/10.1007/s10474-005-0180-4.

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16

Woodward, Charles D. "B2-splines: a local representation for cubic spline interpolation." Visual Computer 3, no. 3 (October 1987): 152–61. http://dx.doi.org/10.1007/bf01962896.

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17

Suenaga, K., and M. Sakai. "A cubic spline approximation of an offset curve of a planar cubic spline." International Journal of Computer Mathematics 78, no. 3 (January 2001): 445–50. http://dx.doi.org/10.1080/00207160108805122.

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18

Korotkiy, Viktor, and Igor' Vitovtov. "Approximation of Physical Spline with Large Deflections." Geometry & Graphics 9, no. 1 (July 22, 2021): 3–19. http://dx.doi.org/10.12737/2308-4898-2021-9-1-3-19.

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Physical spline is a resilient element whose cross-sectional dimensions are very small compared to its axis’s length and radius of curvature. Such a resilient element, passing through given points, acquires a "nature-like" form, having a minimum energy of internal stresses, and, as a consequence, a minimum of average curvature. For example, a flexible metal ruler, previously used to construct smooth curves passing through given coplanar points, can be considered as a physical spline. The theoretical search for the equation of physical spline’s axis is a complex mathematical problem with no elementary solution. However, the form of a physical spline passing through given points can be obtained experimentally without much difficulty. In this paper polynomial and parametric methods for approximation of experimentally produced physical spline with large deflections are considered. As known, in the case of small deflections it is possible to obtain a good approximation to a real elastic line by a set of cubic polynomials ("cubic spline"). But as deflections increase, the polynomial model begins to differ markedly from the experimental physical spline, that limits the application of polynomial approximation. High precision approximation of an elastic line with large deflections is achieved by using a parameterized description based on Ferguson or Bézier curves. At the same time, not only the basic points, but also the tangents to the elastic line of the real physical spline should be given as boundary conditions. In such a case it has been shown that standard cubic Bézier curves have a significant computational advantage over Ferguson ones. Examples for modelling of physical splines with free and clamped ends have been considered. For a free spline an error of parametric approximation is equal to 0.4 %. For a spline with clamped ends an error of less than 1.5 % has been obtained. The calculations have been performed with SMath Studio computer graphics system.

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19

Yoon, K., and S. S. Rao. "Cam Motion Synthesis Using Cubic Splines." Journal of Mechanical Design 115, no. 3 (September 1, 1993): 441–46. http://dx.doi.org/10.1115/1.2919209.

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The application of minimum norm principle, similar to the principle of minimum potential energy, is presented for the general synthesis of cam motion. The approach involves the use of piecewise cubic spline functions for representing the follower displacement. The cubic splines are more convenient and simpler to use compared to general spline functions and also result in smaller peak acceleration and jerk due to the application of the minimum norm principle. A general procedure is presented for application to any cam-follower system. The effectiveness of the approach is illustrated by comparing the results given by the present method with those given by other approaches for a disk cam-translating follower.

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20

Aribowo, Wisnu, and Kazuhiko Terashima. "Cubic Spline Trajectory Planning and Vibration Suppression of Semiconductor Wafer Transfer Robot Arm." International Journal of Automation Technology 8, no. 2 (March 5, 2014): 265–74. http://dx.doi.org/10.20965/ijat.2014.p0265.

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Vibration-free motion in minimal time is desired for industrial robotic applications. Hence, these criteria have to be considered during trajectory planning for a robot arm, wherein polynomial splines are often used for interpolating the trajectory through several via points. Among polynomial splines, the cubic spline is the lowest-degree spline that can provide jerk limitation, a feature that is important for reducing vibration during motion. However, using jerk limitation alone does not eliminate vibration completely and sometimes restricts the performance of industrial robots. This paper proposes an implementation of cubic spline optimization with free via points for reducing motion time, combined with input shaping for suppressing vibration. Experiments are conducted on a semiconductor wafer transfer robot arm to demonstrate the effectiveness of the proposed approach.

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21

Vijayakumar, Hannah. "Two-piece Cubic Spline Functions." Mapana - Journal of Sciences 2, no. 1 (October 2, 2003): 25–33. http://dx.doi.org/10.12723/mjs.3.2.

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.M Prenter defines a cubic Spline function in an interval [a, b] as a piecewise cubic polynomial which is twice continuously differentiable in the entire interval [a, b]. The smooth cubic spline functions fitting the given data are the most popular spline functions and when used for interpolation, they do not have the oscillatory behavior which characterized high-degree polynomials. The natural spline has been shown to be unique function possessing the minimum curvature property of all functions interpolating the data and having square integrable second derivative. In this sense, the natural cubic spline is the smoothest function which interpolates the data. Here Two-piece Natural Cubic Spline functions have been defined. An approximation with no indication of its accuracy is utterly valueless. Where an approximation is intended for the general use, one must , of course, go for the trouble of estimating the error as precisely as possible. In this section, an attempt has been made to derive closed form expressions for the error-functions in the case of Two-piece Spline Functions.

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22

Xu, Weizhi. "Elements of Bi-cubic Polynomial Natural Spline Interpolation for Scattered Data: Boundary Conditions Meet Partition of Unity Technique." Statistics, Optimization & Information Computing 8, no. 4 (December 2, 2020): 994–1010. http://dx.doi.org/10.19139/soic-2310-5070-1083.

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This paper investigates one kind of interpolation for scattered data by bi-cubic polynomial natural spline, in which the integral of square of partial derivative of two orders to x and to y for the interpolating function is minimal (with natural boundary conditions). Firstly, bi-cubic polynomial natural spline interpolations with four kinds of boundary conditions are studied. By the spline function methods of Hilbert space, their solutions are constructed as the sum of bi-linear polynomials and piecewise bi-cubic polynomials. Some properties of the solutions are also studied. In fact, bi-cubic natural spline interpolation on a rectangular domain is a generalization of the cubic natural spline interpolation on an interval. Secondly, based on bi-cubic polynomial natural spline interpolations of four kinds of boundary conditions, and using partition of unity technique, a Partition of Unity Interpolation Element Method (PUIEM) for fitting scattered data is proposed. Numerical experiments show that the PUIEM is adaptive and outperforms state-of-the-art competitions, such as the thin plate spline interpolation and the bi-cubic polynomial natural spline interpolations for scattered data.

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23

Peel, M. C., T. A. McMahon, and G. G. S. Pegram. "Assessing the performance of rational spline-based empirical mode decomposition using a global annual precipitation dataset." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, no. 2106 (April 2009): 1919–37. http://dx.doi.org/10.1098/rspa.2008.0352.

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Empirical mode decomposition (EMD), an adaptive data analysis methodology, has the attractive feature of robustness in the presence of nonlinear and non-stationary time series. Recently, in this journal, Pegram and co-authors ( Pegram et al . 2008 Proc. R. Soc. A 464 , 1483–1501), proposed a modification to the EMD algorithm whereby rational splines replaced cubic splines in the extrema envelope-fitting procedure. The modification was designed to reduce variance inflation, a feature frequently observed in cubic spline-based EMD components primarily due to spline overshooting, by introducing a spline tension parameter. Preliminary results there demonstrated the proof of concept that increasing the spline tension parameter reduces the variance of the resultant EMD components. Here, we assess the performance of rational spline-based EMD for a range of tension parameters and two end condition treatments, using a global dataset of 8135 annual precipitation time series. We found that traditional cubic spline-based EMD can produce decompositions that experience variance inflation and have orthogonality concerns. A tension parameter value of between 0 and 2 is found to be a good starting point for obtaining decomposition components that tend towards orthogonality, as measured by an orthogonality index (OI) metric. Increasing the tension parameter generally results in: (i) a decrease in the range of the OI, which is offset by slight increases in (ii) the median value of OI, (iii) the number of intrinsic mode function components, (iv) the average number of sifts per component, and (v) the degree of amplitude smoothing in the components. The two end conditions tested had little influence on the results, with the reflective case being slightly better than the natural spline case as indicated by the OI. The ability to vary the tension parameter to find an orthogonal set of components, without changing any sifting parameters, is a powerful feature of rational spline-based EMD, which we suggest is a significant improvement over cubic spline-based EMD.

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Bakirov, N. K. "An asymptotically optimal cubic spline." Russian Mathematics 55, no. 4 (March 29, 2011): 5–11. http://dx.doi.org/10.3103/s1066369x11040025.

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25

Ramachandran, Dhanya, and Dr V. Madhukar Mallayya. "Two Variable Cubic Spline Interpolation." IOSR Journal of Mathematics 13, no. 01 (March 2017): 01–05. http://dx.doi.org/10.9790/5728-1301060105.

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26

WHITE, JOHN. "Micromaths: Cubic Spline Curve Fitting." Teaching Mathematics and its Applications 5, no. 1 (1986): 39–45. http://dx.doi.org/10.1093/teamat/5.1.39.

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27

Dyer, S. A., and Xin He. "Cubic-spline interpolation: part 2." IEEE Instrumentation & Measurement Magazine 4, no. 2 (June 2001): 34–36. http://dx.doi.org/10.1109/5289.930984.

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Klassen, R. Victor. "Drawing antialiased cubic spline curves." ACM Transactions on Graphics 10, no. 1 (January 3, 1991): 92–108. http://dx.doi.org/10.1145/99902.99906.

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Kumar, Arun, and L. K. Govil. "On deficient cubic spline interpolants." Journal of Approximation Theory 68, no. 2 (February 1992): 175–82. http://dx.doi.org/10.1016/0021-9045(92)90091-2.

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Farin, G., G. Rein, N. Sapidis, and A. J. Worsey. "Fairing cubic B-spline curves." Computer Aided Geometric Design 4, no. 1-2 (July 1987): 91–103. http://dx.doi.org/10.1016/0167-8396(87)90027-6.

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Xu, Chen-dong, Fa-lai Chen, Jian-song Deng, and Zhou-wang Yang. "Cubic algebraic spline curves design." Applied Mathematics-A Journal of Chinese Universities 26, no. 2 (June 2011): 213–29. http://dx.doi.org/10.1007/s11766-011-2103-1.

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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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XU, Gang, and Guo-Zhao WANG. "Extended Cubic Uniform B-spline and α-B-spline." Acta Automatica Sinica 34, no. 8 (August 2008): 980–84. http://dx.doi.org/10.1016/s1874-1029(08)60047-6.

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CHAND, A. K. B. "NATURAL CUBIC SPLINE COALESCENCE HIDDEN VARIABLE FRACTAL INTERPOLATION SURFACES." Fractals 20, no. 02 (June 2012): 117–31. http://dx.doi.org/10.1142/s0218348x12500119.

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Fractal interpolation functions provide a new light to the natural deterministic approximation and modeling of complex phenomena. The present paper proposes construction of natural cubic spline coalescence hidden variable fractal interpolation surfaces (CHFISs) over a rectangular grid [Formula: see text] through the tensor product of univariate bases of cardinal natural cubic spline coalescence hidden variable fractal interpolation functions (CHFIFs). Natural cubic CHFISs are self-affine or non-self-affine in nature depending on the IFS parameters of univariate natural cubic spline CHFIFs. An upper bound of the error between the natural cubic spline blended coalescence fractal interpolant and the original function is deduced. Convergence of the natural cubic CHFIS to the original function [Formula: see text], and their derivatives are deduced. The effects free variables, constrained free variables and hidden variables are discussed on the natural cubic spline CHFIS with suitably chosen examples.

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CHAND, A. K. B., and G. P. KAPOOR. "CUBIC SPLINE COALESCENCE FRACTAL INTERPOLATION THROUGH MOMENTS." Fractals 15, no. 01 (March 2007): 41–53. http://dx.doi.org/10.1142/s0218348x07003381.

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This paper generalizes the classical cubic spline with the construction of the cubic spline coalescence hidden variable fractal interpolation function (CHFIF) through its moments, i.e. its second derivative at the mesh points. The second derivative of a cubic spline CHFIF is a typical fractal function that is self-affine or non-self-affine depending on the parameters of the generalized iterated function system. The convergence results and effects of hidden variables are discussed for cubic spline CHFIFs.

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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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38

Li, Jie, Yaoyao Tu, and Shilong Fei. "C˜2 Continuous Cubic Hermite Interpolation Splines with Second-Order Elliptic Variation." Tobacco Regulatory Science 7, no. 6 (November 3, 2021): 6317–31. http://dx.doi.org/10.18001/trs.7.6.106.

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In order to solve the deficiency of Hermite interpolation spline with second-order elliptic variation in shape control and continuity, c-2 continuous cubic Hermite interpolation spline with second-order elliptic variation was designed. A set of cubic Hermite basis functions with two parameters was constructed. According to this set of basis functions, the three-order Hermite interpolation spline curves were defined in segments 02, and the parameter selection scheme was discussed. The corresponding cubic Hermite interpolation spline function was studied, and the method to determine the residual term and the best interpolation function was given. The results of an example show that when the interpolation conditions remain unchanged, the cubic Hermite interpolation spline curves not only reach 02 continuity, but also can use the parameters to control the shape of the curves locally or globally. By determining the best values of the parameters, the cubic Hermite interpolation spline function can get a better interpolation effect, and the smoothness of the interpolation spline curve is the best.

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Hou, Xiang Hua, and Hong Hai Liu. "Research on Improved Spline Interpolation Algorithm in Super-Resolution Reconstruction of Video Image." Applied Mechanics and Materials 380-384 (August 2013): 3722–25. http://dx.doi.org/10.4028/www.scientific.net/amm.380-384.3722.

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When low-spline interpolation algorithm is adopted by super-resolution reconstruction for video images, there are some defects, such as saw tooth and blur edge, if the result image is magnified. In this paper, high-order spline interpolation algorithm is introduced and it is optimized. Firstly, the common low-spline interpolation algorithms are analyzed and their shortcomings are pointed out. Then cubic spline interpolation algorithm is discussed. If the image is rotated by cubic spline interpolation algorithm, the magnified image may be not correctly displayed and the image can not be registered in super-resolution reconstruction. Finally, the cubic spline algorithm has been improved. Experimental results show that the improved cubic spline interpolation algorithm can not only eliminate the edge blur and saw tooth, but also do registration in reconstruction when image is rotating.

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Abdul Karim, Samsul Ariffin, and Kong Voon Pang. "Shape Preserving Interpolation UsingC2Rational Cubic Spline." Journal of Applied Mathematics 2016 (2016): 1–14. http://dx.doi.org/10.1155/2016/4875358.

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This paper discusses the construction of newC2rational cubic spline interpolant with cubic numerator and quadratic denominator. The idea has been extended to shape preserving interpolation for positive data using the constructed rational cubic spline interpolation. The rational cubic spline has three parametersαi,βi, andγi. The sufficient conditions for the positivity are derived on one parameterγiwhile the other two parametersαiandβiare free parameters that can be used to change the final shape of the resulting interpolating curves. This will enable the user to produce many varieties of the positive interpolating curves. Cubic spline interpolation withC2continuity is not able to preserve the shape of the positive data. Notably our scheme is easy to use and does not require knots insertion andC2continuity can be achieved by solving tridiagonal systems of linear equations for the unknown first derivativesdi,i=1,…,n-1. Comparisons with existing schemes also have been done in detail. From all presented numerical results the newC2rational cubic spline gives very smooth interpolating curves compared to some established rational cubic schemes. An error analysis when the function to be interpolated isft∈C3t0,tnis also investigated in detail.

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Onn, Nurfarahin, Mohamed Hussein, Collin Howe Hing Tang, Maziah Mohamad, Mohd Zarhamdy Md Zain, and Wei Ying Lai. "Periodic Cubic Spline on Motion of Five-Link Human Bipedal Model Using Nonlinear Predictive Control." Applied Mechanics and Materials 660 (October 2014): 868–72. http://dx.doi.org/10.4028/www.scientific.net/amm.660.868.

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This paper studies about the performance of periodic cubic spline applied on the nonlinear predictive control (NPC) based on receding-horizon cost function optimization. The five-link human bipedal model includes a torso and two legs which has a thigh and a shank in every leg. The periodic cubic spline is used to achieve smooth walking trajectories of every joint in the bipedal model in the sagittal plane. The results show that the tracking error after using periodic cubic spline is smaller than the tracking error before using periodic cubic spline.

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Shumilov, Boris M. "Construction of an Effective Preconditioner for the Even-odd Splitting of Cubic Spline Wavelets." WSEAS TRANSACTIONS ON MATHEMATICS 20 (December 28, 2021): 717–28. http://dx.doi.org/10.37394/23206.2021.20.76.

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In this study, the method for decomposing splines of degree m and smoothness C^m-1 into a series of wavelets with zero moments is investigated. The system of linear algebraic equations connecting the coefficients of the spline expansion on the initial scale with the spline coefficients and wavelet coefficients on the embedded scale is obtained. The originality consists in the application of some preconditioner that reduces the system to a simpler band system of equations. Examples of applying the method to the cases of first-degree spline wavelets with two first zero moments and cubic spline wavelets with six first zero moments are presented. For the cubic case after splitting the system into even and odd rows, the resulting matrix acquires a seven-diagonals form with strict diagonal dominance, which makes it possible to apply an effective sweep method to its solution

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Kačala, Viliam, and Csaba Török. "Optimal Approximation of Biquartic Polynomials by Bicubic Splines." EPJ Web of Conferences 173 (2018): 03012. http://dx.doi.org/10.1051/epjconf/201817303012.

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Recently an unexpected approximation property between polynomials of degree three and four was revealed within the framework of two-part approximation models in 2-norm, Chebyshev norm and Holladay seminorm. Namely, it was proved that if a two-component cubic Hermite spline’s first derivative at the shared knot is computed from the first derivative of a quartic polynomial, then the spline is a clamped spline of classC2and also the best approximant to the polynomial.Although it was known that a 2 × 2 component uniform bicubic Hermite spline is a clamped spline of classC2if the derivatives at the shared knots are given by the first derivatives of a biquartic polynomial, the optimality of such approximation remained an open question.The goal of this paper is to resolve this problem. Unlike the spline curves, in the case of spline surfaces it is insufficient to suppose that the grid should be uniform and the spline derivatives computed from a biquartic polynomial. We show that the biquartic polynomial coefficients have to satisfy some additional constraints to achieve optimal approximation by bicubic splines.

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Tian, Meng, and Hong Ling Geng. "Constrained Control of a Rational Interpolant." Advanced Materials Research 225-226 (April 2011): 170–73. http://dx.doi.org/10.4028/www.scientific.net/amr.225-226.170.

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In this paper, a rational cubic spline interpolation has been constructed using the rational cubic spline with quadratic denominator and the rational cubic spline based on function values. The spline can preserve monotonicity of the data set. The spline not only belongs to in the interpolating interval, but could also be used to constrain the shape of the interpolant curve such as to force it to be the given region. The explicit representation is easily constructed, and numerical experiments indicate that the method produces visually pleasing curves.

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Rakowski, Waldemar. "Application of Cubic Box Spline Wavelets in the Analysis of Signal Singularities." International Journal of Applied Mathematics and Computer Science 25, no. 4 (December 1, 2015): 927–41. http://dx.doi.org/10.1515/amcs-2015-0066.

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Abstract In the subject literature, wavelets such as the Mexican hat (the second derivative of a Gaussian) or the quadratic box spline are commonly used for the task of singularity detection. The disadvantage of the Mexican hat, however, is its unlimited support; the disadvantage of the quadratic box spline is a phase shift introduced by the wavelet, making it difficult to locate singular points. The paper deals with the construction and properties of wavelets in the form of cubic box splines which have compact and short support and which do not introduce a phase shift. The digital filters associated with cubic box wavelets that are applied in implementing the discrete dyadic wavelet transform are defined. The filters and the algorithme à trous of the discrete dyadic wavelet transform are used in detecting signal singularities and in calculating the measures of signal singularities in the form of a Lipschitz exponent. The article presents examples illustrating the use of cubic box spline wavelets in the analysis of signal singularities.

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Munir, N. A. A. A., N. A. Hadi, and M. A. S. Nasir. "C1 Cubic Trigonometric Spline with a Shape Parameter for Positive Shape Preservation." Malaysian Journal of Mathematical Sciences 16, no. 1 (January 31, 2022): 55–66. http://dx.doi.org/10.47836/mjms.16.1.05.

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This paper presents a new construction of C1 cubic trigonometric spline interpolation. Instead of repositioning control points, a shape parameter is introduced in the spline to control the shape and behaviour of the curves. The built basis functions fulfil all the geometric properties of the standard cubic Bezier curve, and the proof is included in this paper. Then, the interpolation of the spline is illustrated using suitable parameter values. Every curve segment comprises four successive control points with a cubic trigonometric spline that carries out all the curve properties. The result showed effective approximation since the developed C1 cubic trigonometric spline produced a smooth and pleasant interpolating curve while preserving the positive data features. The flexibility of the developed spline is compared with the other two existing works: b-spline and bezier-like curves. The analysis shows that the proposed spline gives greater flexibility since it has a broader parameter value range. Therefore, this helps the spline interpolation build opened and closed curves, as incorporated in the paper.Munir, N. A. A. A

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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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Heilat, Ahmed Salem, Hamzeh Zureigat, Raâ€™ed Hatamleh, and Belal Batiha. "New Spline Method for Solving Linear Two-Point Boundary Value Problems." European Journal of Pure and Applied Mathematics 14, no. 4 (November 10, 2021): 1283–94. http://dx.doi.org/10.29020/nybg.ejpam.v14i4.4124.

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In this research, second order linear two-point boundary value problems are treated using new method based on hybrid cubic B-spline. The values of the free parameter,Gamma , are chosen via optimization. The value of the free parameter plays an important role in giving accurate results. Optimization of this parameter is carried out. This method is tested on four examples and a comparison with cubic B-spline, trigonometric cubic B-spline and extended cubic B-spline methods has been carried out. The examples suggest that this method produces more accurate results than the other three methods. The numerical results are presented to illustrate the efficiency of our method.

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Wüst, Sabine, Verena Wendt, Ricarda Linz, and Michael Bittner. "Smoothing data series by means of cubic splines: quality of approximation and introduction of a repeating spline approach." Atmospheric Measurement Techniques 10, no. 9 (September 21, 2017): 3453–62. http://dx.doi.org/10.5194/amt-10-3453-2017.

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Abstract. Cubic splines with equidistant spline sampling points are a common method in atmospheric science, used for the approximation of background conditions by means of filtering superimposed fluctuations from a data series. What is defined as background or superimposed fluctuation depends on the specific research question. The latter also determines whether the spline or the residuals – the subtraction of the spline from the original time series – are further analysed.Based on test data sets, we show that the quality of approximation of the background state does not increase continuously with an increasing number of spline sampling points and/or decreasing distance between two spline sampling points. Splines can generate considerable artificial oscillations in the background and the residuals.We introduce a repeating spline approach which is able to significantly reduce this phenomenon. We apply it not only to the test data but also to TIMED-SABER temperature data and choose the distance between two spline sampling points in a way that is sensitive for a large spectrum of gravity waves.

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KAPOOR, G. P., and SRIJANANI ANURAG PRASAD. "CUBIC SPLINE SUPER FRACTAL INTERPOLATION FUNCTIONS." Fractals 22, no. 01n02 (March 2014): 1450005. http://dx.doi.org/10.1142/s0218348x14500054.

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In the present work, the notion of Cubic Spline Super Fractal Interpolation Function (SFIF) is introduced to simulate an object that depicts one structure embedded into another and its approximation properties are investigated. It is shown that, for an equidistant partition points of [x0, xN], the interpolating Cubic Spline SFIF[Formula: see text] and their derivatives [Formula: see text] converge respectively to the data generating function y(x) ≡ y(0)(x) and its derivatives y(j)(x) at the rate of h2-j+ϵ(0 < ϵ < 1), j = 0, 1, 2, as the norm h of the partition of [x0, xN] approaches zero. The convergence results for Cubic Spline SFIF found here show that any desired accuracy can be achieved in the approximation of a regular data generating function and its derivatives by a Cubic Spline SFIF and its corresponding derivatives.

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