scholarly journals Separation Of Non-Periodic And Periodic 2D Profile Features Using B-Spline Functions

2015 ◽  
Vol 22 (2) ◽  
pp. 289-302 ◽  
Author(s):  
Dariusz Janecki ◽  
Leszek Cedro ◽  
Jarosław Zwierzchowski
Keyword(s):  
Spline Function ◽  
Fourier Transforms ◽  
Spline Functions ◽  
Transition Band ◽  
B Spline ◽  
One Dimensional ◽  
B Splines ◽  
Measured Profile ◽  
Spline Filter ◽  
Filter Frequency

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.

scholarly journals B-splines Algorithms for Solving Fredholm Linear Integro-Differential Equations

2004 ◽  
Vol 1 (2) ◽  
pp. 340-346
Author(s):  
Baghdad Science Journal
Keyword(s):  
Differential Equations ◽  
Spline Function ◽  
Cubic Spline ◽  
Second Order ◽  
Third Order ◽  
B Spline ◽  
Numerical Examples ◽  
B Splines ◽  
The Third ◽  
New Procedures

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.

On the Literature of B-Spline Interpolation Functions

2009 ◽  
pp. 214-222
Author(s):  
Carlo Ciulla
Keyword(s):  
Scientific Community ◽  
Polynomial Interpolation ◽  
Spline Interpolation ◽  
Spline Functions ◽  
Great Effort ◽  
Formal Definition ◽  
B Spline ◽  
B Splines ◽  
Scientific Disciplines ◽  
Definition Of

This chapter reviews the extensive and comprehensive literature on B-Splines. In the forthcoming text emphasis is given to hierarchy and formal definition of polynomial interpolation with specific focus to the subclass of functions that are called B-Splines. Also, the literature is reviewed with emphasis on methodologies and applications of B-Splines within a wide array of scientific disciplines. The review is conducted with the intent to inform the reader and also to acknowledge the merit of the scientific community for the great effort devoted to B-Splines. The chapter concludes emphasizing on the proposition that the unifying theory presented throughout this book has for what concerns two specific cases of B-Spline functions: univariate quadratic and cubic models.

Application of Rational B-Splines to the Synthesis of Cam-Follower Motion Programs

1993 ◽  
Vol 115 (3) ◽  
pp. 621-626 ◽  
Author(s):  
D. M. Tsay ◽  
C. O. Huey
Keyword(s):  
Basis Functions ◽  
Spline Functions ◽  
B Spline ◽  
B Splines ◽  
Motion Constraints ◽  
Cam Follower ◽  
Spline Basis

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.

The Results of the Sub-Pixel Efficacy Region Based B-Spline Interpolation Functions

2009 ◽  
pp. 239-337
Author(s):  
Carlo Ciulla
Keyword(s):  
Spectral Power ◽  
Spline Interpolation ◽  
Image Resolution ◽  
B Spline ◽  
One Dimensional ◽  
B Splines ◽  
Sampling Locations ◽  
Characteristic Features ◽  
Analysis Of Error ◽  
Interpolation Functions

The results obtained processing the MRI database with classic and SRE-based one dimensional quadratic and cubic B-Splines are presented in this chapter. The chapter opens up with information relevant to the image resolution of the MRI database employed for validation. The assessment of the performance of the two classes of interpolators (classic and SRE-based) is conducted both quantitatively and qualitatively. The RSME Ratio is plotted to ascertain which ones of the classic or the SRE-based models deliver the smaller interpolation error. Also, the analysis of error images obtained after processing with either of the two model interpolators and the display of the maps of novel re-sampling locations along with spectral power evolutions corroborates the presentation of the characteristic features of the performances of the interpolation functions treated in this chapter.

Solitary waves of the RLW equation via least squares method

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ozlem Ersoy Hepson ◽  
Idris Dag ◽  
Bülent Saka ◽  
Buket Ay
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Least Square Method ◽  
Least Square ◽  
Spline Functions ◽  
Algebraic Equations ◽  
B Spline ◽  
B Splines ◽  
Rlw Equation ◽  
Set Up

Abstract Integration using least squares method in space and Crank–Nicolson approach in time is managed to set up an algorithm to solve the RLW equation numerically. Trial functions in the least square method consist of a combination of the quartic B-spline functions. Integration of the RLW equation gives a system of algebraic equations. The solutions consisting of a combination of the quartic B-splines are given for some initial and boundary value problems of RLW equation.

The Product of Two B-Spline Functions

2011 ◽  
Vol 186 ◽  
pp. 445-448 ◽  
Author(s):  
Xiang Jiu Che ◽  
Gerald Farin ◽  
Zhan Heng Gao ◽  
Dianne Hansford
Keyword(s):  
Numerical Methods ◽  
Linear System ◽  
Unique Solution ◽  
Coefficient Matrix ◽  
Inner Product ◽  
Spline Functions ◽  
B Spline ◽  
B Splines

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.

scholarly journals Fractional nonlinear Volterra–Fredholm integral equations involving Atangana–Baleanu fractional derivative: framelet applications

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Mutaz Mohammad ◽  
Alexander Trounev
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Numerical Approximation ◽  
Fredholm Integral Equations ◽  
Spline Functions ◽  
Vanishing Moments ◽  
B Spline ◽  
B Splines ◽  
Cas Wavelets

Abstract In this work, we propose a framelet method based on B-spline functions for solving nonlinear Volterra–Fredholm integro-differential equations and by involving Atangana–Baleanu fractional derivative, which can provide a reliable numerical approximation. The framelet systems are generated using the set of B-splines with high vanishing moments. We provide some numerical and graphical evidences to show the efficiency of the proposed method. The obtained numerical results of the proposed method compared with those obtained from CAS wavelets show a great agreement with the exact solution. We confirm that the method achieves accurate, efficient, and robust measurement.

The Extension of Theory and Methodology to B-Splines

2009 ◽  
pp. 223-238
Author(s):  
Carlo Ciulla
Keyword(s):  
Methodological Approach ◽  
The Novel ◽  
B Spline ◽  
One Dimensional ◽  
B Splines ◽  
Sampling Locations ◽  
Starting Point ◽  
Before And After ◽  
The One

The organization of the chapter is similar to that of Chapters VII and X. The methodological approach to extend the unifying theory to the one dimensional quadratic and cubic B-Splines is herein reported along with the most relevant mathematical details. This chapter should be read along with Appendix VI where proofs are given to the assertions herein presented. In either of the two cases: quadratic and cubic B-Spline the math process starts from the calculation of the Intensity-Curvature Functional and continues with the calculation of the Sub-pixel Efficacy Region. Finally, the math process arrives to the calculation of the novel re-sampling locations through the formulas of the unifying theory seen in equations (23) and (33) for the quadratic and the cubic models respectively. The chapter concludes with a section that addresses specifically the theoretical proposition of resilient interpolation for the two classes of B-Splines. This is conducted consistently with Chapters VII and XII of the book choosing to equate the two intensity-curvature terms (before and after interpolation) as the starting point of the math deduction.

A new algorithm based on modified trigonometric cubic B-splines functions for nonlinear Burgers’-type equations

2017 ◽  
Vol 27 (8) ◽  
pp. 1638-1661 ◽  
Author(s):  
Ram Jiwari ◽  
Ali Saleh Alshomrani
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Spline Functions ◽  
Difference Operators ◽  
Content Type ◽  
B Spline ◽  
B Splines ◽  
Burgers Type Equations

Purpose The main aim of the paper is to develop a new B-splines collocation algorithm based on modified cubic trigonometric B-spline functions to find approximate solutions of nonlinear parabolic Burgers’-type equations with Dirichlet boundary conditions. Design/methodology/approach A modification is made in cubic trigonometric B-spline functions to handle the Dirichlet boundary conditions and an algorithm is developed with the help of modified cubic trigonometric B-spline functions. The proposed algorithm reduced the Burgers’ equations into a system of first-order nonlinear ordinary differential equations in time variable. Then, strong stability preserving Runge-Kutta 3rd order (SSP-RK3) scheme is used to solve the obtained system. Findings A different technique based on modified cubic trigonometric B-spline functions is proposed which is quite different from to the schemes developed in Abbas et al. (2014) and Nazir et al. (2016), and the developed algorithms are free from linearization process and finite difference operators. Originality/value To the best knowledge of the authors, this technique is novel for solving nonlinear partial differential equations, and the new proposed technique gives better results than the results discussed in Ozis et al. (2003), Kutluay et al. (1999), Khater et al. (2008), Korkmaz and Dag (2011), Kutluay et al. (2004), Rashidi et al. (2009), Mittal and Jain (2012), Mittal and Jiwari (2012), Mittal and Tripathi (2014), Xie et al. (2008) and Kadalbajoo et al. (2005).

An Example of Two-Dimensional Interpolation Using a Linear Combination of Bicubic B-Splines

2011 ◽  
Vol 57 (3) ◽  
pp. 293-299
Author(s):  
Stanisław Rosłoniec
Keyword(s):  
Linear Combination ◽  
Short Description ◽  
Two Dimensional ◽  
Interpolation Algorithm ◽  
Rectangular Grid ◽  
B Spline ◽  
One Dimensional ◽  
The Real ◽  
B Splines

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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