scholarly journals Image segmentation and preserve the boundary of the image using b-spline basis

2021 ◽  
Vol 5 (2) ◽  
pp. 121-131
Author(s):  
Gajalakshmi N ◽  
Karunanith S
Keyword(s):  
Image Segmentation ◽  
Optimal Solution ◽  
Input Image ◽  
Mean Square ◽  
Knot Insertion ◽  
Spline Collocation ◽  
B Spline ◽  
B Splines ◽  
Number Of Zeros ◽  
Spline Basis

This paper focuses the knot insertion in the B-spline collocation matrix, with nonnegative determinants in all n x n sub-matrices. Further by relating the number of zeros in B-spline basis as well as changes (sign changes) in the sequence of its B-spline coefficients. From this relation, we obtained an accurate characterization when interpolation by B-splines correlates with the changes leads uniqueness and this ensures the optimal solution. Simultaneously we computed the knot insertion matrix and B-spline collocation matrix and its sub-matrices having nonnegative determinants. The totality of the knot insertion matrix and B-spline collocation matrix is demonstrated in the concluding section by using the input image and shows that these concepts are fit to apply and reduce the errors through mean square error and PSNR values

scholarly journals Numerical Solution of Nonlinear Sine-Gordon Equation by Modified Cubic B-Spline Collocation Method

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
R. C. Mittal ◽  
Rachna Bhatia
Keyword(s):  
Numerical Solution ◽  
Collocation Method ◽  
Gordon Equation ◽  
Spline Collocation ◽  
Sine Gordon Equation ◽  
B Spline ◽  
Spline Collocation Method ◽  
B Splines ◽  
Spline Basis ◽  
The Given

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.

Generation and Modification of Non-Uniform B-Spline Surface Approximations to PDE Surfaces Using the Finite Element Method

1990 ◽  
Author(s):  
Joanna M. Brown ◽  
Malcolm I. G. Bloor ◽  
M. Susan Bloor ◽  
Michael J. Wilson
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Spline Approximation ◽  
The Finite Element Method ◽  
B Spline ◽  
B Splines ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Spline Basis ◽  
Element Method

Abstract A PDE surface is generated by solving partial differential equations subject to boundary conditions. To obtain an approximation of the PDE surface in the form of a B-spline surface the finite element method, with the basis formed from B-spline basis functions, can be used to solve the equations. The procedure is simplest when uniform B-splines are used, but it is also feasible, and in some cases desirable, to use non-uniform B-splines. It will also be shown that it is possible, if required, to modify the non-uniform B-spline approximation in a variety of ways, using the properties of B-spline surfaces.

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.

A quartic trigonometric tension b-spline algorithm for nonlinear partial differential equation system

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Özlem Ersoy Hepson
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Higher Order ◽  
Spline Collocation ◽  
Partial Differential ◽  
Content Type ◽  
B Spline ◽  
B Splines ◽  
Coupled Burgers Equation

Purpose The purpose of this study is to construct quartic trigonometric tension (QTT) B-spline collocation algorithms for the numerical solutions of the Coupled Burgers’ equation. Design/methodology/approach The finite elements method (FEM) is a numerical method for obtaining an approximate solution of partial differential equations (PDEs). The development of high-speed computers enables to development FEM to solve PDEs on both complex domain and complicated boundary conditions. It also provides higher-order approximation which consists of a vector of coefficients multiplied by a set of basis functions. FEM with the B-splines is efficient due both to giving a smaller system of algebraic equations that has lower computational complexity and providing higher-order continuous approximation depending on using the B-splines of high degree. Findings The result of the test problems indicates the reliability of the method to get solutions to the CBE. QTT B-spline collocation approach has convergence order 3 in space and order 1 in time. So that nonpolynomial splines provide smooth solutions during the run of the program. Originality/value There are few numerical methods build-up using the trigonometric tension spline for solving differential equations. The tension B-spline collocation method is used for finding the solution of Coupled Burgers’ equation.

BEZIER AND B-SPLINE CURVES WITH KNOTS IN THE COMPLEX PLANE

Fractals ◽  
2011 ◽  
Vol 19 (01) ◽  
pp. 67-86 ◽  
Author(s):  
KONSTANTINOS I. TSIANOS ◽  
RON GOLDMAN
Keyword(s):  
Complex Domain ◽  
Polynomial Algorithms ◽  
Control Points ◽  
Knot Insertion ◽  
Real Numbers ◽  
Koch Curve ◽  
Natural Manner ◽  
B Spline ◽  
B Splines ◽  
Spline Curves

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.

A new structure formulations for cubic B-spline collocation method in three and four-dimensions

2020 ◽  
Vol 9 (1) ◽  
pp. 432-448
Author(s):  
K. R. Raslan ◽  
Khalid K. Ali
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Exact Solutions ◽  
Collocation Method ◽  
Test Problems ◽  
Spline Collocation ◽  
B Spline ◽  
Spline Collocation Method ◽  
Four Dimensions ◽  
B Splines

AbstractIn this work, we introduce a new construct to the cubic B-spline collocation method in the three and four-dimensions. The cubic B-splines method format is displayed in one, two, three, and four-dimensions format. These constructions are of utmost importance in solving differential equations in their various dimensions, which have applications in many fields of science. The efficiency and accuracy of the proposed methods are demonstrated by its application to a few test problems in two, three, and four dimensions. Also, comparing the exact solutions and with the results obtained by using other numerical methods available in the literature as much as possible.

scholarly journals Numerical Solution of the coupled Burgers' equation by Trigonometric B-spline Collocation Method

2021 ◽  
Author(s):  
Yusuf UCAR ◽  
Nuri YAGMURLU ◽  
Mehmet YİĞİT
Keyword(s):  
Numerical Solutions ◽  
Burgers Equation ◽  
The Other ◽  
Test Problems ◽  
Spline Collocation ◽  
Unconditionally Stable ◽  
B Spline ◽  
B Splines ◽  
A New Technique ◽  
Coupled Burgers Equation

In the present study, the coupled Burgers’ equation is going to be solved numerically by presenting a new technique based on collocation finite element method in which trigonometric cubic and quintic B-splines are used as approximate functions. In order to support the present study, three test problems given with appropriate initial and boundary conditions are studied. The newly obtained results are compared with some of the other published numerical solutions available in the literature. The accuracy of the proposed method is discussed by computing the error norms L₂ and L_{∞}. A linear stability analysis of the approximation obtained by the scheme shows that the method is unconditionally stable.

Meshless Local B-Spline Basis Functions-FD Method and its Application for Heat Conduction Problem with Spatially Varying Heat Generation

2013 ◽  
Vol 465-466 ◽  
pp. 490-495 ◽  
Author(s):  
Mas Irfan P. Hidayat ◽  
Bambang Ari-Wahjoedi ◽  
Parman Setyamartana ◽  
Puteri S.M. Megat Yusoff ◽  
T.V.V.L.N. Rao
Keyword(s):  
Heat Conduction ◽  
Heat Generation ◽  
Complex Geometry ◽  
Heat Conduction Problem ◽  
Spline Approximation ◽  
Basis Functions ◽  
Spline Collocation ◽  
B Spline ◽  
Spline Basis ◽  
Spatially Varying

In this paper, a new meshless local B-spline basis functions-finite difference (FD) method is presented for two-dimensional heat conduction problem with spatially varying heat generation. In the method, governing equations are discretized by B-spline approximation in the spirit of FD technique using local B-spline collocation. The key aspect of the method is that any derivative at a point or node is stated as neighbouring nodal values based on the B-spline interpolants. Compared with mesh-based method such as FEM the method is simple and efficient to program. In addition, as the method poses the Kronecker delta property, the imposition of boundary conditions is also easy and straightforward. Moreover, it poses no difficulties in dealing with arbitrary complex domains. Heat conduction problem in complex geometry is presented to demonstrate the accuracy and efficiency of the present method.

scholarly journals NUMERICAL SOLUTION TO LINEAR SINGULARLY PERTURBED TWO POINT BOUNDARY VALUE PROBLEMS USING B-SPLINE COLLOCATION METHOD

2016 ◽  
Vol 4 (1) ◽  
pp. 158-164
Author(s):  
Rajashekhar Reddy
Keyword(s):  
Analytical Solution ◽  
Collocation Method ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Point Boundary ◽  
Spline Collocation ◽  
B Spline ◽  
Numerical Examples ◽  
Spline Collocation Method ◽  
Spline Basis

A Recursive form cubic B-spline basis function is used as basis in B-spline collocation method to solve second linear singularly perturbed two point boundary value problem. The performance of the method is tested by considering the numerical examples with different boundary conditions. Results of numerical examples show the robustness of the method when compared with the analytical solution.

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