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Image Descriptors and Features
Published in Manas Kamal Bhuyan, Computer Vision and Image Processing, 2019
B-spline B-spline are piecewise polynomial functions that can provide local approximations of contours of shapes using a small number of parameters [116]. The B-spline representation results in compression of boundary data, and this representation has been used in shape analysis and synthesis in computer vision applications. Figure 3.19 (left) is a B-spline curve of degree 3 defined by 8 control points. The little dots subdivide the B-spline curve into a number of curve segments. The subdivision of the curve can also be modified. Therefore, B-spline curves have a higher degree of freedom for curve design. As illustrated in Figure 3.19 (right), to design a B-spline curve, we need a set of control points, a set of knots and a set of coefficients, one for each control point. So, all the curve segments are joined together satisfying certain continuity conditions. The degree of a B-spline polynomial can be adjusted to preserve smoothness of the curve to be approximated. Most importantly, B-splines allow local control over the shape of a spline curve.
Computer-Aided Design
Published in Yoseph Bar-Cohen, Advances in Manufacturing and Processing of Materials and Structures, 2018
Nicholaos Bilalis, Emmanuel Maravelakis
The knot vector characterizes the type of B-Spline, and it can be either uniform or nonuniform. In the uniform knot vector, the values between the discrete values of ui are equally distributed, while in the nonuniform know vector, they take random values computed according to the distribution of the control points. The type of the knot vector specifies the shape of the basis functions (Figure 2.6).
Fracture analysis of interfacial crack in piezoelectric bimaterial by XIGA approach using Bézier extraction of NURBS
Published in Mechanics of Advanced Materials and Structures, 2022
NURBS basis function functions along with its properties are briefly described. A 1–D parametric space [0, 1] is considered and knot vectors are a set of non-decreasing real numbers called knots taken for constructing B-spline basis functions. Therefore, a knot vector Ξ is described as with and by using Cox–de Boor recursive formula and its corresponding univariate i B–spline basis function of order p is given by [42]. For p = 0 and, for where p and n represent polynomial order of basic functions and number of control points. Here, ξi and represent i knot and NURBS basis function of order p, respectively.
Extended iso-geometric analysis for modeling three-dimensional cracks
Published in Mechanics of Advanced Materials and Structures, 2019
The basic features of IGA are the NURBS, which are used for approximating the geometry and the displacement fields in the domain. The discretization errors that arise due to geometry approximation get eliminated in IGA. Knot vector (ξ) can be written as ξ = {ξ1, ξ2, ξ3, ……ξn + p + 1}, where ξi represents the ith knot, i denotes the knot index, p denotes the order of the interpolating polynomial and n denotes the number of basis functions required to formulate the B-spline curve. The open knot vector, where the end knots are repeated p + 1 times, has been used for analysis in the present study. The recursive algorithm is used to construct the B-spline basis functions, starting with p = 0, as shown below:
Quadratic B-spline finite element method for a rotating nonuniform Euler–Bernoulli beam
Published in International Journal for Computational Methods in Engineering Science and Mechanics, 2018
Vijay Panchore, Ranjan Ganguli
B-spline curves and surfaces are widely used for design purposes. B-spline curves show the global control over the local control shown by the Bezier curve; the degree of the Bezier curve is defined by the number of control points while the degree of the B-spline curve is independent of the number of control points [1]. The quadratic B-spline interpolation requires four points while the Hermite cubic interpolation requires two points. The continuity between the B-spline curves is achieved with the overlapping of the individual B-spline curves. Since the B-spline approximation meets the required continuity, it can be used for the finite element formulation of a rotating Euler-Bernoulli beam. The weak formulation can be obtained for an interval with the curves contributing in it; three quadratic B-spline curves contribute to a single interval. Galerkin’s method formulates the weak form of the problem and yields a system of symmetric matrices. The natural boundary conditions are satisfied while integrating the weak form and the essential boundary conditions are applied using a transformation matrix while solving the algebraic equations.