27,082 research outputs found
Tchebycheffian B-splines in isogeometric Galerkin methods
Tchebycheffian splines are smooth piecewise functions whose pieces are drawn
from (possibly different) Tchebycheff spaces, a natural generalization of
algebraic polynomial spaces. They enjoy most of the properties known in the
polynomial spline case. In particular, under suitable assumptions,
Tchebycheffian splines admit a representation in terms of basis functions,
called Tchebycheffian B-splines (TB-splines), completely analogous to
polynomial B-splines. A particularly interesting subclass consists of
Tchebycheffian splines with pieces belonging to null-spaces of
constant-coefficient linear differential operators. They grant the freedom of
combining polynomials with exponential and trigonometric functions with any
number of individual shape parameters. Moreover, they have been recently
equipped with efficient evaluation and manipulation procedures. In this paper,
we consider the use of TB-splines with pieces belonging to null-spaces of
constant-coefficient linear differential operators as an attractive substitute
for standard polynomial B-splines and rational NURBS in isogeometric Galerkin
methods. We discuss how to exploit the large flexibility of the geometrical and
analytical features of the underlying Tchebycheff spaces according to
problem-driven selection strategies. TB-splines offer a wide and robust
environment for the isogeometric paradigm beyond the limits of the rational
NURBS model.Comment: 35 pages, 18 figure
Approximation and geometric modeling with simplex B-splines associated with irregular triangles
Bivariate quadratic simplical B-splines defined by their corresponding set of knots derived from a (suboptimal) constrained Delaunay triangulation of the domain are employed to obtain a C1-smooth surface. The generation of triangle vertices is adjusted to the areal distribution of the data in the domain. We emphasize here that the vertices of the triangles initially define the knots of the B-splines and do generally not coincide with the abscissae of the data. Thus, this approach is well suited to process scattered data.\ud
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With each vertex of a given triangle we associate two additional points which give rise to six configurations of five knots defining six linearly independent bivariate quadratic B-splines supported on the convex hull of the corresponding five knots.\ud
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If we consider the vertices of the triangulation as threefold knots, the bivariate quadratic B-splines turn into the well known bivariate quadratic Bernstein-BĂ©zier-form polynomials on triangles. Thus we might be led to think of B-splines as of smoothed versions of Bernstein-BĂ©zier polynomials with respect to the entire domain. From the degenerate Bernstein-BĂ©zier situation we deduce rules how to locate the additional points associated with each vertex to establish knot configurations that allow the modeling of discontinuities of the function itself or any of its directional derivatives. We find that four collinear knots out of the set of five defining an individual quadratic B-spline generate a discontinuity in the surface along the line they constitute, and that analogously three collinear knots generate a discontinuity in a first derivative.\ud
Finally, the coefficients of the linear combinations of normalized simplicial B-splines are visualized as geometric control points satisfying the convex hull property.\ud
Thus, bivariate quadratic B-splines associated with irregular triangles provide a great flexibility to approximate and model fast changing or even functions with any given discontinuities from scattered data.\ud
An example for least squares approximation with simplex splines is presented
Fast space-variant elliptical filtering using box splines
The efficient realization of linear space-variant (non-convolution) filters
is a challenging computational problem in image processing. In this paper, we
demonstrate that it is possible to filter an image with a Gaussian-like
elliptic window of varying size, elongation and orientation using a fixed
number of computations per pixel. The associated algorithm, which is based on a
family of smooth compactly supported piecewise polynomials, the
radially-uniform box splines, is realized using pre-integration and local
finite-differences. The radially-uniform box splines are constructed through
the repeated convolution of a fixed number of box distributions, which have
been suitably scaled and distributed radially in an uniform fashion. The
attractive features of these box splines are their asymptotic behavior, their
simple covariance structure, and their quasi-separability. They converge to
Gaussians with the increase of their order, and are used to approximate
anisotropic Gaussians of varying covariance simply by controlling the scales of
the constituent box distributions. Based on the second feature, we develop a
technique for continuously controlling the size, elongation and orientation of
these Gaussian-like functions. Finally, the quasi-separable structure, along
with a certain scaling property of box distributions, is used to efficiently
realize the associated space-variant elliptical filtering, which requires O(1)
computations per pixel irrespective of the shape and size of the filter.Comment: 12 figures; IEEE Transactions on Image Processing, vol. 19, 201
Sum-factorization techniques in Isogeometric Analysis
The fast assembling of stiffness and mass matrices is a key issue in
isogeometric analysis, particularly if the spline degree is increased. We
present two algorithms based on the idea of sum factorization, one for matrix
assembling and one for matrix-free methods, and study the behavior of their
computational complexity in terms of the spline order . Opposed to the
standard approach, these algorithms do not apply the idea element-wise, but
globally or on macro-elements. If this approach is applied to Gauss quadrature,
the computational complexity grows as instead of as
previously achieved.Comment: 34 pages, 8 figure
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