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A general matrix representation for non-uniform B-spline subdivision with boundary control

Giulio Casciola, Lucia Romani
2007
general formulation of the subdivision matrix generating an endpoint-interpolating B-spline of arbitrary degree.  ...  This consideration prompted us to present an efficient algorithm for the conversion between B-spline bases defined over different knot-partitions, which turns out to be extremely useful for computing a  ...  processes are special cases of the more general problem of transforming a spline between B-spline representations on two arbitrary knot vectors.  ... 
doi:10.6092/unibo/amsacta/2532 fatcat:vzcfp7matrfgljmq5z7vo6rrpa

The conversion matrix between uniform B-spline and Bézier representations

L Romani, M.A Sabin
2004 Computer Aided Geometric Design  
In this work we propose a very simple and efficient procedure for computing in a recursive way the conversion matrices which provide the direct transformation between uniform B-spline and Bézier representations  ...  One of the reasons is that the matrix notation allows an easy conversion between different shape representations and provides a convenient implementation in either hardware or software with available matrix  ...  We like to thank Giulio Casciola for his helpful suggestions. Special thanks also go to the referees.  ... 
doi:10.1016/j.cagd.2004.04.002 fatcat:i5qvcwzqk5a7datpu5b7vne5da

Computation of multi-degree B-splines [article]

Hendrik Speleers
2019 arXiv   pre-print
This enables the use of existing efficient algorithms for B-spline evaluations and refinements in the context of multi-degree splines.  ...  We describe a simple algorithmic construction of a set of basis functions for the space of multi-degree splines, with similar properties to standard B-splines.  ...  The framework for smoothly joining different (B-spline) spaces is quite abstract and can be applied in a more general setting.  ... 
arXiv:1809.01598v2 fatcat:s5rplv34k5eezc637hmq5cpnoi

MONOTONE APPROXIMATION OF AGGREGATION OPERATORS USING LEAST SQUARES SPLINES

G. BELIAKOV
2002 International Journal of Uncertainty Fuzziness and Knowledge-Based Systems  
Linear restrictions on spline coefficients provide necessary and sufficient conditions for spline monotonicity.  ...  The basis for splines is selected in such a way that these restrictions take an especially simple form.  ...  Therefore, the matrix of observation equations (consisting of I rows of the form ( 6 )) can be obtained from the matrix of observation equations in B-spline representation by multiplying it by ) ( 2 1  ... 
doi:10.1142/s0218488502001715 fatcat:wo7fwfir5vb7zfi22muxkypmi4

Generalization of one algorithm for constructing recurrent splines

Galina Tuluchenko, Gennadii Virchenko, Galyna Getun, Viacheslav Martynov, Mykola Tymofieiev
2018 Eastern-European Journal of Enterprise Technologies  
in the approximating properties of the spline in general.  ...  Dicoussar for preliminary determining the length of a link of the spline, made it possible to obtain a smoothing spline, based on the generalized D. A.  ...  second knotonly a value for the function: The basis functions of polynomial (14) will then be assigned by formulae: A matrix for the least squares method takes a traditional form: or according to the  ... 
doi:10.15587/1729-4061.2018.128312 fatcat:xztazuczmfcctkv3ygqenzropi

Low rank tensor methods in Galerkin-based isogeometric analysis

Angelos Mantzaflaris, Bert Jüttler, Boris N. Khoromskij, Ulrich Langer
2017 Computer Methods in Applied Mechanics and Engineering  
This map has a global tensor structure, inherited from the parametric spline geometry representation.  ...  First, we formulate tensor-product B-spline bases as well as the corresponding mass and stiffness matrices as tensors in order to reveal their intrinsic structure.  ...  In the first one, we use the mass matrix for computing the L 2 projection of f (x, y, z) to the (k−refined) B-spline space obtained from the geometry representation.  ... 
doi:10.1016/j.cma.2016.11.013 fatcat:34iodd2w75f4lesjwfwro5oina

Efficient Derivative Computation for Cumulative B-Splines on Lie Groups [article]

Christiane Sommer, Vladyslav Usenko, David Schubert, Nikolaus Demmel, Daniel Cremers
2020 arXiv   pre-print
Lie group cumulative B-splines are a popular way of representing continuous trajectories without singularities.  ...  In this work we present an alternative derivation of time derivatives based on recurrence relations that needs O(k) instead of O(k^2) matrix operations (for a spline of order k) and results in simple and  ...  Qin [17] found that due to their locality property, B-splines are conveniently expressed using a matrix representation.  ... 
arXiv:1911.08860v2 fatcat:ntn3yinsuvdevgi6cemnnx4sju

The singular cases for γ-spline interpolation

G. Farin, D. Hansford, A. Worsey
1990 Computer Aided Geometric Design  
We derive a natural extension of Boehm's free-form y-spline, the G* interpolating y-spline. Primarily, the conditions under which singularities in the spline formulation occur are investigated.  ...  Comparisons are made to the behavior of the interpolating v-spline.  ...  Consider a generic circulant matrix C of order n + 1. That is, a square matrix of the form: for some polynomial p(z).  ... 
doi:10.1016/0167-8396(90)90014-i fatcat:iovvku3ggnelxjcre2yznhen5i

A variation-diminishing generalized spline approximation method

Samuel Karlin, John M Karon
1968 Journal of Approximation Theory  
Our results are based on a representation formula for such splines as linear combinations of certain generalized basic spline functions which were introduced by Karlin in [3] in the study of self-adjoint  ...  SECTION BACKGROUND FOR SPLINES WITH SIMPLE KNOTS Generalized splines on (a,b) associated with the differential expression A,, are defined in Definition 1.1.  ... 
doi:10.1016/0021-9045(68)90003-8 fatcat:6itdphqmibdo5fltlwj4reb5ia

THB-splines: An effective mathematical technology for adaptive refinement in geometric design and isogeometric analysis

Carlotta Giannelli, Bert Jüttler, Stefan K. Kleiss, Angelos Mantzaflaris, Bernd Simeon, Jaka Špeh
2016 Computer Methods in Applied Mechanics and Engineering  
By exploiting a multilevel control structure, we show that truncated hierarchical B-spline (THB-spline) representations support interactive modeling tools, while simultaneously providing effective approximation  ...  The generalization of THB-splines to the more general context of generating systems and also to geometries with arbitrary topologies was recently addressed [46, 48, 49] .  ...  Anh-Vu Vuong for collaborating with us at an earlier stage of this research.  ... 
doi:10.1016/j.cma.2015.11.002 fatcat:saqayplxhfhxnjwnw74peza2bu

A Critical Evaluation of the Resolution Properties of B-Spline and Compact Finite Difference Methods

Wai Yip Kwok, Robert D. Moser, Javier Jiménez
2001 Journal of Computational Physics  
However, B-spline schemes have a more straightforward and robust formulation, particularly near boundaries on nonuniform meshes. c 2001 Elsevier Science  ...  Resolution properties of B-spline and compact finite difference schemes are compared using Fourier analysis in periodic domains, and tests based on solution of the wave and heat equations in finite domains  ...  FIG. 6 . 6 L 2 error in the representation of the complex exponential with wavenumber k for matrix bandwidth (a) 3, (b) 7, (c) 11: --, B-spline collocation; ----, B-spline Galerkin; ---, C 0 finite element  ... 
doi:10.1006/jcph.2001.6919 fatcat:gvhjtojbznf6pptnvmlgwalv6y

Adaptive CAD model (re-)construction with THB-splines

Gábor Kiss, Carlotta Giannelli, Urška Zore, Bert Jüttler, David Großmann, Johannes Barner
2014 Graphical Models  
B-splines.  ...  Furthermore, the local THB-spline evaluation in terms of B-spline patches can be properly combined with commercial geometric modeling kernels in order to convert the multilevel spline representation into  ...  For example, the B-spline representations of THB-splines is precomputed in order to speed up their evaluation.  ... 
doi:10.1016/j.gmod.2014.03.017 fatcat:p6jfrvborvhiradgwffwpmp6tq

Spline Dynamic Matrix: a Novel Representation of Dynamic Models

Matheus S. Escobar, Jorge O. Trierweiler
2012 IFAC Proceedings Volumes  
To cope with these restrictions, this work introduces a new model representation called Spline Dynamic Matrix (SDM), which combines the dynamic matrix model with the inputs parameterized by piecewise cubic  ...  Such representation results in reduced matrixes, allowing variable spacing over simulation horizon.  ...  A c and B c are the state matrixes.  ... 
doi:10.3182/20120710-4-sg-2026.00157 fatcat:45txuv2vgngj7kb3ac5hu6zaei

Page 439 of Mathematical Reviews Vol. , Issue 96a [page]

1996 Mathematical Reviews  
Since there exist contour integral representations for several types of non-polynomial B- splines the author remarks that similar relations may hold in more general situations. H. P.  ...  This recursion formula is applied to derive a generalized two-scale rela- tion for B-splines with multiple knots and to analyze the structure of the autocorrelation symbol ¢/,, which is defined by y/,(  ... 

Compression ofnth-order data arrays by B-splines. Part 1: Theory

Bjørn K. Alsberg, Olav M. Kvalheim
1993 Journal of Chemometrics  
Illustration of the principle behind maximum entropy (and B-splines in general) for compression of a matrix of spectra.  ...  Basis function for B-splines of degree zero The same interval vector h’ as in equation (10) is used to represent knots in B-spline bases and equation (20) is used to generate the B-matrices for k = 0,  ... 
doi:10.1002/cem.1180070105 fatcat:c65q4azckfbrzf3o74o6qoux6y
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