research-article

Multi-degree B-splines: : Algorithmic computation and properties

Authors:

Deepesh Toshniwal,

Hendrik Speleers,

René R. Hiemstra,

Thomas J.R. HughesAuthors Info & Claims

Volume 76, Issue C

https://doi.org/10.1016/j.cagd.2019.101792

Published: 01 January 2020 Publication History

Highlights

•

We consider spaces of univariate multi-degree polynomial splines.

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We derive new properties of a B-spline basis for the multi-degree spline spaces.

•

We examine an algorithmic approach for computing the multi-degree B-spline basis.

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We prove that the algorithm reproduces the multi-degree B-spline basis.

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The algorithm enables efficient access to multi-degree splines in practice.

Abstract

This paper addresses theoretical considerations behind the algorithmic computation of polynomial multi-degree spline basis functions as presented in Toshniwal et al. (2017). The approach in Toshniwal et al. (2017) breaks from the reliance on computation of integrals recursively for building B-spline-like basis functions that span a given multi-degree spline space. The gains in efficiency are indisputable; however, the theoretical robustness needs to be examined. In this paper, we show that the construction of Toshniwal et al. (2017) yields linearly independent functions with the minimal support property that span the entire multi-degree spline space and form a convex partition of unity.

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Digital Library

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Cited By

Chudy FWoźny P(2023)Linear-Time Algorithm for Computing the Bernstein–Bézier Coefficients of B-spline Basis FunctionsComputer-Aided Design10.1016/j.cad.2022.103434154:COnline publication date: 1-Jan-2023
https://dl.acm.org/doi/10.1016/j.cad.2022.103434
Xiao ZShen W(2023)An Efficient Algorithm for Degree Reduction of MD-SplinesAdvances in Computer Graphics10.1007/978-3-031-50078-7_1(3-14)Online publication date: 28-Aug-2023
https://dl.acm.org/doi/10.1007/978-3-031-50078-7_1
Speleers H(2022)Algorithm 1020: Computation of Multi-Degree Tchebycheffian B-SplinesACM Transactions on Mathematical Software10.1145/347868648:1(1-31)Online publication date: 16-Feb-2022
https://dl.acm.org/doi/10.1145/3478686
Show More Cited By

Index Terms

Multi-degree B-splines: Algorithmic computation and properties
1. Mathematics of computing
  1. Mathematical analysis
    1. Functional analysis
    2. Numerical analysis
      1. Computations on polynomials
      2. Interpolation
  2. Probability and statistics
    1. Probabilistic representations
      1. Nonparametric representations
        Spline models

Index terms have been assigned to the content through auto-classification.

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Information

Published In

cover image Computer Aided Geometric Design

Computer Aided Geometric Design Volume 76, Issue C

Jan 2020

78 pages

ISSN:0167-8396

Issue’s Table of Contents

Elsevier B.V.

Publisher

Elsevier Science Publishers B. V.

Netherlands

Publication History

Published: 01 January 2020

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Citations

Cited By

Chudy FWoźny P(2023)Linear-Time Algorithm for Computing the Bernstein–Bézier Coefficients of B-spline Basis FunctionsComputer-Aided Design10.1016/j.cad.2022.103434154:COnline publication date: 1-Jan-2023
https://dl.acm.org/doi/10.1016/j.cad.2022.103434
Xiao ZShen W(2023)An Efficient Algorithm for Degree Reduction of MD-SplinesAdvances in Computer Graphics10.1007/978-3-031-50078-7_1(3-14)Online publication date: 28-Aug-2023
https://dl.acm.org/doi/10.1007/978-3-031-50078-7_1
Speleers H(2022)Algorithm 1020: Computation of Multi-Degree Tchebycheffian B-SplinesACM Transactions on Mathematical Software10.1145/347868648:1(1-31)Online publication date: 16-Feb-2022
https://dl.acm.org/doi/10.1145/3478686
Beccari CCasciola GRomani L(2022)A practical method for computing with piecewise Chebyshevian splinesJournal of Computational and Applied Mathematics10.1016/j.cam.2021.114051406:COnline publication date: 1-May-2022
https://dl.acm.org/doi/10.1016/j.cam.2021.114051
Beccari CCasciola G(2022)Stable numerical evaluation of multi-degree B-splinesJournal of Computational and Applied Mathematics10.1016/j.cam.2021.113743400:COnline publication date: 15-Jan-2022
https://dl.acm.org/doi/10.1016/j.cam.2021.113743
Gabrielides NSapidis N(2022)Numerical Shape Interrogation of Planar Generalized Cubic CurvesComputer-Aided Design10.1016/j.cad.2022.103234146:COnline publication date: 1-May-2022
https://dl.acm.org/doi/10.1016/j.cad.2022.103234
Sande EManni CSpeleers H(2022)Ritz-type projectors with boundary interpolation properties and explicit spline error estimatesNumerische Mathematik10.1007/s00211-022-01286-z151:2(475-494)Online publication date: 1-Jun-2022
https://dl.acm.org/doi/10.1007/s00211-022-01286-z

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