Computing

, Volume 90, Issue 1–2, pp 1–14

The mask of (2b + 4)-point n-ary subdivision scheme

Article
  • 129 Downloads

Abstract

In this paper, we present general formulae for the mask of (2b + 4)-point n-ary approximating as well as interpolating subdivision schemes for any integers \({b\,\geqslant\,0}\) and \({n\,\geqslant\,2}\). These formulae corresponding to the mask not only generalize and unify several well-known schemes but also provide the mask of higher arity schemes. Moreover, the 4-point and 6-point a-ary schemes introduced by Lian [Appl Appl Math Int J 3(1):18–29, 2008] are special cases of our general formulae.

Keywords

Approximating subdivision scheme Interpolating scheme n-Ary schemes a-Ary scheme Mask of scheme 

Mathematics Subject Classification (2000)

65D17 65D07 65D05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aspert N (2003) Non-linear subdivision of univariate signals and discrete surfaces, EPFL Thesis, École Polytechinique Fédérale de Lausanne, Lausanne, SwitzerlandGoogle Scholar
  2. 2.
    Beccari C, Casciola G, Romani L (2007) An interpolating 4-point C 2 ternary non-stationary subdivision scheme with tension control. Comput Aided Geom Design 24: 210–219MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Beccari C, Casciola G, Romani L (2007) A non-stationary uniform tension controlled interpolating 4-point scheme reproducing conics. Comput Aided Geom Design 24: 1–9MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chaikin GM (1974) An Algorithm for high speed curve generation. Comput Graph Image Process 3: 346–349CrossRefGoogle Scholar
  5. 5.
    Choi SW, Lee BG, Lee YJ, Yoon J (2006) Stationary subdivision schemes reproducing polynomials. Comput Aided Geom Design 23(4): 351–360MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Deslauriers G, Dubuc S (1989) Symmetric iterative interpolation processes. Construct Approx 5: 49–68MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Dyn N, Floater MS, Horman K (2005) A C 2 four-point subdivision scheme with fourth order accuracy and its extension. In: Daehlen M, Morken K, Schumaker LL (eds) Mathematical methods for curves and surfaces: Tromso 2004, pp 145–156Google Scholar
  8. 8.
    Dyn N, Levin D, Gregory JA (1987) A 4-point interpolatory subdivision scheme for curve design. Comput Aided Geom Design 4: 257–268MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Hassan MF, Ivrissimitzis IP, Dodgson NA, Sabin MA (2002) An interpolating 4-point C 2 ternary stationary subdivision scheme. Comput Aided Geom Design 19: 1–18MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Jena MK, Shunmugaraj P, Das PC (2003) A non-stationary subdivision scheme for curve interpolation. ANZIAM J 44: 216–235Google Scholar
  11. 11.
    Lian J-a (2008) On a-ary subdivision for curve design: I. 4-point and 6-point interpolatory schemes. Appl Appl Math Int J 3(1): 18–29MATHMathSciNetGoogle Scholar
  12. 12.
    Lian J-a (2008) On a-ary subdivision for curve design: II. 3-point and 5-point interpolatory schemes. Appl Appl Math Int J 3(2): 176–187MATHMathSciNetGoogle Scholar
  13. 13.
    Ko KP (2007) A study on subdivision scheme, Dongseo University Busan South Korea. http://kowon.dongseo.ac.kr/~kpko/publication/2004book.pdf
  14. 14.
    Ko KP, Lee BG, Yoon GJ (2007) A ternary 4-point approximating subdivision scheme. Appl Math Comput 190: 1563–1573MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Ko KP, Lee BG, Tang Y, Yoon GJ (2007) General formula for the mask of (2n + 4)-point symmetric subdivision scheme. http://kowon.dongseo.ac.kr/~kpko/publication/mask_elsevier_second_version-kpko.pdf
  16. 16.
    Mustafa G, Khan F (2009) A new 4-point C 3 quaternary approximating subdivision scheme. Abstr Appl Anal. doi:10.1155/2009/301967
  17. 17.
    Romani L (2009) From approximating subdivision schemes for exponential splines to high-performance interpolating algorithms. J Comput Appl Math 224(1): 383–396MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Weissman A (1990) A 6-point interpolatory subdivision scheme for curve design. M.Sc. Thesis, Tel Aviv UniversityGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsThe Islamia University of BahawalpurBahawalpurPakistan

Personalised recommendations