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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aspert N (2003) Non-linear subdivision of univariate signals and discrete surfaces, EPFL Thesis, École Polytechinique Fédérale de Lausanne, Lausanne, Switzerland
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: 210–219
Beccari C, Casciola G, Romani L (2007) A non-stationary uniform tension controlled interpolating 4-point scheme reproducing conics. Comput Aided Geom Design 24: 1–9
Chaikin GM (1974) An Algorithm for high speed curve generation. Comput Graph Image Process 3: 346–349
Choi SW, Lee BG, Lee YJ, Yoon J (2006) Stationary subdivision schemes reproducing polynomials. Comput Aided Geom Design 23(4): 351–360
Deslauriers G, Dubuc S (1989) Symmetric iterative interpolation processes. Construct Approx 5: 49–68
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 145–156
Dyn N, Levin D, Gregory JA (1987) A 4-point interpolatory subdivision scheme for curve design. Comput Aided Geom Design 4: 257–268
Hassan MF, Ivrissimitzis IP, Dodgson NA, Sabin MA (2002) An interpolating 4-point C 2 ternary stationary subdivision scheme. Comput Aided Geom Design 19: 1–18
Jena MK, Shunmugaraj P, Das PC (2003) A non-stationary subdivision scheme for curve interpolation. ANZIAM J 44: 216–235
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): 18–29
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): 176–187
Ko KP (2007) A study on subdivision scheme, Dongseo University Busan South Korea. http://kowon.dongseo.ac.kr/~kpko/publication/2004book.pdf
Ko KP, Lee BG, Yoon GJ (2007) A ternary 4-point approximating subdivision scheme. Appl Math Comput 190: 1563–1573
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
Mustafa G, Khan F (2009) A new 4-point C 3 quaternary approximating subdivision scheme. Abstr Appl Anal. doi:10.1155/2009/301967
Romani L (2009) From approximating subdivision schemes for exponential splines to high-performance interpolating algorithms. J Comput Appl Math 224(1): 383–396
Weissman A (1990) A 6-point interpolatory subdivision scheme for curve design. M.Sc. Thesis, Tel Aviv University
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by C.C. Douglas.
This work is supported by the Indigenous Ph.D. Scholarship Scheme of Higher Education Commission (HEC) Pakistan.
Rights and permissions
About this article
Cite this article
Mustafa, G., Rehman, N.A. The mask of (2b + 4)-point n-ary subdivision scheme. Computing 90, 1–14 (2010). https://doi.org/10.1007/s00607-010-0108-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00607-010-0108-x