Bivariate interpolation based on univariate subdivision schemes

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Abstract

The paper presents a bivariate subdivision scheme interpolating data consisting of univariate functions along equidistant parallel lines by repeated refinements. This method can be applied to the construction of a surface passing through a given set of parametric curves. Following the methodology of polysplines and tension surfaces, we define a local interpolator of four consecutive univariate functions, from which we sample a univariate function at the mid-point. This refinement step is the basis to an extension of the 4-point subdivision scheme to our setting. The bivariate subdivision scheme can be reduced to a countable number of univariate, interpolatory, non-stationary subdivision schemes. Properties of the generated interpolant are derived, such as continuity, smoothness and approximation order.

Original languageEnglish
Pages (from-to)709-730
Number of pages22
JournalJournal of Approximation Theory
Volume164
Issue number5
DOIs
StatePublished - May 2012

Keywords

  • Approximation order
  • Bivariate interpolation
  • Non-stationary subdivision scheme

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