Linear Multiscale Transforms Based on Even-Reversible Subdivision Operators

Nira Dyn, Xiaosheng Zhuang

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

Abstract

Multiscale transforms for real-valued data, based on interpolatory subdivision operators, have been studied in recent years. They are easy to define and can be extended to other types of data, for example, to manifold-valued data. In this chapter, we define linear multiscale transforms, based on certain linear, non-interpolatory subdivision operators, termed “even-reversible.” For such operators, we prove, using Wiener’s lemma, the existence of an inverse to the linear operator defined by the even part of the subdivision mask and term it “even-inverse.” We show that the non-interpolatory subdivision operators, with spline or pseudo-spline masks, are even-reversible and derive explicitly, for the quadratic and cubic spline subdivision operators, the symbols of the corresponding even-inverse operators. We also analyze properties of the multiscale transforms based on even-reversible subdivision operators, in particular, their stability and the rate of decay of the details.

Original languageEnglish
Title of host publicationApplied and Numerical Harmonic Analysis
PublisherBirkhauser
Pages297-319
Number of pages23
DOIs
StatePublished - 2021

Publication series

NameApplied and Numerical Harmonic Analysis
ISSN (Print)2296-5009
ISSN (Electronic)2296-5017

Keywords

  • Bi-infinite Toeplitz matrix
  • Even-reversible subdivision
  • Interpolatory subdivision
  • Multiscale transform
  • Primal and dual pseudo-spline subdivision
  • Pyramid data
  • Spline subdivision
  • Stability
  • Wiener’s lemma

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