Eno-wavelet transforms for piecewise smooth functions

Tony Chan*, H. M. Zhou

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

27 Scopus citations

Abstract

We have designed an adaptive essentially nonoscillatory (ENO)-wavelet transform for approximating discontinuous functions without oscillations near the discontinuities. Our approach is to apply the main idea from ENO schemes for numerical shock capturing to standard wavelet transforms. The crucial point is that the wavelet coefficients are computed without differencing function values across jumps. However, we accomplish this in a different way than in the standard ENO schemes. Whereas in the standard ENO schemes the stencils are adaptively chosen, in the ENO-wavelet transforms we adaptively change the function and use the same uniform stencils. The ENOwavelet transform retains the essential properties and advantages of standard wavelet transforms such as concentrating the energy to the low frequencies, obtaining maximum accuracy, maintained up to the discontinuities, and having a multiresolution framework and fast algorithms, all without any edge artifacts. We have obtained a rigorous approximation error bound which shows that the error in the ENO-wavelet approximation depends only on the size of the derivative of the function away from the discontinuities. We will show some numerical examples to illustrate this error estimate.

Original languageEnglish (US)
Pages (from-to)1369-1404
Number of pages36
JournalSIAM Journal on Numerical Analysis
Volume40
Issue number4
DOIs
StatePublished - Sep 1 2002

Keywords

  • ENO
  • Image compression
  • Image denoising
  • Signal processing
  • Wavelet

ASJC Scopus subject areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

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