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ENO-wavelet transforms for piecewise smooth functions

Chan, Tony F. and Zhou, H. M. (2002) ENO-wavelet transforms for piecewise smooth functions. SIAM Journal on Numerical Analysis, 40 (4). pp. 1369-1404. ISSN 0036-1429. http://resolver.caltech.edu/CaltechAUTHORS:CHAsiamjna02

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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 ENO-wavelet 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.


Item Type:Article
Additional Information:© 2002 Society for Industrial and Applied Mathematics. Received by the editors April 18, 2000; accepted for publication (in revised form) February 26, 2002; published electronically October 23, 2002. This research was supported in part by grants ONR-N00017-96-1-0277 and NSF DMS-96-26755.
Subject Keywords:ENO, wavelet, image compression, image denoising, signal processing
Issue or Number:4
Record Number:CaltechAUTHORS:CHAsiamjna02
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:CHAsiamjna02
Alternative URL:http://dx.doi.org/10.1137/S0036142900370915
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:564
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:08 Aug 2005
Last Modified:26 Dec 2012 08:40

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