Shift-invariant Gibbs-free denoising algorithm based on wavelet transform footprints
Pier Luigi Dragotti, Martin Vetterli · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 2000
In recent years wavelet have had an important impact on signal processing theory and practice. The effectiveness of wavelets is mainly due to their capability of representing piecewise smooth signals with few non-zero coefficients. Away from discontinuities, the inner product between a wavelet (with a number of zero moments) and a smooth function will be either zero or very small. 8 At singular points, a finite number of wavelets concentrated around the discontinuity lead to non-zero inner products. This ability of wavelet transform to pack the main signal information in few large coefficients is behind the success of wavelet based denoising algorithms. Indeed, traditional approaches simply consist in thresholding the noisy wavelet coefficients, so the few large coefficients carrying the essential information are usually kept while small coefficients mainly containing noise are cancelled. However, wavelet denoising suffers of two main drawbacks: it is not shift-invariant and it exhibits pseudo Gibbs phenomenon around discontinuities. In this work, we present a new denoising algorithm which does not present the pseudo Gibbs phenomenon and which is almost shift-invariant even if we do not use a frame expansion. In our analysis we focus on piecewise polynomial functions. For this class of signals we know that, if wavelets have enough vanishing moments, away from discontinuities the wavelet coefficients are exactly zero. Moreover the wavelet coefficients generated by a discontinuity are highly dependent across scales. Therefore, a good denoising algorithm should take advantage of this dependency. We thus introduce the notion of footprints, which are the traces left by time domain singularities in the wavelet domain. So a footprint is a vector containing all the significan...