Shift - Invariant Gibbs Free Denoising Algorithm based onWavelet Transform FootprintsPier

In recent years wavelet have had an important impact on signal processing theory and practice. The eeectiveness of wavelets is mainly due to their capability of representing piecewise smooth signals with few non-zero coeecients. 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 nite 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 coeecients is behind the success of wavelet based denoising algorithms. Indeed, traditional approaches simply consist in thresholding the noisy wavelet coeecients, so the few large coeecients carrying the essential information are usually kept while small coeecients mainly containing noise are cancelled. However, wavelet denoising suuers 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 coeecients are exactly zero. Moreover the wavelet coeecients 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 signiicant wavelet coeecients generated by a singularity. In our denoising algorithm instead of thresholding the noisy wavelet coeecients independently as in traditional approaches (scalar thresholding), we gather all the noisy coeecients around a discontinuity in a vector and compute the inner product with the closest footprint that represents that discontinuity. If this product is large enough we keep it, otherwise we cancel it (vector thresholding). With this approach we substantially improve the performance of traditional denoising algorithms in terms of SNR and we eliminate the pseudo Gibbs phenomenon since we force the denoised signal to belong to the class of piecewise polynomial functions. The proposed algorithm can potentially be extended to more general signals like piecewise smooth signals and represents an eeective solution to problems like signal denoising.