Artifact free signal denoising with wavelets

Recent years have seen the development of signal denoising algorithms based on the wavelet transform. It has been shown that thresholding the wavelet coefficients of a noisy signal allows us to restore the smoothness of the original signal. However, wavelet denoising suffers a main drawback: around discontinuities the reconstructed signal is smoothed, exhibiting the pseudo-Gibbs phenomenon. We consider the problem of denoising piecewise smooth signals with sharp discontinuities. We propose to apply a traditional wavelet denoising method and to restore the denoised signal using a total variation minimization approach. This second step allows us to remove the Gibbs phenomena and therefore to restore sharp discontinuities, while the other structures are preserved. The main innovation of our algorithm is to constrain the total variation minimization by the knowledge of the remaining wavelet coefficients. In this way, we make sure that the restoration process does not deteriorate the information that has been considered as significant in the denoising step. With this approach we substantially improve the performance of classical wavelet denoising algorithms, both in terms of SNR and in terms of visual artifacts.