We propose a model to reconstruct wavelet coefficients using a total variation minimization algorithm. The approach is motivated by wavelet signal denoising methods, where thresholding small wavelet coefficients leads to pseudo-Gibbs artifacts. By replacing these thresholded coefficients by values minimizing the total variation, our method performs a nearly artifact free signal denoising. In this paper, we detail the algorithm based on a subgradient descent combining a projection on a linear space. The convergence of the algorithm is established and numerical experiments are reported.