A new nonlinear noise reduction method is presented that uses the discrete wavelet transform. Similar to Donoho and Johnstone, we employ thresholding in the wavelet transform domain but, following a suggestion by Coifman, we use an undecimated, shift-invariant, nonorthogonal wavelet transform instead of the usual orthogonal one. This new approach can be interpreted as a repeated application of the original Donoho and Johnstone method for different shifts. The main feature of the new algorithm is a significantly improved noise reduction compared to the original wavelet based approach, both the l₂ error and visually, for a large class of signals. This is shown both theoretically as well as by experimental results.