Data-driven and optimal denoising of a signal and recovery of its derivative using multiwavelets

Multiwavelets are relative newcomers into the world of wavelets. Thus, it has not been a surprise that the used methods of denoising are modified universal thresholding procedures developed for uniwavelets. On the other hand, the specific of a multiwavelet discrete transform is that typical errors are not identically distributed and correlated, whereas the theory of the universal thresholding is based on the assumption of identically distributed and independent normal errors. Thus, we suggest an alternative denoising procedure based on the Efromovich-Pinsker algorithm. We show that this procedure is optimal over a wide class of noise distributions. Moreover, together with a new cristina class of biorthogonal multiwavelets, which is introduced in this paper, the procedure implies an optimal method for recovering the derivative of a noisy signal. A Monte Carlo study supports these conclusions.