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We introduce a new approach for the implementation of minimum mean-square error (MMSE) denoising for signals with decoupled derivatives. Our method casts the problem as a penalized least-squares regression in the redundant wavelet domain. It exploits the link between the discrete gradient and Haar-wavelet shrinkage with cycle spinning. The redundancy of the representation implies that some wavelet-domain estimates are inconsistent with the underlying signal model. However, by imposing additional constraints, our method finds wavelet-domain solutions that are mutually consistent. We confirm the MMSE performance of our method through statistical estimation of Lévy processes that have sparse derivatives.