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The goal of this paper is to shed new light on a wavelet-based denoising method developed by Hadjileontiadis et al. (1997, 2000) which is derived from an iterative denoising algorithm by Coifman and Wickerhauser (1995, 1998). The underlying algorithm is revisited and interpreted as a fixed-point algorithm. This allows us to derive a new version of the algorithm largely increasing computational efficiency.