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In this paper, LLMMSE filtering is performed in the undecimated wavelet domain by means of an adaptive rescaling of detail coefficients. The amplitude of each coefficient is divided by the variance ratio of the noisy coefficient to the noise-free one. All the above quantities are analytically calculated from the speckled image, the speckle variance, and the wavelet filters only, without assuming any model to describe the underlying backscatter. Empirical criteria based on distributions of multiresolution coefficient of variation calculated in the undecimated wavelet domain are introduced to mitigate the rescaling of coefficients in highly heterogeneous areas where the speckle is not fully developed. Experiments carried out on both simulated speckled images and true SAR images demonstrate that the visual quality of results is excellent in terms of both background smoothing and preservation of edge sharpness, textures, and point targets. The absence of decimation in the wavelet decomposition avoids the typical impairments produced by critically-subsampled wavelet-based denoising.