An unsupervised method is first proposed to assess the variance and the spatial correlation coefficients of speckle noise in SAR images. They are obtained as regression coefficients, the former of local standard deviation to local mean, the latter of local unity-lag covariance to local variance, both calculated on homogeneous areas. For this purpose, an automatic procedure has been developed, based on that homogeneous areas produce clusters of scatterpoints that are aligned using the regression line. On true SAR images, the method is capable to carefully reject textured regions, in which speckle may be not fully developed and the variance of the signal is not negligible. On simulated speckled images, an impressive accuracy is obtained. Once the noise parameters are known, adaptive filtering is applied in a multiresolution fashion, to take advantage of increasing SNR of the noisy image at increasing scales, as well as to cope with the spatial correlation of the noise that is halved together with the resolution. Laplacian pyramids are generalized to the noise model by defining ratios of combinations of lowpass image versions, in which the dependence of the noise on the signal is largely removed, together with the nonstationarity of the mean. Experiments on both real and synthetic images demonstrate a high accuracy of results, both for noise estimation and for filtering.