A well-suited approach to calculate the fractal dimension of digital images stems from the power spectrum of a fractional Brownian motion: the ratio between powers at different scales is related to the persistence parameter H and, thus, to the fractal dimension D=3-H. The signal-dependent nature of the speckle noise, however, prevents a correct estimation of fractal dimension from synthetic aperture radar (SAR) images. Here, we propose and assess a novel method to obtain D based on the multi-scale decomposition provided by the normalized Laplacian pyramid (LP), which is a bandpass representation obtained by dividing the layers of a LP by its expanded base band and is designed to force the noise to become signal independent. Extensive experiments on synthetic fractal textures, both noise free and noisy, corroborate the underlying assumptions and show the performances, in terms of both accuracy and confidence of estimation, of pyramid methods compared with the well-established method based on the wavelet transform. Preliminary results on true SAR images from ERS-1 look promising as well.