N. Decoster et al., A wavelet-based method for multifractal image analysis. II. Applications to synthetic multifractal rough surfaces, EUR PHY J B, 15(4), 2000, pp. 739-764
We apply the 2D wavelet transform modulus maxima (WTMM) method to synthetic
random multifractal rough surfaces. We mainly focus on two specific models
that are, a priori, reasonnable candidates to simulate cloud structure in
paper III (S.G. Roux, A. Ameodo, N. Decoster, fur. Phys. J. B 15. 765 (2000
)). As originally proposed by Schertaer and Lovejoy, the first one consists
in a simple power law filtering (known in the mathematical literature as "
fractional integration") of singular cascade measures. The second one is th
e foremost attempt to generate log-infinitely divisible cascades on 2D orth
ogonal wavelet basis. We report numerical estimates of the tau(q) and D(h)
multifractal spectra which are in very good agreement with the theoretical
predictions. We emphasize the 2D WTMM method as a very efficient tool to re
solve multifractal scaling. But beyond the statistical information provided
by the multifractal description, there is much more to learn from the arbo
rescent structure of the wavelet transform skeleton of a multifractal rough
surface. Various statistical quantities such as the self-similarity kernel
and the spacescale correlation functions can be used to characterize very
precisely the possible existence of an underlying multiplicative process. W
e elaborate theoretically and test numerically on various computer syntheti
zed images that these statistical quantities can be directly extracted from
the considered multifractal function using its WTMM skeleton with an arbit
rary analyzing wavelets. This study provides algorithms that are readily ap
plicable to experimental situations.