A wavelet-based method for multifractal image analysis. II. Applications to synthetic multifractal rough surfaces

Abstract:

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. Arnéodo, N. Decoster, Eur. Phys. J. B 15, 765 (2000)). As originally proposed by Schertzer 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 the foremost attempt to generate log-infinitely divisible cascades on 2D orthogonal wavelet basis. We report numerical estimates of the 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 resolve multifractal scaling. But beyond the statistical information provided by the multifractal description, there is much more to learn from the arborescent structure of the wavelet transform skeleton of a multifractal rough surface. Various statistical quantities such as the self-similarity kernel and the space-scale correlation functions can be used to characterize very precisely the possible existence of an underlying multiplicative process. We elaborate theoretically and test numerically on various computer synthetized images that these statistical quantities can be directly extracted from the considered multifractal function using its WTMM skeleton with an arbitrary analyzing wavelets. This study provides algorithms that are readily applicable to experimental situations.