Self-Similar Anisotropic Texture Analysis: the Hyperbolic Wavelet Transform Contribution

Textures in images can often be well modeled using self-similar processes while they may at the same time display anisotropy. The present contribution thus aims at studying jointly selfsimilarity and anisotropy by focusing on a specific classical class of Gaussian anisotropic selfsimilar processes. It will first be shown that accurate joint estimates of the anisotropy and selfsimilarity parameters are performed by replacing the standard 2D-discrete wavelet transform by the hyperbolic wavelet transform, which permits the use of different dilation factors along the horizontal and vertical axis. Defining anisotropy requires a reference direction that needs not a priori match the horizontal and vertical axes according to which the images are digitized, this discrepancy defines a rotation angle. Second, we show that this rotation angle can be jointly estimated. Third, a non parametric bootstrap based procedure is described, that provides confidence interval in addition to the estimates themselves and enables to construct an isotropy test procedure, that can be applied to a single texture image. Fourth, the robustness and versatility of the proposed analysis is illustrated by being applied to a large variety of different isotropic and anisotropic self-similar fields. As an illustration, we show that a true anisotropy built-in self-similarity can be disentangled from an isotropic self-similarity to which an anisotropic trend has been superimposed

On the Fractional Fractal Analysis of Multivariate Pointwise Lipschitz Oscillating Regularity

Classical Lipschitz regularity does not allow to capture possible different oscillating directional pointwise regularity behaviors in coordinate axes of functions f on Rd, d ≥ 2. To overcome this drawback, we use iterated fractional primitives to introduce a notion of multivariate pointwise Lipschitz oscillating regularity. We show a characterization in hyperbolic wavelet bases. As an application, we obtain the fractal print dimension of a given set of multivariate Lipschitz oscillating regularity, from the knowledge of fractional axes oscillating spaces to which f belongs