Multifractal analysis of rectangular pointwise regularity with hyperbolic wavelet bases (Q2057191)

The paper deals with the analysis of directional or anisotropic features for multidimensional signals. The authors define rectangular pointwise regularity through local oscillations of the function over rectangular parallelepipeds. To characterize anisotropic properties of sets the notion of the dimension print is used. It is a set of positive vectors depending on a behavior of some Hausdorf-type measure along different directions. The authors characterize rectangular pointwise regularity in terms of hyperbolic wavelet coefficients and wavelet leaders. The hyperbolic wavelet basis consists of products of shifts and dyadic dilations of one dimension \(r\)-regular wavelet functions and scaling functions that corresponds to diagonal extension matrix with \(2\)'s on the main diagonal and a separable MRA. The authors obtain a numerical procedure that yields information on the dimension print of sets of level rectangular pointwise regularities, expressed in terms of hyperbolic wavelet leaders. An example of anisotropic selfsimilar cascade wavelet series is given, where the anisotropy corresponds to a Sierpinski carpet.