Algorithms of multifractal wavelet analysis in problems of specifying raster images of self-similar structures

Many physical objects exhibit self-similarity and a high degree of self-organization, or, in contrast, possess the properties of dissipativity, irregularity, roughness, chaos, and disorder. The concepts of the theory of fractals and multifractals are used as a theoretical basis for mathematical formalization and quantitative analysis of various solid-state media. The study is devoted to further developing algorithmic approaches of the multifractal wavelet analysis and their application for assessing scaling characteristics of raster images of self-similar physical structures. The wavelet transform modulus maxima (WTMM) method is one of the promising and widespread multifractal techniques based on wavelet transforms. The basic step of WTMM method is presented by constructing the wavelet transform skeleton, which is used for the calculation of partition functions and scaling characteristics. However, the traditional approach leads to the appearance of the so-called “artifacts” at the construction of lines of local extrema. The algorithms for filtering and marking lines of local maxima are proposed within the framework of WTMM. The filtering algorithm is aimed at eliminating “breaking” and “hanging” maxima when constructing continuous lines of local maxima and allows one to work with the correct representation of the wavelet transform skeleton. The marking algorithm uniquely identifies the maximum belonging to a particular line of local extrema and underlines calculating partition functions for negative values of the deformation parameter. A system of computer analysis of raster images was designed in Matlab to perform a multifractal parameterization and visualize the main scaling characteristics. The results of the program application are demonstrated for the test problem of analysis of the fractal image artificially generated with use of brown noise. The developed software tools are applied to estimate the multifractal characteristics of digital images of materials obtained by scanning electron microscopy. The scanning electron microscope (SEM) images are formed line by line by electron beam moving with constant velocity. SEM-images are result of videosignal registration as a sample response under the electron probe action. Hence, one-dimensional WTMM method can be used for specification of complex images by means of analyzing video signal profile. Ferroelectrics is a special class of polar dielectric materials possessing interesting fractal properties. Domain configurations of typical ferroelectrics are the result of self-organization and indicate self-similar behavior. Scaling characteristics are established for the domain structure image of a typical ferroelectric - triglycine sulfate (TGS). The findings suggest, that the SEM-image of a TGS crystal exhibits multifractal properties and is characterized by a spectrum of fractal dimensions, represented by a rather wide range. The analysis of the evolution of the singularities spectrum is carried out for a ferroelectric under temperature exposure. Our results show that the residual polarization contrast during annealing is characterized by a narrowing and a shift of the spectrum to lower values of the Holder exponents in comparison with the initial structure. This result demonstrates that the alignment of the TGS “pseudo-domains” into elongated chains at annealing has a complex organization, even more “ordered” than the initial domain structure observed under equilibrium conditions. In general, the apparatus of multifractal wavelet analysis provides a flexible technique for mathematical diagnostics of the degree of the self-similarity properties of domain configurations and the topographic structure of complex structured physical systems.

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