TY - JOUR

T1 - On using escort distributions in digital image analysis

AU - Ампилова, Наталья Борисовна

AU - Соловьев, Игорь Павлович

AU - Сергеев, Владислав Дмитриевич

PY - 2021/3/2

Y1 - 2021/3/2

N2 - Methods of digital image analysis find wide application both in scientific research and in many branches of industry. During the last decades, interest has grown in images with multifractal structure which are obtained in biology, medicine, chemistry and studying the soil. The mathematics of fractals and fractal geometry are well known and studied. However, despite this, a common approach to designing the methods of practical investigation of such images has not been developed until now. The main purpose of this work is to propose the using of multifractal formalism as the mathematical tool for the statistical description of multifractal sets. Such a description adequately depicts the chaotic behavior of the majority of real systems. The method for calculation of Rényi and singularity spectra based on using parametrized spectra, which are obtained from escort (zooming) distributions of an initial measure, is considered. The method for comparing images based on using vectors of divergences calculated for the sequence of escort distributions is proposed. The role of parametrized spectra as the tool for the approximation of any part of the singularity spectrum is substantiated. An estimation of the rate of growth of the divergence vector is obtained. Main theoretical results are confirmed by numerical experiments with images of biomedical preparations. These show the ability of the implemented methods to find subtle distinctions in image structure for a simple choice of an initial measure.

AB - Methods of digital image analysis find wide application both in scientific research and in many branches of industry. During the last decades, interest has grown in images with multifractal structure which are obtained in biology, medicine, chemistry and studying the soil. The mathematics of fractals and fractal geometry are well known and studied. However, despite this, a common approach to designing the methods of practical investigation of such images has not been developed until now. The main purpose of this work is to propose the using of multifractal formalism as the mathematical tool for the statistical description of multifractal sets. Such a description adequately depicts the chaotic behavior of the majority of real systems. The method for calculation of Rényi and singularity spectra based on using parametrized spectra, which are obtained from escort (zooming) distributions of an initial measure, is considered. The method for comparing images based on using vectors of divergences calculated for the sequence of escort distributions is proposed. The role of parametrized spectra as the tool for the approximation of any part of the singularity spectrum is substantiated. An estimation of the rate of growth of the divergence vector is obtained. Main theoretical results are confirmed by numerical experiments with images of biomedical preparations. These show the ability of the implemented methods to find subtle distinctions in image structure for a simple choice of an initial measure.

KW - Direct multifractal transformation

KW - Escort distribution

KW - Image analysis

KW - Multifractal spectrum

KW - Rényi spectrum

UR - http://www.scopus.com/inward/record.url?scp=85104941636&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/f8f8d4c8-b450-3a8c-a685-f516eb19dd32/

U2 - 10.21595/jme.2021.21851

DO - 10.21595/jme.2021.21851

M3 - Article

VL - 9

SP - 58

EP - 70

JO - Journal of Measurements in Engineering

JF - Journal of Measurements in Engineering

SN - 2335-2124

IS - 1

ER -