Abstract
We discuss entropy characteristics used in various research techniques for investigation of complex dynamical systems including symbolic ones. The dynamics of a system may be studied by analyzing the phase portrait of a system obtained as a digital image. Symbolic dynamics methods allow combining entropy of a given dynamical system with the entropy characteristics of its phase portrait. We apply methods of image analysis based on symbolic dynamics, Rényi entropy, fractal and multifractal characteristics to analyze high resolution images having a complex structure. We also describe the results of applications of described methods to images of biomedical preparations.
Highlights
Scientific investigations of various processes are often based on observations
Symbolic dynamical systems having a presentation by the oriented graph are topological Markov chains, which are very important in different applications
The mathematical formalization for statistical events is expressed in terms of distribution functions, and the entropy is written by using these functions
Summary
Scientific investigations of various processes are often based on observations. P. Symbolic dynamical systems having a presentation by the oriented graph are topological Markov chains, which are very important in different applications. By applying the technique of adaptive partition subdivision, one can construct a sequence of symbolic images that is an approximation of the dynamics of the initial system This method was successfully applied to the approximation of invariant sets, Morse spectrum and invariant measures [10]. When considering phase portraits as digital images, we may interpret the set of frequencies as a measure distribution Basing on this distribution, we may calculate multifractal spectra, Rényi spectra and divergences. The stationary flow (state) of the chain is calculated, which maximizes so called weighted entropy This value interpreted as a “distance” between an initial state, and the stationary one may be used as a classifying sign in image analysis. The described methods were used to analyze and classify digital images of biomedical preparations having a complex structure
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