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The Markov chains usage in the analysis of stochastic estimation error of image inter-frame geometric deformation parameters

DOI 10.18127/j00338486-201909(14)-08

Keywords:

A.G. Tashlinskii – Dr.Sc.(Eng), Professor, Head of Department «Radio Engineering», Ulyanovsk State Technical University
E mail: tag @ulstu.ru
G.L. Safina – Ph.D.(Eng), Associate Professor, Department «Applied Mathematics», National Research Moscow State University of Civil Engineering
E mail: minkinag@mail.ru
M.G. Tsaryov – Senior Lecturer, Department «Radio Engineering», Ulyanovsk State Technical University
E mail: michael.tsaryov@gmail.com
R.O. Kovalenko – Post-graduate Student, Ulyanovsk State Technical University
E mail: r.kovalenko.o@yandex.ru


When estimating the parameters of image interframe geometric deformations under conditions of a priori uncertainty, relay stochastic procedures have found practical application. It is shown that the sequence of estimates of the strain parameters obtained using such procedures is a sequence without aftereffect and is a vector Markov process. Moreover, to analyze the error of stochastic estimation of one parameter, the use of the mathematical apparatus of Markov chains theory is advisable. The study of expressions that allow one to calculate the transition probabilities of the matrix of one-step transitions through the probabilities of the drift of parameter estimates showed that the matrix of one-step transitions for the parameter being estimated has a five-diagonal structure. With a constant step of changing estimates, it does not depend on the iteration number, determining the homogeneity of the Markov chain. However, for the parameter vector, the use of the classical apparatus of Markov chains becomes problematic due to a sharp increase in the size of the transition probability matrix. A priori the choice of the matrix size allows the discretization of the domain of the estimated parameters, but even with this approach, the use of the classical apparatus of Markov chains turns out to be advisable only when evaluating one parameter.
In order to reduce computational costs, one-step transition matrices are proposed, the dimension of which is determined only by the discretization of estimated parameters domain and does not depend on their number. The resulting matrix allows only a recursive way of calculating it from the initial approximation of the parameters to the required iteration. Moreover, the Markov chain of estimates loses the property of homogeneity. Also, accurate information about the probability distribution in the parameter space is lost. Only projections of this spatial distribution are preserved. However, this turns out to be sufficient to find the probability distribution of estimates of interframe deformations parameters at a finite number of iterations.

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