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Modeling of radar images based on stochastic differential equations in partial derivatives


A.V. Korennoy – Dr. Sc. (Eng.), Honored Scientist of RF, Professor, MESC «Zhukovsky–Gagarin Air Force Academy» (Voronezh) E-mail: S.A. Lepeshkin – Ph. D. (Eng.), Deputy Head 95 of Department, Mozhaysky Military Space Academy (Saint Petersburg) E-mail: A.P. Kadochnikov – Post-graduate Student, 95 Department, Mozhaysky Military Space Academy (Saint Petersburg) E-mail: E.A. Yashchenko – Post-graduate Student, 122 Department, MESC «Zhukovsky–Gagarin Air Force Academy» (Voronezh) E-mail:

Nowadays there is a trend of presentation of various nature images, in the form of random fields including radar images. The choice of the mathematical model in the form of a random field, describing the properties of radar images is a necessary stage in the development of pre-processing of image devices in automated systems of radiovision. On the one hand, the selected model should be simple enough to allow for the technical implementation of the automated processing unit, on the other hand, a mathematical model should reflect the properties of radar image as much as possible to provide high-quality processing. The most common systems of radiovision, which allow to obtain radar images of the Earth and the water surface, as well as objects located on it, regardless of the time of day, light conditions, and weather conditions, are radar stations with synthetic aperture, which suggest movement of the carrier of a sensing system. It should be noted that the formation (restoration) of radar images based on the received trajectory signal is mathematically incorrect. This raises the need of regularization of solutions - significantly reducing the scatter of decisions relative to the true value. There is a method for optimal processing of random fields applied to image restoration problems in the automated radar systems, monitoring of the Earth\'s surface, based on the Bayesian approach, wherein the a priori correlation function of the model radar image of restoration is used as parameter of regularization. For a description of a mathematical model of the field of radar image it is proposed to use a unit of stochastic differential equations in partial derivatives, which indicates directly a method for generating field implementations, and allows for the analysis of probabilistic characteristics of the resulting models. It is known that the major equations of mathematical physics are three types of linear partial differential equations of the second order: hyperbolic, parabolic, elliptical. Given the characteristics of systems of radiovision a choice of selection and mathematical description of the radar images models based on linear partial differential equations of the second order, is carried out; expressions for their probability characteristics are determined; parametric models dependencies of implementations and their correlation functions are illustrated. It is shown that by using three types of models may be described by the nature of the various radar images. In this way the correspondence is established between the type of a radar image and its correlation function based on the notion of causality, which requires a causal correlations on both spatial coordinates. Based on these considerations, hyperbolic equations are causal models; parabolic equations - semi causal models; elliptic equations - non-causal models. In order to evaluate the adequacy of the mathematical model of a real radar image, it is necessary to compare the correlation function models and statistical correlation functions of a real radar image. This problem was solved by the least squares method from the condition of minimum mean square error for different values of model parameters. It was found that the proposed three types of models describe a real radar image quite adequately. Moreover, the causal and semi causal models describes well enough a radar image of the earth\'s surface, and non-causal model - radar images of large-scale objects.


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