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# Doubly stochastic models of cylindrical images

Keywords:

V.R. Krasheninnikov – Dr.Sc.(Eng.), Professor, Head of Department «Applied Mathematics and Informatics», Ulyanovsk State Technical University
E-mail: kvrulstu@mail.ru
A.Yu. Subbotin – Post-graduate Student, Department «Applied Mathematics and Informatics», Ulyanovsk State Technical University
E-mail: ashkael@mail.ru

The tasks of image processing and other multivariate data arise when solving many practical problems: Earth and Space monitoring, medical diagnostics, navigation, robotics, etc. To effectively solve these problems, their mathematical formulation is necessary, including a mathematical description, that is, an image model. In most known works, images are viewed as a system of random variables defined on a rectangular grid of dimension two or more [1−3]. Significantly less work is available on images defined on curvilinear surfaces, for example, on a sphere or a cylinder . In addition, the most of models describe homogeneous images. However, many real images have a significant and random heterogeneity. In [5−9], it is proposed to represent the inhomogeneity of images, in the form of «doubly stochastic» models. In this view, the «control» images specify random parameters of the «managed» final image. The heterogeneity of the resulting image is determined by the local features of the control images. In the works mentioned here, the control and controlled images were autoregressive, given on rectangular grids.
In this paper we consider doubly stochastic autoregressive models of images on a cylinder. In these models, an autoregressive model is used, similar to the Habiby model of a rectangular image . If such a rectangular image is rolled into a cylinder, then at the junction there will be sharp jumps of brightness. Therefore, in the proposed model, instead of the rows of the Habibie model, the turns of a cylindrical spiral are used. This makes it possible to obtain a cylindrical image without large jumps. To obtain heterogeneous images, control images are used that are defined on the same spiral, which determine the random autoregressive parameters of the resulting image. In addition, we consider a model in which several images simultaneously influence the parameters of each other's autoregression. That is a kind of feedback between them. The proposed models can also be used as models of quasiperiodic inhomo-geneous processes or systems of such correlated processes obtained by scanning images in a spiral. This can be used to describe and processing of quasiperiodic signals, for example, vibrations of technical objects .

References:
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2. Habibi A. Two-dimensional Bayesian Estimate of Images // Proc. IEEE. 1972. 60(7). P. 878−883.
3. Krasheninnikov V.R., Vasiliev K.K. Multidimensional Image Models and Processing // Intelligent Systems Reference Library 135. Springer International Publishing. 2018. P. 11−64.
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9. Vasiliev K.K., Dementiev V.E., Andriyanov N.A. Filtration and Restoration of Satellite Images Using Doubly Stochastic Random Fields // CEUR Workshop Proceedings. 2017. V. 1814. P. 10−20.
10. Krasheninnikov V.R., Kuvajskova Yu.E. Prognozirovanie dinamiki ob’‘ekta s ispol’zovaniem avtoregressionny’x modelej na czilindre // Radiotexnika. 2016. № 9. S. 36−39.
June 24, 2020
May 29, 2020

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