A.A. Zhilnikov – Ph.D. (Eng.), Lecturer, Department «Logistics of the penitentiary system», Academy of the Federal Penitentiary Service of Russia (Ryazan)

E-mail: ark9876@mail.ru

T.A. Zhilnikov – Ph.D. (Eng.), Associate Professor, Head of Department «Mathematics and information technology management»,  Academy of the Federal Penitentiary Service of Russia (Ryazan)

E-mail: quadrus02@mail.ru

V.I. Zhulev – Dr. Sc. (Eng.), Professor, Head of Department «Information-measuring and biomedical engineering», V.F. Utkin Ryazan State Radio Engineering University; Honored Worker of Higher School of Russian Federation,  Laureate of the Ryazan Region on Science and Technology and the Silver Medal n.a. Academician V.F. Utkin

E-mail: zhulev.v.i@rsreu.ru

Currently, there is a separate area of scientific research, the purpose of which is a noninvasive study, as the internal structure of biological objects, and the distribution of physical fields within them. This isolation of these studies is due to the fact that for existing reasons, such as "remoteness" or "opacity" of a full bodied object, exclude the possibility of mechanical penetration of the primary measuring transducer inside it. Undeniable assistance in solving such research problems is provided by the use of diagnostic computing technologies based on the mathematical apparatus of Radon transform (RT).

RT, proposed by the author in 1917 in the classical scalar representation, is widely and often used in medicine (x-ray, emission, ultrasonic computed tomography), astronomy, optics, geophysics. However, in the presence of a significant inherent potential of development, yet, the practical interest of the transform is limited to the case of two-dimensional transformation of the scalar function of two variables. The reasons for the latter are natural and lie in the fact that often the physical principles underlying the technical implementation of noninvasive research, can simplify the task of reducing the dimensionality of the representation of the original three-dimensional object in space to a set of two-dimensional flat tomographic layers. At the same time, significant difficulties arise in practice when such simplification is not always technically feasible in principle, and in addition, the studied structure (characterizing its function) of the object has not a scalar, but a vector description.

Thus, the detailed mathematical apparatus of the RT in the classical representation is intensively applied, but is almost not considered in the multidimensional aspect in relation to the transformation of vector functions.

With regard to the principles of magnetic methods of nondestructive registration and implementing their technical means, the use of tomographic method is the most natural. However, its use causes a number of difficulties in the registration procedure of Radon image. When using the tomographic method, the need to move away from the particular two-dimensional representation of the mathematical apparatus of the RT to use its general form, modified for vector description.

Thus, the purpose of the study conducted in this paper is to substantiate the possibility of the spread of the mathematical apparatus of RT in the case of vector functions of three variables, with respect to nondestructive registration of the distribution of the force characteristics of the magnetic field in places, inaccessible to mechanical penetration in biological objects.

The formulated goal involves the solution of the problem of modification of the mathematical apparatus of the scalar RT, which should take into account the implicit definition of the force characteristic, which has the vortex nature. The method of evidential reasoning based on RT, vector space transformation, as well as the theory of magnetic field was used in the present study.

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