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

Computed tomography (CT) by its use in biomedicine has proved the high efficiency of this computational diagnosis. The main objective in CT reduced to finding estimate of unknown function initially subjected to during the registration process integration. The search technique is determined by the mathematical apparatus of the special case of the inverse Radon transform and is a solution of the basic integral equation of the first kind. One of the problems of the practical implementation of CT is that the analytical solution proposed by Radon is applicable only to those unknown functions that are valid and satisfy the regularity conditions. So, for example, meet the requirement of the definition of a function at infinity, which in practice is not technically feasible, since the "mechanisms" of direct conversion of primary and integrators in CT is finite in size. As a result, the function of infinite length is replaced by the function of finite length and a solution is sought for its finite definition. The "replacement" produced is in itself a problem that needs to be addressed. In addition, it gives rise to the following problem – even initially smooth functions, rapidly decreasing at infinity, in the finite definition, as a rule, at the boundaries undergo irremovable discontinuities of the first kind and in the final implementation cease to be smooth.

From the point of view of the regularization theory, considerable experience has been gained in solving computational problems due to the discrepancy between the requirements stated for the function (and its Radon image) and the existing practical implementation. The discrepancy is compensated by the introduction of an amendment, which although replaces the exact solution of the main first-order integral equation with an approximate one, but ultimately, makes it resistant to small changes in the initial data of the Radon image. At the same time, the stability of the evaluation of the unknown function to small deviations from the requirements is considered from the point of view of variational calculus.

The aim of this work is to substantiate the possibility of using the finite Radon transform in the final limits for the practical implementation of the CT method. To do this, the problem of determining the regularity of the influence of deviations from the requirements of the regularity conditions on the evaluation of the unknown function is solved. The last will allow formulating approaches to the description of systematic distortions of an assessment with their subsequent elimination.

Thus, the situation that often arises in practice was investigated when the finite implementation Radon image (with the pulse duration, tending to unity) is not rapidly decreasing to the borders of the realization in conditions of finite size integrate transducers. The systematic cause of distortions, which lies in the elimination of the existing constant component in the finite realizations in the course of filtering, is revealed. An approach to eliminate the cause of distortion, suggesting the decomposition of the Fourier series in the half period, is proposed and justified.

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