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The paper develops a method for approximation of the discontinuous functions of two variables by discontinuous interlination splines using arbitrary triangular elements. Experimental data are one-sided traces of a function given along a system of lines (such data are commonly used in remote methods, in particular in tomography). The paper also proposes a method for approximating the discontinuous functions of two variables taking into account triangular elements having one curved side. The proposed methods improve approximation of the discontinuous functions, allowing an application to complex domains of definition and avoiding the Gibbs phenomenon.

In the paper the methods of restoring the internal structure of the object, using new information operators developed by the Ukrainian scientist Professor Lytvyn O.M., namely, interlineation and interflatation, are studied. Interlineation and interfltatation operators restore functions (perhaps approximately) according to their known traces on a given system of lines and planes, respectively. The paper provides a solution to the three-dimensional computer tomography problem by the function interpolation operator. Tomograms obtained from a real computer tomograph and the equations of the planes on which these tomograms lie serve as experimental data. The paper considers the problem of restoring the absorption coefficient inside a three-dimensional object based on its tomograms lying on a system of three groups of parallel planes, which are not necessarily perpendicular to the coordinate axes. In addition, an interflatation operator is constructed on a system of planes, each of which does not necessarily intersect with all others. A method for restoring the internal structure of a three-dimensional body is also being developed, which uses four tomograms and is built using the interpolation of functions of three variables. Also, the paper presents the general forms of densities or absorption coefficients of objects, which are described by functions that can be accurately restored with the help of the specified information. In the work, a method of restoring the internal structure of the body is developed using the operator of blending approximation by Bernstein polynomials. This method is recommended to be used in cases when the experimental data (characteristics of tomograms – geometric parameters of the plane on which the tomogram lies, as well as images on the tomograms) are given with an error, and when the classic interpolation and interfltation operators do not smooth the data, but repeat all the errors in experimental data. Further, the developed new information operators are used to restore a dynamic body. In this paper, the problem of two-dimensional computer tomography is solved not only with the use of new information operators, but also the heterogeneity of the internal structure of the examined body. All proposed methods have high accuracy.

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