MODELING A DYNAMIC THREE-DIMENSIONAL BODY USING INTERFLATATION AND BLENDING APPROXIMATION

Abstract

The paper deals with the problem of reconstruction of the internal structure (density, absorption or attenuation coefficient) of a three-dimensional body by the information about it in the form of tomograms, given on a certain system of planes intersecting the object of study. This problem arises in practice in cases where there is no plane among the planes that are included in the experimental data, which consists of one or another set of points that are of interest to the researcher. For example, such a problem may arise after a patient has undergone examinations on a medical tomograph. After analyzing the obtained tomograms, it becomes necessary to find with their help one or more tomograms in the planes intersecting the body, but not coinciding with any of the given planes. It is noted that the operators of interflatation of functions is a natural generalization of the interpolation operators for the functions of three variables. Therefore, as in the case of interpolation, errors in the experimental data (in this case, in tomograms) are also introduced into the interflatation operators. In mathematics, there is an alternative to interpolation operators, namely approximation operators. These are operators constructed by smoothing experimental data using polynomials, rational functions, trigonometric polynomials, wavelets, and the like. An operator of mixed approximation of a function of three variables is constructed using Bernstein polynomials; the general form of the approximation error by the constructed operator and the estimate of this error are given. In the paper a four-dimensional mathematical model of a three-dimensional body that changes over time is also built and studied. A computational experiment is presented to restore the internal structure of the human heart from tomograms lying on a system of mutually perpendicular planes, which come from an actually operating computer tomograph. The article presents some possibilities of working with tomograms when restoring the internal structure of a three-dimensional body from the known projections of this body (tomograms).