RESTORATION OF THE INTERNAL STRUCTURE OF A DYNAMIC THREEDIMENSIONAL BODY USING BLENDING APPROXIMAT

Abstract

The work is devoted to the problem of restoring the internal structure of a three-dimensional body using information about it in the form of tomograms, which are given on a certain system of planes intersecting the object of study. This problem arises in practice when, among the planes that are included in the experimental data, there is no plane consisting of a particular 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. The article notes that the operators of interflatation of functions is a natural generalization of the operators of interpolation of three variables functions. These operators restore functions (possibly approximately) from their known traces on a given system of planes. It is these experimental data that are used in remote sensing methods, in particular in computed tomography. Thus, interflatation is a mathematical apparatus, naturally associated with the task of reconstructing the characteristics of objects from their known projections. As in the case of interpolation, errors in experimental data (in this case, in tomograms) are also introduced into the interflatation operators. In mathematics, there is an alternative to interpolation operators - approximation operators. These are operators constructed by smoothing experimental data using polynomials, rational functions, trigonometric polynomials, wavelets, and the like. An operator of blending approximation of a three variables function is constructed using Bernstein polynomials; the general form of the approximation error by the constructed operator and the estimate of this error are given. The work also builds and studies a four-dimensional mathematical model of a three-dimensional body that changes over time. A computational experiment is presented to restore the internal structure of a moving human heart from tomograms lying on a system of mutually perpendicular planes, which come from a really operating computer tomograph.