Explicit Formulas for Calculating Fourier Coefficients of Three Variables Using Tomograms

Abstract

The great role of Fourier coefficients in the digital processing of multidimensional signals is known. The classic cubic formulas for the calculation of Fourier coefficients use the values of an approximate function at individual points. Due to the advent of computerized tomograph, new information operators have emerged that do not directly use the function value at individual points in the integration area. These are projections-integrals from an approximate function along a given system of straight lines, tomograms–images of traces of an approximate function of three variables on given planes, etc. The classic Fourier coefficients calculation using projections is based on the formulas used for direct and inverse Radon conversion. For the 2D case, such studies were performed in classical computed tomography circuits, including the direct Fourier method Another approach to calculate 2D Fourier coefficients without using a direct and inverse Radon transform was proposed in 2000 by Oleg M. Lytvyn. This paper deals with the generalization of this method in the 3D case. It is a continuation of the research begun by the authors in 2019. This paper proposes explicit formulas for the limits of integration for the re-integrals arising in the proposed method. Thus, this paper is devoted to the further development of a new scientific direction in the theory of approximate Fourier coefficients of the function of three variables using new information operators.