International Society for Industrial Process Tomography

Discrete form for calculating 3D Fourier coefficients using tomograms

O. M. Lytvyn^1*, O. G. Lytvyn², O. O. Lytvyn¹

¹Ukrainian Engineering-Pedagogical Academy, Kharkiv, Ukraine

²National University of Radioelectronics, Kharkiv, Ukraine

*Email: academ_mail@ukr.net

ABSTRACT

A method is described for solving a 2D problem of X-ray computed tomography, based on the use of finite Fourier sums in which Fourier coefficients are calculated using projections – integrals from the approximate function along a given system of lines cross the object of study. This method approximates discontinuous functions of two variables using finite Fourier sums without the Gibbs phenomenon. Appropriate formulas are obtained by subtracting the discontinuous spline from the investigated discontinuous function. A method of constructing such discontinuous splines is proposed.  The method is generalized to a 3D case and explicit formulas were obtained for calculating 3D Fourier coefficients using integrals from tomograms placed in a given system of planes. That is, the problem of calculating the integrals from tomograms, using projections is an urgent task. The images of the internal structure of 2D or 3D objects are represented as finite Fourier sums and have the following computational properties: the inverse radon transformation is not used when calculating Fourier coefficients; the method allows to obtain guaranteed accuracy; if the internal structure of a 2D object is described by discontinuous functions, propose a method of approximate restoration of the internal structure using discontinuous splines and finite Fourier sums without the Gibbs phenomenon.

Keywords: Fourier coefficients, projections, tomograms, Gibbs phenomenon, finite Fourier sums

Industrial Application: medical tomography, non-destructive testing of micro and macro objects