THE SOLUTION TO THE 3D PROBLEM OF COMPUTER TOMOGRAPHY WITH KNOWN TOMOGRAMS ON THE SYSTEM OF ARBITRARY PLANES

The article is based on restoration method for the internal structure of a three-dimensional body using polynomial interflatation based on known tomograms (traces) lying on a system of arbitrary planes, which is a generalization of the method of body restoration with its known tomograms on a system of three groups of parallel planes. A definition of tomograms in the mathematical sense has been provided and the algorithm of transition of tomogram images into the functional dependence has been outlined.

Theorems on interflatation properties and errors of the built operator have been formulated and proved. A test case for the construction of the interflatation operator for the quadratic function has been demonstrated, and the computational experiment involved the development of a number of programs in MathCad. The experiment has provided visualization results for an exact solution and a solution obtained experimentally for the case when the exact function is known. It has been shown that the constructed structure approximates this function exactly, which is not the case of classic interpolation operators.

The suggested method makes it possible to solve the problem of three-dimensional computer tomography for a fundamentally new data collection scheme. For example, it permits the use of the fan scheme for data collection in each of the planes in which the tomograms lie. 

1. Natterer F. Mathematical aspects of computerized tomography. New York, Wiley, 1986. 221 p.

2. Herman, G. T. Image Reconstruction from Projections. The Fundamentals of Computerized Tomography. New York – London, Academic Press, 1980. 316 p.

3. Helgason S. The Radon Transform. Boston, Birkhäuser, 1980. 195 p. Doi: 10.1007/978-1-4899-6765-7

4. Sergienko I. V., Litvin O. M., Pershina Yu. I. Mathematical modeling in computer tomography with the use of function interflatation. Kharkiv, 2008. 160 p. (in Ukrainian).

5. Litvin O. M., Pershina Yu. I. Mathematical model of internal structure restoration for a 3D object with the use of function interflatation. Dopovidi Nacional’noi’ akademii’ nauk Ukrai’ny = Reports of the National Academy of Sciences of Ukraine, 2005, no. 1, pp. 20–24 (in Ukrainian).

6. Litvin O. M. Function interlination and some of its applications. Kharkiv, Basis Publ., 2002. 544 p. (in Ukrainian).

7. Likhachev A. V., Pickalov V. V. A new method for deriving unknown additive background in projection in threedimensional tomography. Computational Mathematics and Mathematical Physics, 2002, vol. 42, no. 3, pp. 341–352.

8. Trofimov O. E., Tyurenkova L. V. On one method for image restoration based on multiangle tomogram. Novosibirsk, IAE SO AN SSSR, 1989. 28 p. (in Russian).

9. Pikalov V. V., Lihachev A. V. Comparison of spiral tomography methods. Vyichislitelnyie metodyi i programmirovaniya = Numerical Methods and Programming, 2004, no. 5, pp. 170–183 (in Russian).

10. Nikolskii S. M. Boundary properties of functions determined on the angular point domain. Matematicheskii sbornik = Sbornik: Mathematics, 1958, vol. 45 (87), no. 2, pp. 181–194 (in Russian).