Аналіз результатів обчислювального експерименту відновлення розривних функцій двох змінних за допомогою проекцій: Fìz.-mat. model. ìnf. tehnol. 2021, 33:12-17

• Date: 2021
• Author Affiliations: Oleg Lytvyn — Українська інженерно-педагогічна академія, вул. Університетська, 16, 61003, Харків; Oleg Lytvyn — Українська інженерно-педагогічна академія, вул. Університетська, 16, 61003, Харків; Oleksandra Lytvyn — Харківський національний університет радіоелектроніки, пр. Науки,14, 61166, Харків
• Keywords: keywords
• Main Authors: Lytvyn, Oleg, Lytvyn, Oleksandra
• Format: Article
• Language: Ukrainian
• Published: Інститут прикладних проблем механіки і математики ім. Я. С. Підстригача НАН України 2021
• Subjects: комп’ютерна томографія; розривна функція; розривний сплайн; сума Фур’є
• Online Access: https://www.fmmit.lviv.ua/index.php/fmmit/article/view/194
• Journal Title: Physico-mathematical modeling and informational technologies

This article presents the main statements of the method of approximation of discontinuous functions of two variables, describing an image of the surface of a 2D body or an image of the internal structure of a 3D body in a certain plane, using projections that come from a computer tomograph. The method is based on the use of discontinuous splines of two variables and finite Fourier sums, in which the Fourier coefficients are found using projection data. The method is based on the following idea: an approximated discontinuous function is replaced by the sum of two functions – a discontinuous spline and a continuous or differentiable function. A method is proposed for constructing a spline function, which has on the indicated lines the same discontinuities of the first kind as the approximated discontinuous function, and a method for finding the Fourier coefficients of the indicated continuous or differentiable function. That is, the difference between the function being approximated and the specified discontinuous spline is a function that can be approximated by finite Fourier sums without the Gibbs phenomenon. In the numerical experiment, it was assumed that the approximated function has discontinuities of the first kind on a given system of circles and ellipses nested into each other. The analysis of the calculation results showed their correspondence to the theoretical statements of the work. The proposed method makes it possible to obtain a given approximation accuracy with a smaller number of projections, that is, with less irradiation. References Lytvyn, O. M., Lytvyn, O. G., Lytvyn, O. O., Mezhuyev, V. I. (2020). The Method of Reconstructing Discontinuous Functions Using Projections Data and Finite Fourier Sums. The IX International Scientific and Practical Conference «Information Control Systems &Technologies (ICST-2020)». DOI https://doi.org/10.1109/acitt.2019.8779938 Lytvyn, O. M., Lytvyn, O. G. (2018). On an approach to the approximation of discontinuous functions using projections and finite Fourier sums. Computational methods and systems of information transformation: coll. etc. V-th scientific-technical. conf., Lviv. Lytvyn, O. M., Lytvyn, O. G. (2018). A method for restoring discontinuous functions of a special kind using projections and finite Fourier sums. 7th International Scientific and Technical Conference "Information Systems and Technologies IST-2018". Lytvyn, O. M. (2000). Periodic splines and a new method for solving the flat problem of X-ray computed tomography // System analysis, management and information technology: Bulletin of the Kharkiv State. Polytechnic. un-tu. Collection of scientific works. Issue 125. Kharkiv: KhDPU. Gottlieb, S., Jung, J.-H., Kim, S. (2011). A Review of David Gottlieb’s Work on the Resolution of the Gibbs Phenomenon, Commun. Comput. Phys., 9(3), 497–519. DOI https://doi.org/10.4208/cicp.301109.170510s