Polynomial interpolation with known projections on an arbitrary system of N groups of lines consisting of M parallel lines
Keywords:
computed tomography, projection, polynomial interpolation functions of two variablesAbstract
Given N groups of lines, each consisting of M parallel lines. Each line of the group overlaps with all other lines from (N–1) group. It is believed that in the points of intersection of these lines there are given finite function values f(x, y) continuous together with its derivatives of the first order, the carrier of which is a square [0, 1]´[0, 1]. The projections are considered to be known, where the integrals along each of the n´m line, coming from CT. In fact, these integrals are along line segments that cross media. This paper solves the following problem: to build an operator approximation function f(x, y), which not only interpolates the function in these nodes, but also has given projections. The results of this study can be used in non-destructive testing of important parts in mechanical engineering.
References
Popov D. A. Reconstruction of characteristic functions in two-dimensional Radon tomography / D. A. Popov // Advances mathematic. sciences. - 1998. - T.53, edition 1 (319). – p. 115 - 198.
Litvin O.M. Mathematical modeling few projections computer tomography based interlineation and mixed approximation of functions. Abstract dis. upon. . . candidate of Phys. - Mat. Sciences: K. 2009. – p. 20
Litvin O. M. Operators interpolation with specified values of the integral / O. M. Litvin, O. O. Litvin / / development and application of mathematical methods in scientific and technical research. Abstracts ext. Ukrainian Scientific Conference, Lviv, October 5-7, 1995 - Lviv, 1995. - p. 113.
Litvin O. M. About one approach to solving the problem of a flat X-ray computed tomography / O. M. Litvin, O.O. Litvin / / Nonlinear boundary value problems of mathematical physics and their applications, Proc. scientific. tr. Institute of Mathematics of NAS of Ukraine, Kyiv, 1996. - S. 170-173.
Litvin O. M. The method of solving the plane problem of X-ray computed tomography / O. M. Litvin, O. O. Litvin / / Extras. National Academy of Sciences of Ukraine. - 1998. - № 12. - P.29-33.
Litvin O. M. Interlineation functions and some of its applications / O. M. Litvin - H.: Osnova, 2002. - 544p.
A. Litvin Methods of calculation. Related Topics / OM Lytvyn - Kyiv: Naukova Dumka, 2005. – 332 p.
Litvin O. O Methods of operators with given projections along intersecting direct that interpolates f(x, y) the points of intersection of these lines / O. O. Litvin E. L. Hurdey // Problemy mashinostoenia. – 2013. – № 3. – P. .
Natterer F. Mathematical Aspects Computer tomography: Trans. with Eng. / F. Natterer. - Moscow: Mir, 1990. – 279p.
Trinity J. H. Statestycheskaya theory tomography / J. H. Trinity. - M.: Radio and Communications, 1989. – 240 p.
Radon I. Determination of functions from their integrals along certain varieties / I. Radon // In the book: Helgason R. Radon transform. M. Mir, 1983. – 135 p.
Helhason S. Radon transform. Translated from Eng. / S. Helgason - Mir, 1983. – 152p.
Hermen H. Recover images from projections. Basics reconstruction tomography / G. Hermen - Moscow: Mir , 1983. - 352p.
Tereshchenko S. Methods of computerized tomography / S. Tereshchenko - Moscow: Fizmatgiz, 2004. – 320p.
Fedorov G. Computational emission tomography / G Fedorov, S. Tereshchenko - Energoatomizdat, 1990, 184p.
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