Информационная система "Конференции"

Abstracts

Numerical solution of differential and integral equations

Tomographic methods for reconstructing the internal structure of various objects are widely used in medicine, in some branches of industry, and also in physical research. Usually, a ray tomography approximation is dealt with. This approximation implies that, to each measurement $f_i$, a line integral of a sought function $g(x,y,z)$ along a certain straight line corresponds. When the number of measurements is infinite the relationship between the projection data $f$ and function $g$ is expressed with the well-known in integral geometry P or D transforms [1]. In practice a number of measurements is often insufficient for the valid approximation of the equations inverting these transforms. In such case the problem of finding $g$ is reduced to the inversion of the system of linear algebraic equations $f=Ag$. For the ray tomography each equation of this system is an approximation of the integral from along the straight line in the three-dimensional space.

The function $g(x,y,z)$ is determined on a grid in the three- dimensional space. The nodes of the grid are not the same ones, which are defined on the set of straight lines. The problem of interpolation arises due to this reason. The different ways of this problem solution lead to different methods of projection matrix A construction. Some of them have been considered and compared in the present paper. They belong to the zero-order accuracy (for instance, nearest neighbor method) and the first-order accuracy (three-linear interpolation) in respect to space interpolation. The usage of higher orders accuracy schemes for solving three-dimensional tomography problem results in too long computation time.

Quantitative characteristics of accuracy of 3D-tomograms reconstructions were obtained for some approximations in numerical simulations. The modifications of ART algorithm [2-3] were used for the solution of corresponding systems of linear algebraic equations.

References

Note. Abstracts are published in author's edition

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