Algorithms of 3D Cone Beam Tomography For Incomplete Data

Trofimov O., Kasjanova S., Badazhkov D.

Institute of Automation and Electrometry Siberian Division of Russian Academy of Sciences.

Prospect academica Koptyuga 1, 630090 Novosibirsk, Russia.

E-mail: Trofimov@iae.nsk.su

In X-ray computerized tomography, we often have to do with incomplete data. Also often we have situations when we are interested in values of the density not in all points of the object, but only in some domain. The typical example is the control of quality of welded joints of pipes. We need not determine the density in all points of the pipe, but only near by the welded joint. It is desirable also to limit a range of directions of X-rays. In similar situations, the methods of three-dimensional cone beam tomography may be useful.
In classical computerized x-ray tomography, a three-dimensional object usually is being considered as a set of thin cuts, and the two-dimensional Radon inversion formulae are used for reconstruction of the density function. But for analysis of some objects another schema is more natural, when an x-ray source is moving along a three-dimensional curve. Here we have a mathematical problem of reconstruction of a function of three variables from its integrals along the lines that cross the 3D-curve.