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\par }\pard \qj \li360\ri-464\widctlpar\aspalpha\aspnum\faauto\adjustright\rin-464\lin360\itap0 {\b\lang1033\langfe1049\langnp1033                         Radiation  Tomography Through Generalized Radon Transform  
\par                           D.S. Anikonov,    IM SB RAS; Russia, 630090, Novosibirsk, 
\par                                     Ak. Koptyug prosp., 4; e-mail: anik@math.nsc.ru
\par }\pard \qj \li0\ri-464\widctlpar\aspalpha\aspnum\faauto\adjustright\rin-464\lin0\itap0 {\b\lang1033\langfe1049\langnp1033   
\par }\pard \qj \li360\ri-464\widctlpar\aspalpha\aspnum\faauto\adjustright\rin-464\lin360\itap0 {\fs22\lang1033\langfe1049\langnp1033 
\par    Here the  tomography problem means find
ing information about the internal structure of an unknown medium by analysis of radiation signal passing through the medium. Generally speaking, the following kinds of signal may be considered: photon, neutron, proton, electron flows, etc.    
\par    From mathematical point of view, many concrete   problems of  tomography  can be considered as a general problem of integral geometry which is also called a problem of inversion of generalized  Radon Transform.
\par    For example, the classical X-ray tomography proble
m based on the ray approximation proves to be the well-known problem which has been solved by mathematician  Radon in 1917.  More complex similar problems arise in vector and tensor tomography. However, such approaches mean neglect of particle scattering 
with the order greater than one. The latter assumption narrows practical applications of the technology based on the ray approximation. 
\par    Attempts of studying  a X-ray tomography problem with purely Compton single scattering imply a problem of integral ge
ometry in which integration is produced over a cone, and certain investigations are based on the inevitability of an so-called conical Radon Transform.
\par     The approach connected with a transport equation allows one to consider radiation tomography problem
s under rather wide restrictions, particularly for any scattering. In this way the concrete settings of the tomography problems appear to be the inversion of Radon Transform in the case when integration is produced along a straight line, but the unknown i
ntegrand additionally depends on variables characterized all straight lines in a space.
\par     Thus, many various ways to tomography problems imply a such common problem as inversion of generalized Radon Transform and may be studied simultaneously.  
\par    However, a such general problem of integral geometry hardly can be solved completely. Thus, a following problem may be considered: \'93 How to determine a part of unknown information at least?\'94
 For example, a concrete version of such problem may be formulated as a question: \'93How to find a surface where the integrand is discontinuous?\'94
 Note that it is interpreted as a problem of flaw detection. This version of a problem of integral geometry has been successfully investigated. Particularly, the theorem of uniqueness is proved and the stable algorithm is created and tested.     
\par    This approach corresponds to general idea for determination of a part of unknown information with minimization of restrictions, as possible.  Perhaps, this way may occur fruitful in investigation of many inverse problems.  
\par 
\par 
\par 
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\par }}