Abstract
It is known that in the class of all continuous functions, a line is not a set of injectivity of the circular Radon transform (CRT). The problem arises: to find an additional condition that allows one to reconstruct an unknown function f∈C(R2) (not necessarily with compact support) using the circular Radon transform (CRT) with detectors placed on a line. In this article an additional condition is found. This problem is equivalent to the injectivity of a transform that consist of CRT and the weighted CRT. Also, in this article, a new iterative inversion formula for the transform is presented. This formula allows the reconstruction of an unknown function by local data. Such inversions are of theoretical importance in many areas of mathematics and are required in mathematical models of Thermo and photo-acoustic tomography, radar imaging, and others.
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