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- Title
RECONSTRUCTION OF A FUNCTION DEFINED ON R<sup>2</sup> FROM ITS CIRCULAR TRANSFORMS, CENTERED ON AN ARC.
- Authors
Aramyan, Rafik
- Abstract
The problem of complete characterization of injectivity sets of the circular Radon transform on the space of compactly supported continuous functions is solved only for two-dimensional case. In particular, any small curve in the plane which is not contained in a straight line is an injectivity set. The problem becomes more complicated without compactness of support condition. The problem arises to find an additional condition that allows one to reconstruct an unknown function f ∈ C (R 2) (not necessarily with compact support) using the circular Radon transform (CRT) with detectors placed on an arc. In this article, an additional condition is found. This problem is equivalent to the injectivity of a transform that consists of CRT and 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 using the local data of the circular integrals of the function over circles centered on an arc. Such inversions are of theoretical importance in many areas of mathematics and are required in the mathematical models of thermo and photoacoustic tomography, radar imaging, and others.
- Subjects
TRIGONOMETRIC functions; RADON transforms; INTEGRAL functions; PLANE curves; CONTINUOUS functions; MATHEMATICAL models
- Publication
Journal of Mathematical Sciences, 2024, Vol 280, Issue 1, p98
- ISSN
1072-3374
- Publication type
Article
- DOI
10.1007/s10958-023-06748-9