Inversion of the two-data circular Radon transform centered on a curve on C ( R 2 ) ${\cal C}(\mathbf {R}^2)$

Abstract

More often, in the mathematical literature, the injectivity of the spherical Radon transform (SRT) for compactly supported functions is considered. In this article, an additional condition, for the reconstruction of an unknown function $f\in C\left(R^2\right)$ (the support can be noncompact) using the circular Radon transform (CRT) over circles centered on a smooth simple curve is found. It is proved that this problem is equivalent to the injectivity of a so-called two-data CRT over circles centered on a smooth curve (can be a segment). Also, we present an inversion formula of the transform that uses the local data of the circular integrals to reconstruct the unknown function. Such inversions are the mathematical base of modern modalities of imaging, such as thermo- and photoacoustic tomography and radar imaging, and have theoretical significance.