Abstract:
It is known that a smooth surface (even it is the boundary of a convex body) is a set of non-injectivity of the spherical Radon transform (SRT) in the space of continuous functions in $\mathbb{R}^{d}$. In this article, for the reconstruction of a function $f$ defined on $\mathbb{R}^3$ (the support can be non-compact), using the spherical Radon transform over spheres centered on a spherical domain, the injectivity of the so-called two-data spherical Radon transform is considered. An inversion formula of the transform that uses the local data of the spherical integrals to reconstruct the unknown function is presented. Such inversions are the mathematical base of the thermo- and photoacoustic tomography and radar imaging, and have theoretical significance in many areas of mathematics.
Key words and phrases:spherical Radon transform, inverse problems, integral transform, thermo-acoustic tomography.
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