Inversion of the attenuated X-ray transforms: method of Riesz potentials (Q2173550)

Summary: The attenuated X-ray transform arises from the image reconstruction in single-photon emission computed tomography. The theory of attenuated X-ray transforms is so far incomplete, and many questions remain open. This paper is devoted to the inversion of the attenuated X-ray transforms with nonnegative varying attenuation functions \(\mu\), integrable on any straight line of the plane. By constructing the symmetric attenuated X-ray transform \(A_\mu\) on the plane and using the method of Riesz potentials, we obtain the inversion formula of the attenuated X-ray transforms on \(L^p(\mathbb{R}^2)$ $(1\leq p <2)\) space, with nonnegative attenuation functions \(\mu\), integrable on any straight line in \(\mathbb{R}^2\). These results are succinct and may be used in the type of computerized tomography with attenuation.