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null (Ed.)In two dimensions, we consider the problem of inversion of the attenuated \begin{document}$ X $$\end{document}-ray transform of a compactly supported function from data restricted to lines leaning on a given arc. We provide a method to reconstruct the function on the convex hull of this arc. The attenuation is assumed known. The method of proof uses the Hilbert transform associated with \begin{document}$$ A $$\end{document}$-analytic functions in the sense of Bukhgeim.more » « less
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This paper presents a new algorithm for X-ray Computerized Tomography (CT) based on Bukhgeim’s theory of analytic maps. The reconstruction relies on a Cauchy-type integral formula, where the integration over the boundary replaces the integration in the back- projection operator used in existing algorithms. From the numerical computation stand point, the proposed method recovers the attenuation coefficient at arbitrarily points by utilizing the boundary integration without internal global meshes. This means that it achieves high-parallel efficiency, and it reduces computational resources. Some numerical examples are presented to show feasibility of the proposed algorithm.more » « less
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This work concerns the numerical realization of a Cauchy-type integral formula for sequence valued analytic functions in the sense of Bukhgeim, and its applications to the source reconstruction problem in inverse radiative transport through a non-absorbing and non-scattering medium. The inverse source problem is mathematically equivalent to the classical X-ray Computed Tomography (CT), where a function is to be determined from its line integrals. The proposed algorithms have the added advantage to extend to the source determination problems in media with absorbing and scattering properties. Such extensions cannot be achieved in the existing X-ray CT algorithms. The numerical experiments demonstrate the feasibility of our new tomographic algorithms.more » « less
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Abstract We revisit the inverse source problem in a two dimensional absorbing and scattering medium and present a direct reconstruction method, which does not require iterative solvability of the forward problem, using measurements of the radiating flux at the boundary. The attenuation and scattering coefficients are known and the unknown source is isotropic. The approach is based on the Cauchy problem for a Beltrami-like equation for the sequence valued maps, and extends the original ideas of Bukhgeim from the non-scattering to the scattering media. We demonstrate the feasibility of the method in a numerical experiment in which the scattering is modeled by the two dimensional Henyey–Greenstein kernel with parameters meaningful in optical tomography.more » « less
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