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In this paper, the stationary photon transport equation has been extended by analytical continuation fromℝ  3{\mathbb{R}^{3}}    toℂ  3{\mathbb{C}^{3}}   . A solution to the inverse problem posed by this equation is obtained on a hyper-sphere and a hyper-cylinder as X-ray and Radon transforms, respectively. We show that these results can be transformed into each other, and they agree with known results. Numerical reconstructions of a three-dimensional Shepp–Logan head phantom using the obtained inverse formula illustrate the analytical results obtained in this manuscript.

An inversion formula for the transport equation in ℝ 3 using complex analysis in several variables