X-reconstruction: 3D Tomographic Reconstruction in Astrophysics. I. Mathematical Formulation for Systems in the Optically Thin Regime

We present a novel methodology for reconstructing 3D structures of astronomical objects. Astronomical observations record radiation integrated along the line of sight, so intrinsically 3D structures are usually inferred only from 2D images on the plane of the sky. Recovering 3D structure from such data is therefore a central but difficult problem. In this first work of the series, we develop the foundations of X-reconstruction, a tomographic framework that recasts optically thin radiative transfer as a Radon-type integral and exploits axisymmetry to enable closed-form inversions. Starting from the radiative transfer equation, we show that, in the pure-emission and pure-absorption limits, the observables can be expressed as simple line-of-sight integrals of a volumetric source function, which we present in a unified form amenable to Radon and Abel analysis. For axisymmetric systems viewed edge-on, the 3D inverse problem can be solved in Fourier space using a Hankel transform combined with the projection-slice theorem. We derive analytical transforms for several basic axisymmetric elements and verify numerically that both a Fourier–Hankel–Abel cycle and a direct Abel inversion method accurately recover the volumetric source distributions. An example reconstruction of a bipolar-outflow-like configuration illustrates how realistic structures can be decomposed into and reconstructed from these analytic building blocks, opening a concrete path toward 3D inference from 2D data for astrophysical images of axisymmetric systems.