The identification problem for the attenuated X-ray transform (Q2927714)

The attenuated X-ray transform given by NEWLINE\[NEWLINEX_af(x,\theta)=\int e^{-Ba(x+t\theta,\theta)}f(x+t\theta)\mathrm dt,\quad x\in\mathbb R^2,\quad \theta\in S^1,\tag{1.1}NEWLINE\]NEWLINE in the plane with a source \(f\) and the attenuation \(a\) is studied for recovery problem. Starting from the attenuated X-ray transform, given by (1.1), the linearization of the identification problem is computed, which can also be computed by transport equation. The singularities of g can be recovered from the non-trapping for the Hamiltonian flow, by considering microlocal point. The actual injectivity and stability of I are then proved. The uniqueness and non-uniqueness for the sufficient and explicit sufficient conditions are proved with certain analyticity conditions. Analyzing microlocal consequences and the study of radial \(a\) and \(f\), the linearization mapping for an infinite dimensional, uniqueness for radial cases are proved.