Spectral reconstruction of an attenuated Radon transform ($L_k=k{\partial }_z+k^{-1}{\partial }_{\bar{z}}$)

Complexifying the directional transport equation yields spectral solutions analytic inside and outside the unit circle. An integrating factor built from known attenuation removes its multiplication term. The limiting Green kernels encode weighted line measurements and transverse Hilbert transforms. Their jump gives a Riemann-Hilbert problem; its solution recovers the source from an asymptotic coefficient. A general primary treatment of this spectral construction is www.stat.uchicago.edu/~guillaumebal/PAPERS/AttRadTh.pdf .

Let $F(\rho ,\tau ,\theta )={\int }_{\tau }^{\infty }f(s\mathbf{d}+\rho \mathbf{n})ds$, $S=\mathcal{R}f$, and $A={\int }_{\mathbb{R}}{e}^{-F}gd\tau$ be the directed attenuated Radon transform. With the signed Cauchy boundary operators, set $J=e^F[e^{P^{-}S}{P}^{-}(e^{-P^{-}S}A)+e^{-P^{+}S}{P}^{+}(e^{P^{+}S}A)]$. An integrating factor turns the attenuated equation into unattenuated directional transport; its spectral solution has jump $-J$ on the unit circle. A normalized Cauchy integral formula and the coefficient at spectral infinity then prove the reconstruction formula. Known attenuation and full directed angular data determine the source; attenuation itself is not reconstructed by this pair.

For $w=x_1+i{x}_2$ and $0<|k|\ne 1$, the operator $D_k=(k+k^{-1}){\partial }_{x_1}/2+(k-k^{-1}){\partial }_{x_2}/(2i)$ has the displayed decaying fundamental solution. A rotated real-linear coordinate turns it into a multiple of a Wirtinger derivative, reducing the kernel identity to ${\partial }_{\bar{z}}(1/(\pi z))=\delta$. The kernel is analytic in $k$ separately inside and outside the unit circle, is $O(k)$ at zero and $O(k^{-1})$ at infinity, and develops a jump at the real directional-transport limit.