= Spectral reconstruction of an attenuated Radon transform
{title2=$L_k=k\partial_z+k^{-1}\partial_{\bar z}$}
For known attenuation $a(x)$ and a scalar source $f(x)$ in the plane, start with the directional <transport equation> $(\boldsymbol\theta\cdot\nabla+a)\psi=f$. Introduce a complex spectral parameter $k$ and <Wirtinger derivatives> $\partial_z=(\partial_{x_1}-i\partial_{x_2})/2$, $\partial_{\bar z}=(\partial_{x_1}+i\partial_{x_2})/2$:
$$
L_k=k\partial_z+k^{-1}\partial_{\bar z}.
$$
At $k=e^{i\theta}$ this is the original directional <derivative> $\cos\theta\,\partial_{x_1}+\sin\theta\,\partial_{x_2}$. Away from the unit circle, its <fundamental solution> supplies analytic spectral families. Choosing $h_k$ with $L_kh_k=a$ turns $(L_k+a)\psi_k=f$ into $L_k(e^{h_k}\psi_k)=e^{h_k}f$, removing attenuation by an <integrating factor>.<On the attenuated Radon transform with full and partial measurements by Guillaume Bal>[§§2.1–2.2: transport equation and spectral Riemann-Hilbert construction]{ref}
The boundary values from inside and outside the unit circle have a jump determined by directed <attenuated Radon transform> data and transverse <Hilbert transforms>. A normalized <Riemann-Hilbert problem> reconstructs the spectral family. In the exterior normalization $\psi_k=\psi_{-1}/k+O(k^{-2})$, substituting into the transformed equation gives $f=\partial_z\psi_{-1}$. Thus the desired source is recovered from a spectral coefficient, rather than from a direct angular average.
This construction assumes sufficient regularity and decay, known attenuation and the required directed angular measurements. General angle-dependent sources have a different identifiable content; partial angular data require additional hypotheses. The <projection formula for inverse attenuated Radon transform> makes one full-data normalization explicit.
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