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Guillaume Bal studied reconstruction from the attenuated Radon transform with full and partial measurements. His analysis distinguishes complete-data inversion from partial-angle reconstruction requiring additional conditions. The work uses a Riemann-Hilbert problem and operator decompositions, showing how incomplete ray measurements can still determine a source when attenuation and admissible functions are suitably controlled.^[ref]

For known attenuation $a(x)$ and a scalar source $f(x)$ in the plane, start with the directional transport equation $(\mathbf{\theta }\cdot \nabla +a)\psi =f$. Introduce a complex spectral parameter $k$ and Wirtinger derivatives ${\partial }_z=({\partial }_{x_1}-i{\partial }_{x_2})/2$, ${\partial }_{\bar{z}}=({\partial }_{x_1}+i{\partial }_{x_2})/2$: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 $L_k{h}_k=a$ 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.^[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.