Radon transform ($\mathcal{R}f(s,\omega )$)

The Radon transform integrates a function over affine hyperplanes:

The Radon transform records line integrals of a planar density. Apply the transverse Hilbert transform and a derivative to each projection, then average the filtered values over all lines through the reconstruction point. With $Hh(\rho )={\pi }^{-1}\mathrm{PV}\int h(r)/(\rho -r)dr$, the filter has Fourier multiplier $|\kappa |$. The Fourier inversion theorem, or the jump of the complexified transport Cauchy kernel, proves the displayed formula. The factor is $1/(2\pi )$ when directions are integrated over a half-circle. The growing Fourier multiplier explains amplification of measurement noise and the need for regularization of an inverse problem.

The integral of a source over a directed line, weighted by exponential absorption between each source point and a detector. For line coordinates $x=p\mathbf{n}+s\mathbf{d}$ and a detector at the negative end, it is $\int f(p,s)\exp [-{\int }_{-\infty }^s\mu (p,r)dr]ds$. The integrating factor of a transport equation derives the weight. Reversing the detector direction changes the weight for spatially varying attenuation, so the two directed measurements need not coincide. Setting $\mu =0$ recovers the Radon transform.

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.

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.