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.

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.