 Filtered backprojection

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.