= Filtered backprojection
{title2=$f(x)=(4\pi)^{-1}\int_0^{2\pi}(\partial_\rho H\mathcal Rf)(x\cdot n_\theta,\theta)d\theta$}
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 $Hh(\rho)=\pi^{-1}\operatorname{PV}\int h(r)/(\rho-r)dr$, 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>.
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