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Transmission measurements determine line integrals of an absorption coefficient after taking the logarithm of incident intensity divided by transmitted intensity. Under the ideal straight-ray model, these measurements are a Radon transform; filtered backprojection reconstructs a planar slice. Angular coverage, calibration and noise control matter because differentiating measured projections amplifies high frequencies. In contrast, single-photon emission computed tomography reconstructs an emitting source through a possibly attenuating medium.

The two boundary values have jumpThe line-tail contributions cancel. Because the complexified transport Cauchy kernel is holomorphic on both sides of the unit circle and vanishes at spectral infinity, the Cauchy integral formula reconstructs it from this jump:There is no additional entire term: its inside and outside boundary values would agree, so it would define an entire function vanishing at infinity. Expand outside the circle as $\mu ={\mu }_1/\lambda +O({\lambda }^{-2})$. ThenIn the original coordinate $w=x_1+i{x}_2$, the transport equation is $(\lambda {\partial }_w+{\lambda }^{-1}{\partial }_{\bar{w}})\mu =f$. Its constant term at infinity is ${\partial }_w{\mu }_1=f$. Sincewe obtain the inverse Radon transform in filtered backprojection form:Opposite directions represent the same unoriented line. Reversing $\theta$ by $\pi$ reverses $\rho$; the Hilbert transform changes sign under this reversal, and its derivative changes it back. Thus the integrand agrees for opposite directions and the factor becomes $1/(2\pi )$ if the angular integral is over $[0,\pi )$. With the convention $\hat{h}(\kappa )=\int e^{-i\kappa \rho }h(\rho )d\rho$, the Hilbert transform has Fourier multiplier $-i\mathrm{sgn}\kappa$; the derivative makes the filter $|\kappa |$. This also identifies the formula with the standard ramp-filter reconstruction.

In ideal computed tomography, the unknown is an absorption coefficient $a(x)$. Along a straight ray the transmitted intensity satisfies $dI/d\tau =-aI$, soThus calibrated transmission measurements supply the Radon transform of the slice. The inversion above reconstructs the coefficient by filtered backprojection: filter each projection in its transverse variable and then accumulate it over the rays through each image point. This turns a collection of line-integral measurements into a spatial map, which is the key usefulness of the result.

The ideal derivation assumes known ray geometry, sufficiently complete angular sampling and a straight-ray attenuation model. The ramp Fourier multiplier grows at high frequency, so measurement noise is amplified. Practical reconstruction therefore uses discretization and regularization of an inverse problem, for example a window on the ramp filter. Limited angles or missing detector positions cannot be treated as if they were complete noiseless Radon transform data.