= Solution
In ideal <computed tomography>, the unknown is an absorption coefficient $a(x)$. Along a straight ray the transmitted intensity satisfies $dI/d\tau=-aI$, so
$$
\log\frac{I_{\mathrm{in}}}{I_{\mathrm{out}}}=\int_{\mathbb R}a(\tau\boldsymbol d+\rho\boldsymbol n)d\tau=Ra(\rho,\theta).
$$
Thus 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.
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