= Solution
A <CT number> is a dimensionless relative attenuation measure, conventionally $H=1000(\mu_{\rm CT}-\mu_{{\rm water},\rm CT})/\mu_{{\rm water},\rm CT}$. The <X-ray> spectrum used in <computed tomography> has an effective energy far below $511\,\mathrm{keV}$ and is polychromatic. Its <photoelectric absorption> contribution depends strongly on composition, whereas water and soft-tissue attenuation at the <positron emission tomography> energy is mainly <Compton scattering>. Thus a <CT number> is neither a $511\,\mathrm{keV}$ <linear attenuation coefficient> nor an energy-independent <mass attenuation coefficient>.
The usual conversion calibrates a piecewise linear or bilinear relation between <CT numbers> and $\mu(511\,\mathrm{keV})$. The air–water branch models predominantly density differences in lung and soft tissue; a separate water–bone branch accounts for the different energy dependence of mineralized tissue. The calibration depends on the <computed tomography> spectrum. After geometric registration, the resulting <linear attenuation coefficient> map is forward projected along each <coincidence line of response> to obtain $\exp(\int\mu_{511}\,ds)$. If a mass coefficient is stored, its conversion to a linear coefficient must also include <mass density>. Unusual high-atomic-number materials can violate the simple tissue calibration, so the conversion is a physical material model, not a universal arithmetic rescaling.<CT-based PET attenuation correction study (1998)>[original study: segmentation, scaling and hybrid conversion from CT energies to 511 keV]{ref}<CT-to-PET bone attenuation calibration study (2012)>[Materials and Methods: piecewise conversion and air–water versus water–bone mixture branches]{ref}
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