  The accurate interpretation of experiments with matter at extreme densities
and pressures is a notoriously difficult challenge. In a recent work
[T.~Dornheim et al., Nature Comm. (in print), arXiv:2206.12805], we have
introduced a formally exact methodology that allows extracting the temperature
of arbitrarily complex materials without any model assumptions or simulations.
Here, we provide a more detailed introduction to this approach and analyze the
impact of experimental noise on the extracted temperatures. In particular, we
extensively apply our method both to synthetic scattering data and to previous
experimental measurements over a broad range of temperatures and wave numbers.
We expect that our approach will be of high interest to a gamut of
applications, including inertial confinement fusion, laboratory astrophysics,
and the compilation of highly accurate equation-of-state databases.
