  We present a novel procedure to estimate the Equation of State of the
intergalactic medium in the quasi-linear regime of structure formation based on
Ly$\alpha$ forest tomography and apply it to 21 high quality quasar spectra
from the UVES\_SQUAD survey at redshift $z=2.5$. Our estimation is based on a
full tomographic inversion of the line of sight. We invert the data with two
different inversion algorithms, the iterative Gauss-Newton method and the
regularized probability conservation approach, which depend on different priors
and compare the inversion results in flux space and in density space. In this
way our method combines fitting of absorption profiles in flux space with an
analysis of the recovered density distributions featuring prior knowledge of
the matter distribution. Our estimates are more precise than existing
estimates, in particular on small redshift bins. In particular, we model the
temperature-density relation with a power law and observe for the temperature
at mean density $T_0 = 13400^{+1700}_{-1300}\,\mathrm{K}$ and for the slope of
the power-law (polytropic index) $\gamma = 1.42 \pm 0.11$ for the power-law
parameters describing the temperature-density relation. Moreover, we measure an
photoionization rate $\Gamma_{-12} = 1.1^{+0.16}_{-0.17}$. An implementation of
the inversion techniques used will be made publicly available.
