  Despite the wealth of $Planck$ results, there are difficulties in
disentangling the primordial non-Gaussianity of the Cosmic Microwave Background
(CMB) from the secondary and the foreground non-Gaussianity (NG). For each of
these forms of NG the lack of complete data introduces model-dependencies.
Aiming at detecting the NGs of the CMB temperature anisotropy $\delta T$, while
paying particular attention to a model-independent quantification of NGs, our
analysis is based upon statistical and morphological univariate descriptors,
respectively: the probability density function $P(\delta T)$, related to
${\mathrm v}_{0}$, the first Minkowski Functional (MF), and the two other MFs,
${\mathrm v}_{1}$ and ${\mathrm v}_{2}$. From their analytical Gaussian
predictions we build the discrepancy functions $\Delta_{k}$ ($k=P,0,1,2$) which
are applied to an ensemble of $10^{5}$ CMB realization maps of the $\Lambda$CDM
model and to the $Planck$ CMB maps. In our analysis we use general Hermite
expansions of the $\Delta_{k}$ up to the $12^{th}$ order, where the
coefficients are explicitly given in terms of cumulants. Assuming hierarchical
ordering of the cumulants, we obtain the perturbative expansions generalizing
the $2^{nd}$ order expansions of Matsubara to arbitrary order in the standard
deviation $\sigma_0$ for $P(\delta T)$ and ${\mathrm v}_0$, where the
perturbative expansion coefficients are explicitly given in terms of complete
Bell polynomials. The comparison of the Hermite expansions and the perturbative
expansions is performed for the $\Lambda$CDM map sample and the $Planck$ data.
We confirm the weak level of non-Gaussianity ($1$-$2$)$\sigma$ of the
foreground corrected masked $Planck$ $2015$ maps.
