  We present an analysis of the morphological classification of 89 luminous
infrared galaxies (LIRGs) from the Great Observatories All-sky LIRG Survey
(GOALS) sample using non-parametric coefficients and compare their morphology
as a function of wavelength. We rely on images obtained in the optical (B- and
I-band) as well as in the infrared (H-band and 5.8$\mu$m). Our classification
is based on the calculation of $Gini$ and the second order of light ($M_{20}$)
non-parametric coefficients which we explore as a function of stellar mass
($M_\star$), infrared luminosity ($L_{IR}$) and star formation rate (SFR). We
investigate the relation between $M_{20}$, the specific SFR (sSFR) and the dust
temperature ($T_{dust}$) in our galaxy sample. We find that $M_{20}$ is a
better morphological tracer than $Gini$, as it allows to distinguish systems
formed by double systems from isolated and post-merger LIRGs. The
multi-wavelength analysis allows us to identify a region in the $Gini$-$M_{20}$
parameter space where ongoing mergers reside, regardless of the band used to
calculate the coefficients. In particular when measured in the H-band, this
region can be used to identify ongoing mergers, with a minimal contamination
from LIRGs in other stages. We also find that while the sSFR is positively
correlated with $M_{20}$ when measured in the mid-infrared, i.e. star-bursting
galaxies show more compact emission, it is anti-correlated with the B-band
based $M_{20}$. We interpret this as the spatial decoupling between obscured
and un-obscured star formation, whereby the ultraviolet/optical size of a LIRGs
experience an intense dust enshrouded central starburst is larger than in the
one in the mid-infrared since the contrast between the nuclear to the extended
disk emission is smaller in the mid-infrared. This has important implications
for high redshift surveys of dusty sources. [abridged]
