  The aim of this paper is to clear up the problem of the connection between
the 3D geometric moments invariants and the invariant theory, considering a
problem of describing of the 3D geometric moments invariants as a problem of
the classical invariant theory. Using the remarkable fact that the groups
$SO(3)$ and $SL(2)$ are locally isomorphic, we reduced the problem of deriving
3D geometric moments invariants to the well-known problem of the classical
invariant theory. We give a precise statement of the 3D geometric invariant
moments computation, introducing the notions of the algebras of simultaneous 3D
geometric moment invariants, and prove that they are isomorphic to the algebras
of joint $SL(2)$-invariants of several binary forms. To simplify the
calculating of the invariants we proceed from an action of Lie group $SO(3)$ to
an action of its Lie algebra $\mathfrak{sl}_2$. The author hopes that the
results will be useful to the researchers in the fields of image analysis and
pattern recognition.
