  Geometric moment invariants (GMIs) have been widely used as basic tool in
shape analysis and information retrieval. Their structure and characteristics
determine efficiency and effectiveness. Two fundamental building blocks or
generating functions (GFs) for invariants are discovered, which are dot product
and vector product of point vectors in Euclidean space. The primitive
invariants (PIs) can be derived by carefully selecting different products of
GFs and calculating the corresponding multiple integrals, which translates
polynomials of coordinates of point vectors into geometric moments. Then the
invariants themselves are expressed in the form of product of moments. This
procedure is just like DNA encoding proteins. All GMIs available in the
literature can be decomposed into linear combinations of PIs. This paper shows
that Hu's seven well known GMIs in computer vision have a more deep structure,
which can be further divided into combination of simpler PIs. In practical
uses, low order independent GMIs are of particular interest. In this paper, a
set of PIs for similarity transformation and affine transformation in 2D are
presented, which are simpler to use, and some of which are newly reported. The
discovery of the two generating functions provides a new perspective of better
understanding shapes in 2D and 3D Euclidean spaces, and the method proposed can
be further extended to higher dimensional spaces and different manifolds, such
as curves, surfaces and so on.
