Title: Isomorphism between Differential and Moment Invariants under Affine Transform,
Abstract: The invariant is one of central topics in science, technology and
engineering. The differential invariant is essential in understanding or
describing some important phenomena or procedures in mathematics, physics,
chemistry, biology or computer science etc. The derivation of differential
invariants is usually difficult or complicated. This paper reports a discovery
that under the affine transform, differential invariants have similar
structures with moment invariants up to a scalar function of transform
parameters. If moment invariants are known, relative differential invariants
can be obtained by the substitution of moments by derivatives with the same
order. Whereas moment invariants can be calculated by multiple integrals, this
method provides a simple way to derive differential invariants without the need
to resolve any equation system. Since the definition of moments on different
manifolds or in different dimension of spaces is well established, differential
invariants on or in them will also be well defined. Considering that moments
have a strong background in mathematics and physics, this technique offers a
new view angle to the inner structure of invariants. Projective differential
invariants can also be found in this way with a screening process.