Isomorphism between Differential and Moment Invariants under Affine Transform

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.