On the inverse problem of rotation moment invariants

Numerous papers have been devoted to the moment invariants (MI). They usually describe their derivation, various invariant properties, numerical behavior, and their power to serve as the features in many pattern recognition tasks. Only few papers have dealt with the independence and completeness of the sets of MI’s. However, both these properties are of fundamental importance from theoretical as well as practical point of view. In our recent paper [1] we presented a general approach to deriving rotation moment invariants. MI’s were constructed as products of appropriate powers of complex moments. We also proposed how to construct the basis (independent and complete set) of the invariants of this kind. In this paper we prove stronger theorem. We show the basis described in Ref. [1] is a basis of all possible rotation moment invariants (not only of those constructed as products of moment powers). In other words, knowing the invariants from this basis we can calculate for instance traditional Hu moment invariants [2], Zernike moment invariants [3], and Fourier–Mellin invariants [4], to name a few. This theorem can be equivalently formulated as the solution of the inverse problem. We show in this paper that all moments can be recovered from the invariant basis.