There is currently a considerable interest in methods of invariant 3D image recognition. Indeed, very often information about 3D objects can be obtained by computer tomographic reconstruction, 3D magnetic resonance imaging, passive 3D sensors or active range finders. Due to that algorithms of systematic derivation of 3D moment invariants should he developed for 3D colour object recognition. In this work we proposed an elegant theory which allows to describe many such invariants. Our theory is based on the theory of triplet numbers and quaternions. We propose triplet-quaternion-valued invariants, which are related to the descriptions of objects as the zero sets of implicit polynomials. These are global invariants which show great promise for recognition of complicated objects. Triplet-quaternion-valued invariants have good discriminating power for computer recognition of 3D colour objects using statistical pattern recognition methods. For fast computation of triplet-quaternion-valued invariants we use modular arithmetic of Galois fields and rings, which maps calculation of invariants to fast number theoretical Fourier-Galois-Hamilton-transform.