Complete Moment Invariants and Pose Determination for Orthogonal Transformations of 3D Objects

It is well known that for the simpler problem of constructing translation invariants of grey scale images (1D, 2D or 3D) central moments can be used. There are plain closed formulae expressing them in terms of the ordinary geometrical moments. Moreover, central moments are ordinary moments of the properly normalized image. In this paper we present moment invariants for the more involved problem of rotations and re ections of 3D density objects, having exactly the same qualities as those mentioned above of central moments. The mathematical analysis of this problem is complicated mainly due to noncommutativity of the group of 3D rotations SO(3). However, by constructing basis functions using harmonic polynomials, rather than monomials, we achieve a decomposition of the action of SO(3) in irreducible representations acting on invariant subspaces, thus simplifying the problem. Using a suitable generating function for harmonic polynomials we work out a novel and very compact description of these subspaces. In addition, we introduce the notion of \spherical moments" denoting inner products of the basis functions with an object, and we encode them using the same generating function. In conjunction with the Cayley-Klein parameterization of SO(3) we obtain a simple relationship between the encoded spherical moments of two rotated/re ected versions of a 3D object. This relationship enables us to express the spherical moments of a uniquely normalized object in terms of the spherical moments of the not normalized (actual) object, just as we can express central moments in terms of ordinary moments. In doing so we don't lose any information and since moments uniquely characterize an object with compact support we see that we have constructed complete moment invariants. The normalization process itself is carried out using exclusively moments of third order and yields at the same time unique pose determination.