Object Recognition Based on Moment (or Algebraic) Invariants

Toward the development of an object recognition and positioning system, able to deal with arbitrary shaped objects in cluttered environments, we introduce methods for checking the match of two arbitrary curves in 2D or surfaces in 3D, when each of these subobjects (i.e., regions) is in arbitrary position, and we also show how to efficiently compute explicit expressions for the coordinate transformation which makes two matching subobjects (i.e., regions) coincide. This is to be used for comparing an arbitrarily positioned subobject of sensed data with objects in a data base, where each stored object is described in some “standard” position. In both cases, matching and positioning, results are invariant with respect to viewer coordinate system, i.e., invariant to the arbitrary location and orientation of the object in the data set, or, more generally, to affine transformations of the objects in the data set, which means translation, rotation, and different stretchings in two (or three) directions, and these techniques apply to both 2D and 3D problems. The 3D Euclidean case is useful for the recognition and positioning of solid objects from range data, and the 2D affine case for the recognition and positioning of solid objects from projections, e.g., from curves in a single image, and in motion estimation. The matching of arbitrarily shaped regions is done by computing for each region a vector of centered moments. These vectors are viewpointdependent, but the dependence on the viewpoint is algebraic and well known. We then compute moment invariants, i.e., algebraic functions of the moments that are invariant to Euclidean or affine transformations of the data set. We present a new family of computationally efficient algorithms, based on matrix computations, for the evaluation of both Euclidean and affine algebraic moment invariants of data sets. The use of moment invariants greatly reduces the computation required for the matching, and hence initial object recognition. The approach to determining and computing these moment invariants is different than those used by the vision community previously. The method for computing the coordinate transformation which makes the two matching regions coincide provides an estimate of object position. The estimation of the matching transformation is based on the same matrix computation techniques introduced for the computation of invariants, it involves simple manipulations of the moment vectors, it neither requires costly iterative methods, nor going back to the data set. The use of geometric invariants in this application is equivalent to specifying a center and an orientation for an arbitrary data constellation in a region. These geometric invariant methods appear to be very important for dealing with the situation of a large number of different possible objects in the presence of occlusion and clutter. As we point out in this paper, each moment invariant also defines an algebraic invariant, i.e., an invariant algebraic function of the coefficients of the best fitting polynomial to the data. Hence, this paper also introduces a new design and computation approach to algebraic invariants.