Recognition of Planar Objects Using the Density of Affine Shape

In this paper, we will study the recognition problem for finite point configurations, in a statistical manner. We study the statistical theory of shape for ordered finite point configurations, or otherwise stated, the uncertainty of geometric invariants. Here, a general approach for defining shape and finding its density, expressed in the densities for the individual points, is developed. No approximations are made, resulting in an exact expression of the uncertainty region. In particular, we will concentrate on the affine shape, where often analytical computations is possible. In this case confidence intervals for invariants can be obtained from a priori assumptions on the densities of the detected points in the images. However, the theory is completely general and can be used to compute the density of any invariant (Euclidean, affine, similarity, projective, etc.) from arbitrary densities of the individual points. These confidence intervals can be used in such applications as geometrical hashing, recognition of ordered point configurations, and error analysis of reconstruction algorithms. Finally, an example will be given, illustrating the theory for the problem of recognizing planar point configurations from images taken by an affine camera. This case is of particular importance in applications, where details on a conveyor belt are captured by a camera, with image plane parallel to the conveyor belt and extracted feature points from the images are used to sort the objects.