Recognition of Planar Point Configurations using the Density of Affine Shape

We study the statistical theory of shape for ordered finite point configurations, or otherwise stated, the uncertainty of geometric invariants. Such studies have been made for affine invariants, where a bound on errors is used instead of errors described by density functions, and a first-order approximation gives an ellipsis as uncertainty region. 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. Similar results have been obtained for the special case of the density of the cross ratio. In particular, we concentrate on the affine shape, where often analytical computations are 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, 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 is given, illustrating an application of the theory for the problem of recognising 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