In this paper, 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 in e.g. [GHJ92], [Wer93], where in the former case a bound on errors are used instead of errors described by density functions, and in the latter case 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, see [May95,åst96].
In particular, we will 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. Another approach towards this problem, in the case of similarity transformations, can be found in [Ken89]. For the special case of normally distributed feature points in a plane and similarity transformations, see [BOO86], [MD89].
Finally, an example will be 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.
- error analysis
- distribution of invariants