Abstract

Affine transformations of the plane have been used in a number of model-based recognition systems, in order to approximate the effects of perspective projection. The mathematics underlying these methods is for exact data, where there is no positional uncertainty in the measurement of feature points. In practice, various heuristics are used to adapt the methods to real data with uncertainty. In this paper, we provide a precise analysis of affine point matching under uncertainty. We obtain an expression for the range of affine-invariant values that are consistent with a given set of four points, where each data point lies in a disk of radius e. This analysis reveals that the range of affine-invarint values depends on the actual x-y-positions of the data points. That is, when there is uncertainty in the data then the representation is no longer invariant with respect to the Cartesian coordinate system. This is problematic for the geometric hashing method, because it means that the precomputed lookup table used by that method is not correct when there is positional uncertainty in the sensor data. We analyze the effect that this has on the probability that the geometric hashing method will find false positive matches of a model to an image, and contrast this with a similar analysis of the alignment method.

Citations