A Study of Affine Matching With Bounded Sensor Error

Affine transformations of the plane have been used by modelbased recognition systems to approximate the effects of perspective projection. Because the underlying mathematics are based on exact data, in practice various heuristics are used to adapt the methods to real data where there is positional uncertainty. This paper provides a precise analysis of affine point matching under uncertainty. We obtain an expression for the range of affine-invariant values consistent with a given set of four points, where each data point lies in an ∃-disc. This range is shown to depend on the actual x- y-positions of the data points. Thus given uncertainty in the data, the representation is no longer invariant with respect to the Cartesian coordinate system. This is problematic for methods, such as geometric hashing, that depend on the invariant properties of the representation. We also analyze the effect that uncertainty has on the probability that recognition methods using affine transformations will find false positive matches. We find that such methods will produce false positives with even moderate levels of sensor error.