Abstract
Affine transformations of the plane have been used in a number of model-based recognition systems. 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 that are consistent with a given set of four points, where each image point lies in an ∈-disc of uncertainty. This range is shown to depend on the actualx-y-positions of the data points. In other words, given uncertainty in the data there are no representations that are invariant with respect to the Cartesian coordinate system of the data. This is problematic for methods, such as geometric hashing, that are based on affine-invariant representations. 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 there is a significant probability of false positives with even moderate levels of sensor error, suggesting the importance of good verification techniques and good grouping techniques.
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This report describes research done in part at the Artificial Intelligence Laboratory of the Massachusetts Institute of Technology. Support for the laboratory's research is provided in part by an ONR URI grant under contract N00014-86-K-0685, and in part by DARPA under Army contract number DACA76-85-C-0010 and under ONR contracts N00014-85-K-0124 and N00014-91-J-4038. WELG is supported in part by NSF contract number IRI-8900267. DPH is supported at Cornell University in part by NSF grant IRI-9057928 and matching funds from General Electric and Xerox, and in part by AFOSR under contract AFOSR-91-0328.
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Grimson, W.E.L., Huttenlocher, D.P. & Jacobs, D.W. A study of affine matching with bounded sensor error. Int J Comput Vision 13, 7–32 (1994). https://doi.org/10.1007/BF01420793
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DOI: https://doi.org/10.1007/BF01420793