Abstract
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.
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 contract N00014-85-K-0124. 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 Kodak, and in part by AFOSR under contract AFOSR-91-0328.
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References
Ballard, D.H., 1981, “Generalizing the Hough Transform to Detect Arbitrary Patterns,” Pattern Recognition 13(2): 111–122.
Basri, R. & S. Ullman, 1988, “The Alignment of Objects with Smooth Surfaces,” Second Int. Conf. Comp. Vision, 482–488.
Besl, P.J. & R.C. Jain, 1985, “Three-dimensional Object Recognition,” ACM Computing Surveys, 17(1):75–154.
Costa, M., R.M. Haralick & L.G. Shapiro, 1990, “Optimal Affine-Invariant Point Matching,” Proc. 6th Israel Conf. on AI, pp. 35–61.
Cyganski, D. & J.A. Orr, 1985, “Applications of Tensor Theory to Object Recognition and Orientation Determination”, IEEE Trans. PAMI, 7(6):662–673.
Efimov, N.V., 1980, Higher Geometry, translated by P.C. Sinha. Mir Publishers, Moscow.
Ellis, R.E., 1989, “Uncertainty Estimates for Polyhedral Object Recognition,” IEEE Int. Conf. Rob. Aut., pp. 348–353.
Forsythe, D., J.L. Mundy, A. Zisserman, C. Coelho, A. Heller, & C. Rothwell, 1991, “Invariant Descriptors for 3-D Object Recognition and Pose”, IEEE Trans. PAMI, 13(10):971–991.
Grimson, W.E.L., 1990, Object Recognition by Computer: The role of geometric constraints, MIT Press, Cambridge.
Grimson, W.E.L., 1990, “The Combinatorics of Heuristic Search Termination for Object Recognition in Cluttered Environments,” First Europ. Conf. on Comp. Vis., pp. 552–556.
Grimson, W.E.L. & D.P. Huttenlocher, 1990, “On the Sensitivity of the Hough Transform for Object Recognition,” IEEE Trans. PAMI 12(3):255–274.
Grimson, W.E.L. & D.P. Huttenlocher, 1991, “On the Verification of Hypothesized Matches in Model-Based Recognition”, IEEE Trans. PAMI 13(12):1201–1213.
Grimson, W.E.L. & D.P. Huttenlocher, 1990, “On the Sensitivity of Geometric Hashing”, Proc. Third Int. Conf. Comp. Vision, pp. 334–338.
Grimson, W.E.L., D.P. Huttenlocher, & D.W. Jacobs, 1991, “Affine Matching With Bounded Sensor Error: A Study of Geometric Hashing & Alignment,” MIT AI Lab Memo 1250.
Huttenlocher, D.P. and S. Ullman, 1987, “Object Recognition Using Alignment”, Proc. First Int. Conf. Comp. Vision, pp. 102–111.
Huttenlocher, D.P. & S. Ullman, 1990, “Recognizing Solid Objects by Alignment with an Image,” Inter. Journ. Comp. Vision 5(2):195–212.
Jacobs, D., 1991, “Optimal Matching of Planar Models in 3D Scenes,” IEEE Conf. Comp. Vis. and Patt. Recog. pp. 269–274.
Klein, F., 1939, Elementary Mathematics from an Advanced Standpoint: Geometry, MacMillan, New York.
Korn, G.A. & T.M. Korn, 1968, Mathematical Handbook for Scientists and Engineers, McGraw-Hill, New York.
Lamdan, Y., J.T. Schwartz & H.J. Wolfson, 1988, “Object Recognition by Affine Invariant Matching,” IEEE Conf. Comp. Vis. and Patt. Recog. pp. 335–344.
Lamdan, Y., J.T. Schwartz & H.J. Wolfson, 1990, “Affine Invariant Model-Based Object Recognition,” IEEE Trans. Rob. Aut., vol. 6, pp. 578–589.
Lamdan, Y. & H.J. Wolfson, 1988, “Geometric Hashing: A General and Efficient Model-Based Recognition Scheme,” Second Int. Conf. Comp. Vis. pp. 238–249.
Lamdan, Y. & H.J. Wolfson, 1991, “On the Error Analysis of ‘Geometric Hashing',” IEEE Conf. Comp. Vis. and Patt. Recog. pp. 22–27.
Thompson, D. & J.L. Mundy, 1987, “Three-Dimensional Model Matching From an Unconstrained Viewpoint”, Proc. IEEE Conf. Rob. Aut. pp. 280.
Van Gool, L., P. Kempenaers & A. Oosterlinck, 1991, “Recognition and Semi-Differential Invariants,” IEEE Conf. Comp. Vis. and Patt. Recog. pp. 454–460.
Wayner, P.C., 1991, “Efficiently Using Invariant Theory for Model-based Matching,” IEEE Conf. Comp. Vis. and Patt. Recog. pp. 473–478.
Weiss, I., 1988, “Projective Invariants of Shape,” DARPA IU Workshop pp. 1125–1134.
Wolfson, H.J., 1990, “Model Based Object Recognition by Geometric Hashing,” First Europ. Conf. Comp. Vis. pp. 526–536.
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Grimson, W.E.L., Huttenlocher, D.P., Jacobs, D.W. (1992). A study of affine matching with bounded sensor error. In: Sandini, G. (eds) Computer Vision — ECCV'92. ECCV 1992. Lecture Notes in Computer Science, vol 588. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-55426-2_34
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DOI: https://doi.org/10.1007/3-540-55426-2_34
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