Abstract
We analyze the amount of information needed to carry out various model-based recognition tasks, in the context of a probabilistic data collection model. We focus on objects that may be described as semi-algebraic subsets of a Euclidean space, and on a wide class of object transformations, including perspective and affine transformations of 2D objects, and perspective projections of 3D objects. Our approach borrows from computational learning theory. We draw close relations between recognition tasks and a certain learnability framework. We then apply basic techniques of learnability theory to derive upper bounds on the number of data features that (provably) suffice for drawing reliable conclusions. The bounds are based on a quantitative analysis of the complexity of the hypotheses class that one has to choose from. Our central tool is the VC-dimension, which is a well studied parameter measuring the combinatorial complexity of families of sets. It turns out that these bounds grow linearly with the task complexity, measured via the VC-dimension of the class of objects one deals with.
This work was supported by the Technion fund for the promotion of research and by the Smoler research fund.
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References
Blumer, A., A. Ehrenfeucht, D. Haussler and M.K. Warmuth, 1989, “Learnability and The Vapnik-Chervonenkis Dimension”, JACM, 36(4), pp. 929–965.
S. Ben-David and M. Lindenbaum, 1993, “Localization vs. Identification of Semi-Algebraic Sets”, Proceedings of the 6th ACM Conference on Computational Learning Theory, pp. 327–336.
Collins, G.E., 1975, “Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition”, Proceedings of the 2nd GI Conf. On Automata Theory and Formal Languages, Springer Lec. Notes Comp. Sci. 33, pp. 515–532.
Goldberg P. and M. Jerrum, 1993, “Bounding the Vapnik-Chervonenkis Dimension of Concept Classes Parametrized by Real Numbers”, Proceedings of the 6th ACM Conference on Computational Learning Theory, pp. 361–368.
Gottschalk P.G., J.L. Turney, and T.N. Mudge, 1989, “Efficient Recognition of Partially Visible Objects Using a Logarithmic Complexity Matching Technique”, Int. J. of Rob. Res., 8(6), pp. 110–131.
Grimson, W.E.L., and D.P. Huttenlocher, 1991, “On the Verification of Hypothesized Matches in Model-Based Recognition”, IEEE Trans. on Pattern Analysis and Mach. Intel., PAMI-13(12), pp. 1201–1213.
Grimson, W.E.L., D.P. Huttenlocher, and D.W. Jacobs, 1992, “A study of affine Matching with Bounded sensor error”, Second Europ. Conf. Comp. Vision, pp. 291–306.
M. Lindenbaum, “Bounds on Shape Recognition Performance”, 1993. Proceedings of the 7th ICIAP, Italy.
M. Lindenbaum and S. Ben-David, “Applying VC-dimension Analysis to Object Recognition”, 1993, CIS report No. 9330, Computer Science Department, Technion.
Maybank, S.J., 1993, Probabilistic Analysis of the Application of the Cross Ratio to Model Based Vision”, Manuscript.
Milnor, J., 1964, “On the Betti Numbers of Real Varieties”, Proc. Amer. Math. Soc. 15, pp. 275–280.
Mundy, J.L. and A.J. Heller, 1990, “The Evolution and Testing of Model-Based Object Recognition Systems”, Proc. 3rd ICCV, pp. 268–282.
Vapnik, V.N. and A.Y. Chervonenkis, 1971, “On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities”, Theory of Probability and its applications, 16(2), pp. 264–280.
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© 1994 Springer-Verlag Berlin Heidelberg
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Lindenbaum, M., Ben-David, S. (1994). Applying VC-dimension analysis to object recognition. In: Eklundh, JO. (eds) Computer Vision — ECCV '94. ECCV 1994. Lecture Notes in Computer Science, vol 800. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57956-7_29
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DOI: https://doi.org/10.1007/3-540-57956-7_29
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