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Differential and Numerically Invariant Signature Curves Applied to Object Recognition

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Abstract

We introduce a new paradigm, the differential invariant signature curve or manifold, for the invariant recognition of visual objects. A general theorem of É. Cartan implies that two curves are related by a group transformation if and only if their signature curves are identical. The important examples of the Euclidean and equi-affine groups are discussed in detail. Secondly, we show how a new approach to the numerical approximation of differential invariants, based on suitable combination of joint invariants of the underlying group action, allows one to numerically compute differential invariant signatures in a fully group-invariant manner. Applications to a variety of fundamental issues in vision, including detection of symmetries, visual tracking, and reconstruction of occlusions, are discussed.

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Authors and Affiliations

  1. Department of Mathematics, University of Pennsylvania, Philadelphia, PA, 19066-1102. E-mail

    Eugenio Calabi

  2. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455. E-mail

    Peter J. Olver & Steven Haker

  3. Department of Mathematics, University of St. Thomas, St. Paul, MN, 55105-1096.E-mail

    Chehrzad Shakiban

  4. Department of Electrical Engineering, University of Minnesota, Minneapolis, MN, 55455. E-mail

    Allen Tannenbaum

Authors
  1. Eugenio Calabi
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  2. Peter J. Olver
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  3. Chehrzad Shakiban
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  4. Allen Tannenbaum
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  5. Steven Haker
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Calabi, E., Olver, P.J., Shakiban, C. et al. Differential and Numerically Invariant Signature Curves Applied to Object Recognition. International Journal of Computer Vision 26, 107–135 (1998). https://doi.org/10.1023/A:1007992709392

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