International Journal of Computer Vision

, Volume 26, Issue 2, pp 107–135

Differential and Numerically Invariant Signature Curves Applied to Object Recognition

  • Eugenio Calabi
  • Peter J. Olver
  • Chehrzad Shakiban
  • Allen Tannenbaum
  • Steven Haker
Article

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.

object recognition symmetry group differential invariant joint invariant signature curve Euclidean group equi-affine group numerical approximation curve shortening flow snake 

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Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Eugenio Calabi
    • 1
  • Peter J. Olver
    • 2
  • Chehrzad Shakiban
    • 3
  • Allen Tannenbaum
    • 4
  • Steven Haker
    • 2
  1. 1.Department of MathematicsUniversity of PennsylvaniaPhiladelphia
  2. 2.School of MathematicsUniversity of MinnesotaMinneapolis
  3. 3.Department of MathematicsUniversity of St. ThomasSt. Paul
  4. 4.Department of Electrical EngineeringUniversity of MinnesotaMinneapolisMN

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