International Journal of Computer Vision

, Volume 26, Issue 2, pp 107–135

Differential and Numerically Invariant Signature Curves Applied to Object Recognition


  • Eugenio Calabi
    • Department of MathematicsUniversity of Pennsylvania
  • Peter J. Olver
    • School of MathematicsUniversity of Minnesota
  • Chehrzad Shakiban
    • Department of MathematicsUniversity of St. Thomas
  • Allen Tannenbaum
    • Department of Electrical EngineeringUniversity of Minnesota
  • Steven Haker
    • School of MathematicsUniversity of Minnesota

DOI: 10.1023/A:1007992709392

Cite this article as:
Calabi, E., Olver, P.J., Shakiban, C. et al. International Journal of Computer Vision (1998) 26: 107. doi:10.1023/A:1007992709392


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 recognitionsymmetry groupdifferential invariantjoint invariantsignature curveEuclidean groupequi-affine groupnumerical approximationcurve shortening flowsnake
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© Kluwer Academic Publishers 1998