Differential and Numerically Invariant Signature Curves Applied to Object Recognition

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

Research output: Contribution to journalArticlepeer-review

166 Scopus citations


We introduce a new paradigm, the differential invariant signature curve or manifold, for the invariant recognition of visual objects. A general theorem of E. 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.

Original languageEnglish (US)
Pages (from-to)107-135
Number of pages29
JournalInternational Journal of Computer Vision
Issue number2
StatePublished - 1998

Bibliographical note

Funding Information:
⁄Supported in part by NSF Grant DMS 92-03398. †Supported in part by NSF Grant DMS 95-00931. ‡Supported in part by NSF Grant ECS-9122106, by the Air Force Office of Scientific Research F49620-94-1-00S8DEF, by the Army Research Office DAAH04-93-G-0332, DAAH04-94-G-0054, and AFOSR-MURI.


  • Curve shortening flow
  • Differential invariant
  • Equi-affine group
  • Euclidean group
  • Joint invariant
  • Numerical approximation
  • Object recognition
  • Signature curve
  • Snake
  • Symmetry group


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