Differential and Numerically Invariant Signature Curves Applied to Object Recognition

  • Authors:

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

  • Affiliations:

  • Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19066-1102. E-mail: calabi@math.upenn.edu;School of Mathematics, University of Minnesota, Minneapolis, MN 55455. E-mail: olver@ima.umn.edu/ http://www.math.umn.edu/&sim/olver;Department of Mathematics, University of St. Thomas, St. Paul, MN 55105-1096. E-mail: c9shakiban@stthomas.edu;Department of Electrical Engineering, University of Minnesota, Minneapolis, MN 55455. E-mail: tannenba@ee.umn.edu;School of Mathematics, University of Minnesota, Minneapolis, MN 55455. E-mail: haker@math.umn.edu

  • Venue:
  • International Journal of Computer Vision
  • Year:
  • 1998

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Abstract

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