Differential and Numerically Invariant Signature Curves Applied to Object Recognition |
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Authors: | Calabi Eugenio Olver Peter J. Shakiban Chehrzad Tannenbaum Allen Haker Steven |
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Affiliation: | (1) Department of Mathematics, University of Pennsylvania, Philadelphia, PA, 19066-1102. E-mail;(2) School of Mathematics, University of Minnesota, Minneapolis, MN, 55455. E-mail;(3) Department of Mathematics, University of St. Thomas, St. Paul, MN, 55105-1096.E-mail;(4) Department of Electrical Engineering, University of Minnesota, Minneapolis, MN, 55455. E-mail |
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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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Keywords: | 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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