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Cartan's Moving Frame Method and Its Application to the Geometry and Evolution of Curves in the Euclidean, Affine and Projective Planes

Cartan's Moving Frame Method and Its Application to the Geometry and Evolution of Curves in the Euclidean, Affine and Projective Planes,Olivier D. Fau

Cartan's Moving Frame Method and Its Application to the Geometry and Evolution of Curves in the Euclidean, Affine and Projective Planes   (Citations: 24)
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Conference: Applications of Invariance in Computer Vision , pp. 11-46, 1993
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    • ...While geometric invariants have long been used to solve problems in computer vision and image processing [10, 26, 27, 30, 40], designing robust algorithms that are tolerant to noise and image occlusion remains an open problem...

    Shuo Fenget al. Classification of Curves in 2D and 3D via Affine Integral Signatures

    • ...Together, they are sufficient to describe a planar curve up to a projective transformation [8]...
    • ...Faugeras [8] describes very nicely the Euclidean, affine and projective geometry of plane curves, with explicit formulas for projective length and curvature...
    • ...Since x(σ ) and λ(σ )x(σ ) are equivalent curves, we can force p(σ ) to be zero by choosing the value of λ(σ ) as λ(σ ) = exp ( 1 σ 0 p(τ )dτ) [8]...

    Thomas Lewineret al. Projective Splines and Estimators for Planar Curves

    • ...In [20, 10, 2, 6, 91, 72], the characterization of submanifolds via their differential invariant signatures was applied to the problem of object recognition and symmetry detection, [16, 17, 30, 87]...

    Peter J. Olver. Lectures on Moving Frames

    • ...Together, they are sufficient to describe a planar curve up to a projective transformation ([7])...
    • ...However, Faugeras [7] describes very nicely the Euclidian, affine and projective geometry and evolutions of plane curves, with explicit formulas for projective length and curvature...
    • ...If one takes σ as a new parameter for the curve, then H(σ )=1 . The function H(t) has the following alternative definition: Suppose that the curve is parameterized by affine arc length s, then, following [7],...
    • ...Projective curvature. Since x(σ) and λ(σ)x(σ) are equivalent curves, we can force p(σ) to be zero by choosing λ(σ) = exp ( 1 3 � σ 0 p(τ )dτ ) [7]...

    Thomas Lewineret al. Projective Estimators for Point/Tangent Representations of Planar Curv...

    • ...To perform this parametrization the group metric, g, is defined by the expression: dr = gd p for any parametrization p where r is obtained via the relation: r = � p 0 g(ξ )dξ . We have, of course, Img [C( p) ]= Img [C(r )]. Based on the group metric and arc-length, the group curvature can be computed using either Lie theory or Cartan’s moving frame method (Cartan 1935; Faugeras 1993)...
    • ...The fact that for transitive group actions, an object can be fully reconstructed modulo group transformations from a prescribed and finite collection of differential invariants is a consequence of a general theorem by Elie Cartan (Cartan 1935; Faugeras 1993; Calabi et al. 1996)...
    • ...We then present the method of the moving of a curve, an ingenious formalism developed by Élie Cartan (Cartan 1935; Faugeras 1993; Guggenheimer 1977; Spivak 1979)...
    • ...For a transitive group action, a plane curve can be fully reconstructed (modulo group action) from a prescribed and finite collection of differential invariants (Cartan 1935; Faugeras 1993)...

    Tamar Flashet al. Affine differential geometry analysis of human arm movements

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