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Parabolic Polygons and Discrete Affine Geometry

Parabolic Polygons and Discrete Affine Geometry. M.Craizer, T.Lewiner, J.M.Morvan Departamento de Matemática – PUC-Rio Université Claude Bernard-Lyon-France. Motivation: affine geometry. length. length. radius. radius. ...projective geometry. Geometry. Euclidean. Affine. translation.

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Parabolic Polygons and Discrete Affine Geometry

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  1. Parabolic Polygons and Discrete Affine Geometry M.Craizer, T.Lewiner, J.M.Morvan Departamento de Matemática – PUC-Rio Université Claude Bernard-Lyon-France

  2. Motivation: affine geometry length length radius radius ...projective geometry Geometry Euclidean Affine translation rotation shearing

  3. Motivation: reconstruction • Tangent at sample points • available or easily computable • surely improve reconstruction • Intrinsic in the model

  4. Summary • The Parabolic Polygon Model • Planar curves : points + tangents • Affine invariant • Properties • Affine length estimation • Affine curvature estimation • Application • Affine curve reconstruction

  5. Geometry • Euclidean geometry (rotations, translations) • → length, curvature • → straight line polygon: point, edges • Affine geometry (rotations, translations + shearing) • → affine length, affine curvature • → parabolic polygon: point + tangents, edges

  6. Affine geometry of curves

  7. Discrete curve model • AND tangents • Ordered sample points

  8. Elementary parabola • Support triangle

  9. Parabolic Polygons • Parabola = flat affine curve • Polygon with parabolic arcs

  10. Affine Invariance

  11. Affine length estimator • affine length of an arc of the curve • = • affine length of the arc of parabola

  12. Affine curvature estimator ni Estimated from 3 samples Curvature concentrated at the vertices

  13. Estimators convergence :ellipse Length Curvature

  14. Estimators convergence :hyperbola Length Curvature

  15. Affine Curve Reconstruction • Variation of: • L. H. Figueiredo and J. M. Gomes.Computational morphology of curves.Visual Computer (11), 1994. • Connect to the affine closest pointpreventing high curvatures

  16. Affine vs Euclidean Reconstruction Points + tangents Only points

  17. Affine Reconstruction:Invariance Points + tangents Only points

  18. Affine Reconstruction:inflection points • Curvature threshold todetect inflection points

  19. Conclusion & Ongoing works Intrinsic use of tangent in the curve model Affine invariant Differential estimators Affine curve reconstruction • Surface model • Cubic splines at inflection points • Projective invariance • Applications to object detection and matching

  20. Thank you foryour attention! http://www.mat.puc-rio.br/~craizer http://www.matmidia.mat.puc-rio.br/

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