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Parabolic Polygons and Discrete Affine Geometry

Explore the affine and discrete geometry of parabolic polygons, focusing on length and curvature estimations for curve reconstruction. Affine invariance properties and reconstruction techniques are discussed, with applications in object detection and matching.

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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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