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. 2 /10 Motivation: affine geometry length radius Geometry Euclidean translation rotation shearing Affine...projective geometry

3. 3 /10 Motivation: reconstruction Tangent at sample points  available or easily computable  surely improve reconstruction Intrinsic in the model

4. 4 /10 Summary The Parabolic Polygon Model  Planar curves : points + tangents  Affine invariant Properties  Affine length estimation  Affine curvature estimation Application  Affine curve reconstruction

5. 5 /10 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. 6 /10 Affine geometry of curves

7. 7 /10 Discrete curve model AND tangentsOrdered sample points

8. 8 /10 Elementary parabola Support triangle

9. 9 /10 Parabolic Polygons Polygon with parabolic arcs Parabola = flat affine curve

10. 10 /10 Affine Invariance

11. 11 /10 Affine length estimator affine length of an arc of the curve = affine length of the arc of parabola

12. 12 /10 Affine curvature estimator Estimated from 3 samples Curvature concentrated at the vertices nini

13. 13 /10 Estimators convergence : ellipse LengthCurvature

14. 14 /10 Estimators convergence : hyperbola LengthCurvature

15. 15 /10 Affine Curve Reconstruction Connect to the affine closest point preventing high curvatures Variation of: L. H. Figueiredo and J. M. Gomes. Computational morphology of curves. Visual Computer (11), 1994.

16. 16 /10 Affine vs Euclidean Reconstruction Points + tangentsOnly points

17. 17 /10 Affine Reconstruction: Invariance Points + tangentsOnly points

18. 18 /10 Affine Reconstruction: inflection points Curvature threshold to detect inflection points

19. 19 /10 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 for your attention! http://www.mat.puc-rio.br/~craizer http://www.matmidia.mat.puc-rio.br/