Parabolic Polygons and Parabolic Polygons and Discrete Affine Discrete Affine

GeometryGeometry

M.Craizer, T.Lewiner, J.M.MorvanDepartamento de Matemática – PUC-RioUniversité Claude Bernard-Lyon-France

2/10 Motivation: affine geometryMotivation: affine geometry

length length

radius radius

GeometryEuclideantranslation

rotationshearing

Affine...projective geometry

3/10 Motivation: reconstructionMotivation: reconstruction

Tangent at sample points available or easily computable surely improve reconstruction

Intrinsic in the model

4/10 SummarySummary

The Parabolic Polygon Model Planar curves : points + tangents Affine invariant

Properties Affine length estimation Affine curvature estimation

Application Affine curve reconstruction

5/10 GeometryGeometry

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

6/10 Affine geometry of curvesAffine geometry of curves

7/10 Discrete curve modelDiscrete curve model

AND tangentsOrdered sample points

8/10 Elementary parabolaElementary parabola

Support triangle

9/10 Parabolic PolygonsParabolic Polygons

Polygon with parabolic arcs

Parabola = flat affine curve

10/10 Affine InvarianceAffine Invariance

11/10 Affine length estimatorAffine length estimator

affine length of an arc of the curve=

affine length of the arc of parabola

12/10 Affine curvature estimatorAffine curvature estimator

Estimated from 3 samplesCurvature concentrated at the vertices

ni

13/10

Estimators convergence :Estimators convergence :ellipseellipse

Length Curvature

14/10

Estimators convergence :Estimators convergence :hyperbolahyperbola

Length Curvature

15/10 Affine Curve ReconstructionAffine Curve Reconstruction

Connect to the affine closest pointpreventing high curvatures

Variation of:L. H. Figueiredo and J. M. Gomes.

Computational morphology of curves.Visual Computer (11), 1994.

16/10

Affine vs Euclidean Affine vs Euclidean ReconstructionReconstruction

Points + tangents Only points

17/10

Affine Reconstruction:Affine Reconstruction:InvarianceInvariance

Points + tangents Only points

18/10

Affine Reconstruction:Affine Reconstruction:inflection pointsinflection points

Curvature threshold todetect inflection points

19/10 Conclusion & Ongoing worksConclusion & Ongoing works

Intrinsic use of tangent in the curve modelAffine invariantDifferential estimators

Affine curve reconstruction

Surface model Cubic splines at inflection points Projective invariance Applications to object detection and matching

Thank you forThank you foryour attention!your attention!

http://www.mat.puc-rio.br/~craizerhttp://www.matmidia.mat.puc-rio.br/