Parabolic Polygons and
Discrete Affine Geometry
M.Craizer, T.Lewiner, J.M.Morvan
Departamento de Matemática – PUC-Rio
Université Claude Bernard-Lyon-France
2/10
Motivation: affine geometry
Euclidean
Affine Geometry
translation
rotation
shearing
...projective geometry
Motivation: reconstruction
3/10
Tangent at sample points
 available
or easily computable
 surely improve reconstruction
Intrinsic in the model
Summary
4/10
The Parabolic Polygon Model
 Planar
curves : points + tangents
 Affine invariant
Properties
 Affine
length estimation
 Affine curvature estimation
Application
 Affine
curve reconstruction
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
6/10
Affine geometry of curves
Discrete curve model
7/10
Ordered sample points
AND tangents
Elementary parabola
8/10
Support triangle
Parabolic Polygons
9/10
Parabola = flat affine curve
Polygon with parabolic arcs
10/10
Affine Invariance
11/10
Affine length estimator
affine length of an arc of the curve
=
affine length of the arc of parabola
12/10
Affine curvature estimator
ni
Estimated from 3 samples
Curvature concentrated at the vertices
Estimators convergence :
ellipse
13/10
Length
Curvature
Estimators convergence :
hyperbola
14/10
Length
Curvature
15/10
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 point
preventing high curvatures
Affine vs Euclidean
Reconstruction
16/10
Points + tangents
Only points
Affine Reconstruction:
Invariance
17/10
Points + tangents
Only points
18/10
Affine Reconstruction:
inflection points
Curvature threshold to
detect inflection points
Conclusion & Ongoing works
19/10
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

Thank you for
your attention!
http://www.mat.puc-rio.br/~craizer
http://www.matmidia.mat.puc-rio.br/